On elementary four-wave interactions in dispersive media

Abstract

The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

1. Introduction

The objective of the present study is to provide an account of the solutions of the system:

(1.1)\begin{equation} \left. \begin{gathered} \mathrm{i} \frac{\mathrm{d} b_1}{\mathrm{d} t} = (\omega_1 + T_{11}|b_1|^2 + 2 T_{12}|b_2|^2 + 2 T_{13}|b_3|^2 + 2 T_{14}|b_4|^2) b_1 + 2 T b_2^* b_3 b_4,\\ \mathrm{i} \frac{\mathrm{d} b_2}{\mathrm{d} t} = (\omega_2 + 2 T_{21}|b_1|^2 + T_{22}|b_2|^2 + 2 T_{23}|b_3|^2 + 2 T_{24}|b_4|^2) b_2 + 2 T b_1^* b_3 b_4,\\ \mathrm{i} \frac{\mathrm{d} b_3}{\mathrm{d} t} = (\omega_3 + 2 T_{31}|b_1|^2 + 2 T_{32}|b_2|^2 + T_{33}|b_3|^2 + 2 T_{34}|b_4|^2) b_3 + 2 T b_4^* b_1 b_2,\\ \mathrm{i} \frac{\mathrm{d} b_4}{\mathrm{d} t} = (\omega_4 + 2 T_{41}|b_1|^2 + 2 T_{42}|b_2|^2 + 2 T_{43}|b_3|^2 + T_{44}|b_4|^2) b_4 + 2 T b_3^* b_1 b_2, \end{gathered} \right\} \end{equation}

where $b_i(t)$, $i=1,\ldots,4$, are complex valued functions of time $t$, where an asterisk stands for complex conjugate and where the real-valued coefficients $\omega _i$, $T_{ij}=T_{ji}$ and $T$ are independent of time. System (1.1) is completed by initial conditions:

(1.2)\begin{equation} b_i(0) = \sqrt{q_i}\mathrm{e}^{\mathrm{i} \varphi_i}, \quad q_i \geq 0, \quad \varphi_i \in [0,2{\rm \pi}]. \end{equation}

The positive $q_i$ are the initial wave actions.

Equations similar to (1.1) were first derived by Armstrong et al. (1962) in nonlinear optics (time being replaced by a spatial direction) and by Benney (1962) in hydrodynamics. Physically, (1.1) governs the evolution of an isolated quartet of weakly nonlinear waves with wave vectors $(\boldsymbol {k}_1,\boldsymbol {k}_2,\boldsymbol {k}_3, \boldsymbol {k}_4)$ satisfying

(1.3)\begin{equation} \boldsymbol{k}_1 + \boldsymbol{k}_2 = \boldsymbol{k}_3 + \boldsymbol{k}_4, \quad \boldsymbol{k}_i \not= \boldsymbol{k}_j, \end{equation}

propagating with linear frequency $\omega _i = \omega (\boldsymbol {k}_i)>0$ in a conservative dispersive medium with non-decay dispersion law, i.e. where three-wave quadratic interactions are excluded. The coupling coefficients $T_{ij}$ and $T$ depend on the wave vectors and on the physical properties of the medium under consideration. Defining the frequency mismatch

(1.4)\begin{equation} \Delta \omega = \omega_1 + \omega_2 - \omega_3 - \omega_4, \end{equation}

the interaction (1.3) is said to be resonant if $\Delta \omega = 0$.

Systems similar to (1.1) were also derived by Bretherton (1964) and Inoue (1975) from scalar model equations, by Boyd & Turner (1978) in plasma physics and by Chen & Snyder (1989) in nonlinear optics. Stiassnie & Shemer (2005) deduced (1.1) from the Zakharov equation (Zakharov 1966, 1968; Krasitskii 1990, 1994) that governs the evolution of discrete or continuous spectra of weakly nonlinear gravity waves, and which is generally used as the starting point for weak turbulence statistical theory (Yuen & Lake 1982; Zakharov, L'vov & Falkovich 1992; Zakharov 1999; Janssen 2004; Nazarenko & Lukaschuk 2016), even though Hasselmann (1962) in his pioneering work used primitive equations.

Complementary to (1.1), an isolated wave triad $(\boldsymbol {k}_1,\boldsymbol {k}_2,\boldsymbol {k}_3)$ satisfying

(1.5)\begin{equation} \boldsymbol{k}_1 + \boldsymbol{k}_2 = 2 \boldsymbol{k}_3, \quad \boldsymbol{k}_i \not= \boldsymbol{k}_j, \end{equation}

also interacts nonlinearly at third order and is governed by

(1.6)\begin{equation} \left. \begin{gathered} \mathrm{i} \frac{\mathrm{d} c_1}{\mathrm{d} t} = (\omega_1 + T_{11}|c_1|^2 + 2 T_{12}|c_2|^2 + 2 T_{13}|c_3|^2 ) c_1 + T c_2^* c_3^2,\\ \mathrm{i} \frac{\mathrm{d} c_2}{\mathrm{d} t} = (\omega_2 + 2 T_{21}|c_1|^2 + T_{22}|c_2|^2 + 2 T_{23}|c_3|^2 ) c_2 + T c_1^* c_3^2,\\ \mathrm{i} \frac{\mathrm{d} c_3}{\mathrm{d} t} = (\omega_3 + 2 T_{31}|c_1|^2 + 2 T_{32}|c_2|^2 + T_{33}|c_3|^2 ) c_3 + 2 T c_3^* c_1 c_2, \end{gathered} \right\} \end{equation}

as first derived by Benney (1962) for resonant gravity-wave interactions:

(1.7)\begin{equation} \omega_1 + \omega_2 = 2 \omega_3, \end{equation}

and deduced by Shemer & Stiassnie (1985) from the Zakharov equation. In nonlinear optics, a spatial analogue of (1.6) has been derived by Cappellini & Trillo (1991), who also pointed out that the three-wave system (1.6) cannot be deduced from the four-wave system (1.1) by simple algebraic relations between the complex amplitudes $b_i(t)$ and $c_i(t)$.

System (1.1) may be solved following a procedure introduced by Armstrong et al. (1962) for wave triads in quadratic interaction and extended to quartets in cubic interaction by Bretherton (1964): it consists of reducing (1.1) to a single scalar equation governing the evolution of an auxiliary variable, say $q(t)$. Bretherton proved that $q(t)$ is ‘a periodic function of [t], with period depending on the initial conditions but which is only in exceptional cases infinite’, but did not give explicit solutions. This was achieved by Inoue (1975), Boyd & Turner (1978), Turner (1980), Chen & Snyder (1989) and Stiassnie & Shemer (2005) using elliptic functions. Some of the ‘exceptional cases’ mentioned by Bretherton correspond to solutions now called ‘pump’ and ‘breathers’; some of these were given by Inoue (1975) and Turner (1980), but some others were missing.

Following a similar procedure, periodic solutions of (1.6) involving elliptic functions were found by Shemer & Stiassnie (1985) and Cappellini & Trillo (1991). These latter authors, who also found breather solutions, proved that the problem may be recast to an elegant one-degree integrable Hamiltonian system; see also Trillo & Wabnitz (1991). This approach allows one to plot phase portraits representing level-lines of the Hamiltonian from which interesting qualitative and quantitative results can be deduced: isolated points surrounded by closed orbits in phase space correspond respectively to stable equilibria and to periodic solutions, while saddle points connected by separatrices correspond respectively to unstable equilibria and to non-periodic solutions (Cappellini & Trillo 1991; Trillo & Wabnitz 1991; Andrade & Stuhlmeier 2023b). A similar Hamiltonian formulation is used in the present study, but phase portraits are not necessary for our purpose.

Steady equilibria of (1.1) or of (1.6) are of fundamental importance. By steady equilibria, we mean solutions with steady amplitudes $|b_i(t)|^2 = |b_i(0)|^2$. Benney (1962) noticed that the simplest of these is the finite-amplitude travelling wave discovered by Stokes in 1847. The existence of finite-amplitude bichromatic wavetrains $(\boldsymbol {k}_1,\boldsymbol {k}_2)$ was first proved by Phillips (1960), Longuet-Higgins & Phillips (1962) and Benney (1962). These steady bichromatic waves were one of the essential ingredients for the development of the theory of weakly nonlinear wave interactions (see also the historical survey by Phillips (1981)): in a few words, we recall that Phillips (1960) and Longuet-Higgins (1962) proved that the propagation of a finite-amplitude bichromatic gravity wavetrain, say $(\boldsymbol {k}_1,\boldsymbol {k}_3)$, leads initially by cubic (or ‘tertiary’) nonlinear resonant interaction to the spontaneous linear growth of a third wave $\boldsymbol {k}_2$ satisfying (1.5) and (1.7). As noticed by Phillips (1967), this behaviour may be deduced from (1.6) considering $|c_2| \ll |c_1|,|c_3|$.

The previous mechanism has to be distinguished from quadratic (or ‘secondary’) nonlinear interactions inside wave triads for which it has been proved by Galeev & Karpman (1963) and Hasselmann (1967) that a finite-amplitude wave $\boldsymbol {k}_3$ is exponentially unstable to a couple of infinitesimal disturbances $(\boldsymbol {k}_1 , \boldsymbol {k}_2)$ if (see also Craik 1985, p. 131)

(1.8)\begin{equation} \boldsymbol{k}_1 + \boldsymbol{k}_2 = \boldsymbol{k}_3,\quad \omega_1 + \omega_2 = \omega_3. \end{equation}

Sometimes called ‘decay instability’, this mechanism cannot, however, operate in gravity waves, as proved by Phillips (1960) and Hasselmann (1962).

Back to cubic resonant interactions between three waves satisfying (1.5), (1.6) and (1.7), the continuous energy transfer from $(\boldsymbol {k}_1,\boldsymbol {k}_3)$ to $\boldsymbol {k}_2$ discovered by Phillips excludes therefore the possibility of steady states at resonance. However, the existence of non-resonant steady states in a system of three waves $(\boldsymbol {k}_1, \boldsymbol {k}_2, \boldsymbol {k}_3)$ with finite constant amplitudes $|b_i|$ has been established near resonance by Cappellini & Trillo (1991), Liao, Xu & Stiassnie (2016) and Andrade & Stuhlmeier (2023b).

In the case of ‘non-degenerate’ quartets satisfying (1.1) and (1.3), the existence of steady equilibria has been established at resonance in numerical simulations by Liu & Liao (2014), observed experimentally by Liu et al. (2015) and identified off-resonance by Andrade & Stuhlmeier (2023a) using a Hamiltonian approach. The bidirectional standing wave of Okamura (1985) is also a particular case of such steady quartets. It was clear from these studies that the existence of steady equilibria is conditioned by certain constraints between the finite amplitudes $|b_i|$, but a general explicit formulation of these compatibility conditions was missing.

Concerning stability, it has been known since Zakharov (1966) and Phillips (1967) that the modulational instability discovered by Benjamin & Feir (1967) may be interpreted as the cubic interaction between a Stokes wave $\boldsymbol {k}_3$ with constant finite amplitude $|b_3|$ disturbed by a couple of ‘satellites’ with infinitesimal amplitudes $|b_1|, |b_2| \ll |b_3|$ and wave vectors $(\boldsymbol {k}_1,\boldsymbol {k}_2)$ satisfying (1.5), slightly off resonance (see reviews in Yuen & Lake 1982; Shemer & Stiassnie 1991; Janssen 2004; Zakharov & Ostrovsky 2009). It has also been known since Okamura (1984) and Ioualalen & Kharif (1994) that a standing wave or a bichromatic wavetrain $(\boldsymbol {k}_1,\boldsymbol {k}_2)$ with constant finite amplitudes $|b_1|, |b_2|$ may also be destabilized by a couple of infinitesimal satellites with wave vectors $(\boldsymbol {k}_3,\boldsymbol {k}_4)$ satisfying (1.3), at resonance or slightly off. A theory for this kind of modulational instability has been presented in Leblanc (2009).

The present study focuses only on four-wave interactions (1.3) inside a single quartet governed by (1.1). Motivated by the recent work of Andrade & Stuhlmeier (2023a) who established some interesting links between various aspects mentioned above, but who restricted their analysis to quartets with symmetric initial conditions $|b_1(0)|= |b_2(0)|$ and $|b_3(0)|= |b_4(0)|$, the present study aims at answering the questions that remain open, putting the various pieces of the puzzle together and finding the missing ones. In some sense, our work may be viewed as the extension to non-degenerate quartets (1.3) of the analyses of Shemer & Stiassnie (1985), Cappellini & Trillo (1991) and Andrade & Stuhlmeier (2023b) for the ‘degenerate’ case (1.5).

The paper is organized in two parts: the first one is generic to dispersive media (§§ 2–4), the second specific to deep-water gravity waves (§§ 5–7). More precisely: reduction to a single equation is carried out in § 2; steady equilibria and their stability are investigated in § 3; unsteady solutions are presented in § 4; models of steady and periodic solutions on deep water are described in § 5; examples of pump and breathers are elaborated in §§ 6 and 7. Results are summarized in § 8 and complements are given in the appendices.

2. Bretherton equation

Multiplying each equation of (1.1) respectively by $b_1^*,\ldots,b_4^*$ and adding the complex conjugate equations yields (Bretherton 1964; Stiassnie & Shemer 2005)

(2.1)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} |b_1|^2 = \frac{\mathrm{d}}{\mathrm{d} t} |b_2|^2 =- \frac{\mathrm{d}}{\mathrm{d} t} |b_3|^2 =- \frac{\mathrm{d}}{\mathrm{d} t} |b_4|^2 =- 4 T |b_1|\,|b_2|\,|b_3|\,|b_4| \sin p, \end{equation}

where the relative phase $p(t)$ is defined as

(2.2)\begin{equation} p(t) = p_1(t) + p_2(t) - p_3(t) - p_4(t), \quad p_i(t) = \arg b_i(t). \end{equation}

Individual phases $p_i(t)$ are governed by (Inoue 1975; Andrade & Stuhlmeier 2023a)

(2.3)\begin{equation} \frac{{\rm d} p_i}{{\rm d}t} =- \omega_i + T_{ii} |b_i|^2 - 2\sum_{j=1}^4 T_{ij} |b_j|^2 - \frac{2T}{|b_i|^2} |b_1|\,|b_2||b_3|\,|b_4| \cos p. \end{equation}

The first three equalities in (2.1) yield integrals of motions known as the Manley–Rowe relations (Manley & Rowe 1956) that may be written, following Bretherton (1964) and Stiassnie & Shemer (2005), as

(2.4)\begin{equation} |b_1(t)|^2- q_1 = |b_2(t)|^2- q_2 = q_3 - |b_3(t)|^2 = q_4 - |b_4(t)|^2 \equiv q(t), \end{equation}

where, by construction, the (positive or negative) relative action $q(t)$ satisfies initially

(2.5)\begin{equation} q(0)=0. \end{equation}

Furthermore, four-wave interactions governed by (1.1) are bounded since from (2.4) (conditions for ‘explosive’ four-wave interactions are detailed in Turner (1980), Verheest (1982) and Safdi & Segur (2007); none of these are fulfilled here)

(2.6)\begin{equation} -\min(q_1,q_2) \leq q(t) \leq \min (q_3,q_4). \end{equation}

From (2.1) and (2.4) we get also (here and below, $\dot q(t) = \mathrm {d} q / \mathrm {d} t$)

(2.7)\begin{equation} \dot q =- 4 T \sqrt{(q_1+q)(q_2+q)(q_3-q)(q_4-q)} \sin p, \end{equation}

while, from (2.2) and (2.3),

(2.8)\begin{equation} \dot p =- (2A q + B) - 2T F(q) \cos p, \end{equation}

where

(2.9)\begin{equation} \left. \begin{gathered} A = \tfrac{1}{2}(T_{11} + T_{22} + T_{33} + T_{44}) + 2 (T_{12} - T_{13} - T_{14} - T_{23} - T_{24} + T_{34}),\\ B = \Delta \omega + B_1 q_1 + B_2 q_2 - B_3 q_3 - B_4 q_4, \quad \Delta \omega = \omega_1 + \omega_2 - \omega_3 - \omega_4,\\ B_1 = T_{11} + 2(T_{12} - T_{13} - T_{14}), \quad B_2 = T_{22} + 2(T_{12} - T_{23} - T_{24}),\\ B_3 = T_{33} + 2(T_{34} - T_{13} - T_{23}), \quad B_4 = T_{44} + 2(T_{34} - T_{14} - T_{24}) \end{gathered} \right\} \end{equation}

and

(2.10)\begin{align} F(q) = \left( \frac{1}{q_1+q} + \frac{1}{q_2+q} - \frac{1}{q_3-q} - \frac{1}{q_4-q} \right) \sqrt{(q_1+q)(q_2+q)(q_3-q)(q_4-q)}. \end{align}

If $T=0$, (2.7) and (2.8) yield, together with (2.5)

(2.11)\begin{equation} q(t) = 0, \quad p(t) = (\Delta \omega + B_1 q_1 + B_2 q_2 - B_3 q_3 - B_4 q_4) t + p_0. \end{equation}

We shall therefore consider $T \not = 0$ from now on. The initial phase mismatch $p_0= p(0)$ introduced above reads also, from (1.2)

(2.12)\begin{equation} p_0 = \arg b_1(0) + \arg b_2(0) - \arg b_3(0) - \arg b_4(0) = \varphi_1 + \varphi_2 - \varphi_3 - \varphi_4. \end{equation}

Equations (2.7) and (2.8) may be written in canonical form:

(2.13)\begin{equation} \dot p =- \partial H/ \partial q, \quad \dot q = \partial H / \partial p, \end{equation}

where the Hamiltonian $H(\,p,q)$, defined up to an additive constant, may be written as

(2.14)\begin{equation} H(\,p,q) = G(q) - G(0) + 4T \sqrt{(q_1+q)(q_2+q)(q_3-q)(q_4-q)} \cos p, \end{equation}

with

(2.15)\begin{align} G(q) &= q \Delta \omega + \tfrac{1}{2} (T_{11}(q_1+q)^2 + T_{22}(q_2+q)^2 + T_{33}(q_3-q)^2 + T_{44}(q_4-q)^2) \nonumber\\ & \quad + 2T_{12}(q_1+q)(q_2+q) + 2T_{13}(q_1+q)(q_3-q) + 2T_{14}(q_1+q)(q_4-q) \nonumber\\ & \quad + 2T_{23}(q_2+q)(q_3-q) + 2T_{24}(q_2+q)(q_4-q) + 2T_{34}(q_3-q)(q_4-q). \end{align}

The degree of the Hamiltonian system (2.13) is one so that it is integrable. Since $H(\,p,q)$ is conserved along the flow of $(\,p,q)$, we have, using (2.5) and (2.14)

(2.16)\begin{equation} H(\,p(t),q(t)) = H(\,p(0),0) = 4T \sqrt{q_1q_2q_3q_4} \cos p_0. \end{equation}

Since $p_0$ intervenes through its cosine, we shall consider $p_0 \in [0,{\rm \pi} ]$ from now on. (The Hamiltonian derived by Andrade & Stuhlmeier (2023a) is a particular case of (2.14) in which, with the present notations, $q_1=q_2$ and $q_3=q_4$.)

Squaring (2.7) and taking (2.14) and (2.16) into account leads to

(2.17)\begin{equation} \dot q^2 = 16 T^2 (q_1+q)(q_2+q)(q_3-q)(q_4-q) - (G(q) - G(0) - H(\,p_0,0))^2 , \end{equation}

or equivalently to the Bretherton equation:

(2.18)\begin{align} \dot q^2 = f(q), \quad f(q) = 16 T^2 (q_1+q)(q_2+q)(q_3-q)(q_4-q) -(Aq^2 + Bq + C)^2, \end{align}

where $A$ and $B$ are defined in (2.9) and where

(2.19)\begin{equation} C =- 4T \sqrt{q_1q_2q_3q_4} \cos p_0. \end{equation}

Equations similar to (2.18) have been derived by Inoue (1975), Boyd & Turner (1978), Chen (1989) and Stiassnie & Shemer (2005). (Substituting $q(t) = 4T Z(t)$ in (2.18) yields equation (3.9) in Stiassnie & Shemer (2005).) Expanding $f(q)$ gives

(2.20)\begin{equation} f(q) = a q^4 + b q^3 + c q^2 +d q + e, \end{equation}

with

(2.21)\begin{equation} \left. \begin{gathered} a = 16T^2 - A^2, \\ b = 16T^2 (q_1 + q_2 - q_3 - q_4) - 2AB,\\ c = 16T^2 (q_1 q_2 + q_3 q_4 - (q_1 + q_2)(q_3 + q_4)) - B^2 - 2AC, \\ d = 16T^2 ((q_1 + q_2)q_3 q_4 - q_1 q_2 (q_3 + q_4)) - 2BC, \\ e = 16T^2 q_1 q_2 q_3 q_4 \sin^2 p_0. \end{gathered} \right\} \end{equation}

According to Jeffreys & Jeffreys (1956, p. 667), Pars (1965, p. 4) or Craik (1985, p. 138), (2.18) may therefore be interpreted as the energy conservation equation of a unit-mass Newtonian particle $q(t)$ initially at $q(0)=0$ moving rectilinearly in a quartic potential $U(q)$ defined by

(2.22)\begin{equation} U(q) =- {\tfrac{1}{2}}\, f(q). \end{equation}

3. Steady solutions and their stability

3.1. Steady equilibria

From the definition adopted in § 1, a steady equilibrium is a solution of (1.1) with constant amplitudes $|b_i|$. From the Manley–Rowe relations (2.1), it corresponds to a solution of (2.18) such that $\dot q (t) = 0$ for all time. Since $q(0) = 0$, a steady equilibrium therefore corresponds in our problem to the null solution $q(t) = 0$, $\forall t \geq 0$, for which (2.18) implies $f(0)=0$. But since (2.18) is equivalent to ‘Newton's second law’:

(3.1)\begin{equation} \ddot{q} = {\tfrac{1}{2}}\, f'(q) =- U'(q), \end{equation}

steady equilibria also satisfy $U'(0) = 0$ (here and below, $f'(q) = \mathrm {d} f / \mathrm {d} q$), i.e. they are critical points of the potential energy (Arnold 1989, p. 99). For (2.20), conditions $f(0) = 0$ and $f'(0) = 0$ are respectively equivalent to $e=0$ and $d=0$. Since we now consider $T \not = 0$, condition $e=0$ yields either $q_1 q_2 q_3 q_4 = 0$, or $p_0 = 0$ or ${\rm \pi}$, while condition $d=0$ becomes

(3.2)\begin{equation} 2 T ((q_1 + q_2)q_3 q_4 - q_1 q_2 (q_3 + q_4)) + B \sqrt{q_1q_2q_3q_4} \cos p_0 = 0. \end{equation}

Suppose first $q_1 q_2 q_3 q_4 = 0$ and choose, without lost of generality, $q_4=0$. Condition (3.2) implies $q_1 q_2 q_3 = 0$. Therefore, a second wave must have a zero initial wave action, say $q_3 = 0$. From system (1.1) with $|b_i|^2 = q_i$, $i=1,2$, we get the steady bichromatic wave first considered by Phillips (1960) and Longuet-Higgins & Phillips (1962) (see also Zakharov 1967; Hogan, Gruman & Stiassnie 1988; Leblanc 2009):

(3.3)\begin{equation} \left. \begin{gathered} b_1(t)=\sqrt{q_1} \mathrm{e}^{\mathrm{i} \varphi_1} \mathrm{e}^{-\mathrm{i} \varOmega_1 t}, \quad \varOmega_1 = \omega_1 + T_{11}q_1 + 2 T_{12}q_2, \\ b_2(t)=\sqrt{q_2} \mathrm{e}^{\mathrm{i} \varphi_2} \mathrm{e}^{-\mathrm{i} \varOmega_2 t}, \quad \varOmega_2 = \omega_2 + T_{22}q_2 + 2 T_{21}q_1, \end{gathered} \right\} \end{equation}

where we recall that $\varphi _i = p_i(0)$. These solutions were given by Benney (1962) who also noticed that if in addition $q_2=0$ one recovers the Stokes wave:

(3.4)\begin{equation} b_1(t)=\sqrt{q_1} \mathrm{e}^{\mathrm{i} \varphi_1} \mathrm{e}^{-\mathrm{i} \varOmega_1 t}, \quad \varOmega_1 = \omega_1 + T_{11}q_1. \end{equation}

Turning now to the case $q_1 q_2 q_3 q_4 \not = 0$ and $p_0 = 0$ or ${\rm \pi}$, we get from (3.2):

(3.5)\begin{equation} \Delta \omega + B_1 q_1 + B_2 q_2 - B_3 q_3 - B_4 q_4 = 2 T \frac{q_1 q_2 (q_3 + q_4) - (q_1 + q_2)q_3 q_4 }{\sqrt{q_1q_2q_3q_4} \cos p_0}. \end{equation}

Since from (2.7) $p(t) = p_0$ at equilibrium, we conclude that:

Theorem 3.1 The wave quartet defined by

(3.6)\begin{equation} \left. \begin{gathered} b_i(t)=\sqrt{q_i} \mathrm{e}^{\mathrm{i} \varphi_i}\mathrm{e}^{-\mathrm{i} \varOmega_i t}, \quad q_i > 0, \quad i=1,\ldots,4,\\ \varOmega_i = \omega_i - T_{ii} q_i + 2\sum_{j=1}^4 T_{ij} q_j + \frac{2T}{q_i}\sqrt{q_1q_2q_3q_4} \cos p_0,\\ p_0 = \varphi_1 + \varphi_ 2 - \varphi_3 - \varphi_ 4 = 0 \mbox{ or } {\rm \pi}, \end{gathered} \right\} \end{equation}

and satisfying the compatibility condition (3.5) is a steady solution of (1.1) with (1.2).

3.2. Linear stability

Analogy with Newtonian dynamics allows us to formulate a simple criterion for linear stability of steady equilibria. Indeed, (3.1) yields, after linearization around $q=0$,

(3.7)\begin{equation} \ddot q - \gamma^2 q = 0, \quad \gamma^2 = {\tfrac{1}{2}} f''(0) =- U''(0), \end{equation}

where from (2.20) with (2.21) and (2.22), we have $\gamma ^2 = c$. Therefore, the null solution is linearly stable if $c < 0$ and unstable otherwise; growth is exponential if $c > 0$ or algebraic if $c = 0$.

Consider first the linear stability of the bichromatic solution (3.3) for which $q_3 = q_4 = 0$. In that case, $c=16T^2 q_1 q_2 - B^2$, where $B$ is defined in (2.9). Therefore, the bichromatic wavetrain (3.3) is exponentially unstable if

(3.8)\begin{equation} 16 T^2 q_1 q_2 > ( \Delta \omega + B_1 q_1 + B_2 q_2)^2. \end{equation}

Recalling that $q_i=|b_i|^2$ and that $B_i$ are defined in (2.9), we recover the criterion derived in Leblanc (2009, equation (15)) and recovered by Andrade & Stuhlmeier (2023a). We also recall that (3.8) characterizes respectively ‘type B’ and ‘class-Ia’ instabilities following the respective classifications of Okamura (1984) for one-dimensional standing waves and of Ioualalen & Kharif (1994) for steady bichromatic wavetrains.

Turning now to the stability of steady wave quartets with non-zero initial wave actions $(q_1, q_2, q_3, q_4)$ satisfying (3.5) and $\cos p_0 = \pm 1$, the linear stability criterion becomes, after replacing $B$ by the right-hand side of (3.5) and $C$ by (2.19) onto the expression of $c$ in (2.21):

Theorem 3.2 The steady wave quartet (3.6) satisfying (3.5) is exponentially unstable if

(3.9)\begin{align} \frac{A}{2T} \frac{\cos p_0}{\sqrt{q_1 q_2 q_3 q_4}} > \frac{1}{4} \left( \frac{1}{q_1} + \frac{1}{q_2} - \frac{1}{q_3} - \frac{1}{q_4}\right)^2 + \frac{(q_1 + q_2)(q_3 + q_4) - q_1 q_2 - q_3 q_4 }{q_1 q_2 q_3 q_4}, \end{align}

algebraically unstable if equality holds, otherwise linearly stable.

3.3. The particular case of equal wave actions

Consider the particular case of the steady wave quartet with equal wave actions $q_1 = q_2 = q_3 = q_4 \equiv q_0>0$ and initial phase mismatch $p_0 = 0$ or ${\rm \pi}$. We recall that the existence of such steady interactions is conditioned by (3.5) which reduces in the present case to

(3.10)\begin{equation} \Delta \omega + q_0 \Delta B = 0, \end{equation}

where $\Delta B = B_1 + B_2 - B_3 - B_4$, i.e. from (2.9):

(3.11)\begin{equation} \Delta B = T_{11} + T_{22} - T_{33} - T_{44} + 4(T_{12} - T_{34}). \end{equation}

At resonance, (3.10) implies $\Delta B = 0$, while off resonance we get $q_0 = - \Delta \omega / \Delta B$. Since $q_0>0$, $\Delta \omega$ and $\Delta B$ must have opposite signs for the quartet to exist. The corresponding instability criterion may be easily deduced from (3.9). Therefore:

Criterion 3.1 Steady wave quartets (3.6) with $q_1 = q_2 = q_3 = q_4 \equiv q_0$ exist if $q_0=-\Delta \omega /\Delta B>0$. At resonance, they exist for any $q_0>0$ if $\Delta B = 0$. In both cases, they are exponentially unstable if

(3.12)\begin{equation} (A/T) \cos p_0 > 4 \quad (\,p_0=0 \mbox{ or } {\rm \pi}). \end{equation}

3.4. Lagrange theorem and Lyapunov stability

In his treatise on analytical mechanics published in 1788, Lagrange presented his famous principle on the stability of equilibrium positions, which was rigorously proved by Lejeune–Dirichlet in 1846 and generalized by Lyapunov in 1892 (see Loria & Panteley 2017). Lagrange theorem may be stated as (Gantmacher 1975, pp. 166–173; Arnold 1989, p. 99):

Lagrange theorem. If $U(0)=0$ is a strict local minimum of the potential $U(q)$, then the null solution of (2.18) is Lyapunov stable.

For our purpose, stability in the sense of Lyapunov is defined by:

Lyapunov stability. The null solution is Lyapunov stable if for each $\varepsilon > 0$ there exists $\delta > 0$ such that, if $|\dot q (0)| < \delta$ initially, then $\sup (|q(t)|, |\dot q (t)|) < \varepsilon$ for all $t\geq 0$.

In the present case, $U(q) = - {\tfrac {1}{2}}(a q^4 + b q^3 + c q^2 + d q + e)$. But, as stated previously, $q=0$ is a steady equilibrium if $U(0)=0$ and $U'(0)=0$, or equivalently $d=e=0$. Therefore, at equilibrium, $U(q) = - {\tfrac {1}{2}}q^2(a q^2 + b q + c)$. But since in that case $U''(0)=-c$, then $U(q)$ admits a strict local minimum at $q=0$ if $c<0$. Therefore in our case, a linearly stable equilibrium is also Lyapunov stable. This means that if $c<0$, any sufficiently small disturbance will remain bounded.

Of course it would be necessary to be precise about what is meant by ‘sufficiently small’ but we shall not pursue that direction for at least two reasons: firstly, because my study is restricted to the interactions inside a single quartet while interactions with other waves may be destabilizing (see e.g. Okamura 1985); secondly, because higher-order interactions are not taken into account (see e.g. Andrade & Stuhlmeier 2023a). As a consequence, we have to keep in mind that stability is only indicative because limited to the discrete four-wave interactions considered in the present study.

By contrast, linear instability criteria are meaningful in the nonlinear regime as it is known that the existence of an exponentially growing solution of the linearized equation (3.7) implies instability of the null solution in the nonlinear equation (3.1) (see Verhulst 1996, p. 88). Furthermore, if other interactions were taken into account, various instability mechanisms would compete without mutual cancellation.

Therefore, the various instability criteria presented in the present study have to be considered as sufficient conditions for instability.

4. Exact unsteady solutions for a quartic potential

4.1. General properties

We have seen in § 2 that solutions of (2.18) with (2.5) are bounded. Furthermore, following Inoue (1975), we get from (2.18) the following inequalities:

(4.1)\begin{equation} \left. \begin{gathered} f(0) = 16T^2 q_1 q_2 q_3 q_4 \sin^2 p_0 \geq 0,\\ f(-q_1) \leq 0, \quad f(-q_2) \leq 0, \quad f(q_3) \leq 0, \quad f(q_4) \leq 0. \end{gathered} \right\} \end{equation}

Therefore we can conclude that, by continuity, $f$ admits at least two real roots, say $\xi _-$ and $\xi _+$, verifying $\xi _- \leq 0 \leq \xi _+$; $f$ is therefore a polynomial function of degree at least equal to two: either quartic, cubic or quadratic.

Excluding the case $\xi _- = \xi _+ = 0$ corresponding to root $0$ with multiplicity at least equal to 2 corresponding to a steady solution since $d=0$ and $e=0$ as explained in § 3.1, we restrict from now on our discussion to the cases where either $\xi _- \leq 0 < \xi _+$ or $\xi _- < 0 \leq \xi _+$. Without lost of generality, suppose that $\xi _-$ and $\xi _+$ are the closest roots from $0$. Equation (2.18) shows that unsteady solutions exist if $f(q) \geq 0$; since $q(0) = 0$ and $f(0) \geq 0$, Jeffreys & Jeffreys (1956, pp. 667–668), Bretherton (1964) and Pars (1965, pp. 4–6) showed that $\xi _- \leq q(t) \leq \xi _+$, $\forall t \geq 0$, and that the particle ‘velocity’ vanishes on the boundaries of this interval: $\dot q ( \xi _\pm ) = 0$. If $\xi _-$ and $\xi _+$ are both simple roots, they are turning points, i.e. the sign of $\dot q(t)$ changes on turning points and the direction of the particle is reversed; therefore $q(t)$ is periodic. If the multiplicity of either $\xi _-$ or $\xi _+$, say $\xi _-$, is strictly greater than one, then $\lim _{t\to \pm \infty } q(t) = \xi _-$ and the solution is a breather. Finally if $\xi _-$ and $\xi _+$ are both double roots, then either $\lim _{t\to \pm \infty } q(t) = \xi _\pm$ or $\lim _{t\to \pm \infty } q(t) = \xi _\mp$ and the solution is a pump. Finally, from (2.6), we have also

(4.2)\begin{equation} -\min(q_1,q_2) \leq \xi_- \leq q(t) \leq \xi_+ \leq \min (q_3,q_4). \end{equation}

Equation (2.18) with (2.5) may be integrated (see e.g. Craik 1985, p. 138):

(4.3)\begin{equation} t \equiv t(q) ={\pm} \int_0^q \frac{{\rm d} \xi}{\sqrt{f(\xi)}} ={\pm} \int_0^q \frac{{\rm d} \xi}{\sqrt{-2 U(\xi)}} , \end{equation}

from which $q \equiv q(t)$ may formally be obtained by inversion. The sign indeterminacy above shows that for each fixed values of the parameters, (2.18) admits two solutions, say $\{ Q_- (t),\, Q_+ (t) \}$, such that $Q_-(t) = Q_+(-t)$. Now, let $Q_+$ be the solution such that at the initial time $\dot Q_+ (0) \geq 0$. Therefore, $\dot Q_-(0) = - \dot Q_+ (0) \leq 0$. The solution to choose is determined because of (2.7) from which we get

(4.4)\begin{equation} \dot q (0) =-4T \sqrt{q_1 q_2 q_3 q_4} \sin p_0. \end{equation}

Thus, if $-4T \sqrt {q_1 q_2 q_3 q_4} \sin p_0 \geq 0$, then $q(t)=Q_+ (t)$. Else $q(t)=Q_-(t)=Q_+(-t)$. Finally, if $q(t)$ is periodic, the period $\tau$ is

(4.5)\begin{equation} \tau = 2 \int_{\xi_-}^{\xi_+} \frac{{\rm d} \xi}{\sqrt{f(\xi)}} = 2 \int_{\xi_-}^{\xi_+} \frac{{\rm d} \xi}{\sqrt{-2 U(\xi)}} . \end{equation}

In the quartic case, $f$ in (2.20) with $a \not =0$ may be factorized as

(4.6)\begin{equation} f(q) = a (q-\xi_1)(q-\xi_2)(q-\xi_3)(q-\xi_4), \end{equation}

where $\xi _1$, $\xi _2$, $\xi _3$ and $\xi _4$ are the four roots of $f$. (The labelling of the roots $\xi _i$ is independent of the labelling of the initial wave actions $q_i$.) Since $f$ is real-valued, roots are either real or complex conjugate by pairs. The roots of a quartic polynomial may formally be obtained by quadrature with Ferrari's method (published by Cardan in 1545) but formulae, which are too lengthy to be reported here, are implemented in computer algebra systems. Simple expressions are given in Appendix A in the case $q_1=q_2=q_3=q_4$.

The nature of the solutions of the Bretherton equation (2.18) depends on the sign of $a$ and on the nature of the roots, as illustrated in figure 1 (see also Turner 1980). If $a=0$, the potential is either cubic or quadratic and the solutions are postponed to Appendix B.

Figure 1. The possible configurations for bounded unsteady solutions of (2.18) satisfying (2.5). Motion occurs in the potential well defined by $q \in [\xi _-, \xi _+]$, where $\xi _\pm$ such that $\xi _- \leq 0 < \xi _+$ or $\xi _- < 0 \leq \xi _+$ are the nearest roots around zero between which $U(q) = - \tfrac {1}{2} f(q) \leq 0$. Quartic potential (4.6) with $a>0$: (a) periodic solution (4.8); (b) breather solution (4.14); (c) pump solution (4.16). Quartic potential (4.6) with $a<0$: (d) periodic solution (4.21); (e) breather solution (4.23); (f) rational breather solution (4.25); (g) periodic solution (4.27). Cubic potential (B1) with $b>0$: (h) periodic solution (B2); (i) breather solution (B4). Any other possibility may be deduced by symmetry with respect to the vertical axis. The case of quadratic potential with periodic solution (B6) has been omitted.

4.2. The case $a > 0$

4.2.1. Periodic solution

If $f$ admits four distinct real roots $\xi _1,\ldots, \xi _4$ such that (figure 1a)

(4.7)\begin{equation} \xi_1 < \xi_2 \leq 0 < \xi_3 < \xi_4 \quad \mbox{or} \quad \xi_1 < \xi_2 < 0 \leq \xi_3 < \xi_4, \end{equation}

then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{I}(t), Q_{I}(- t) \}$ with

(4.8)\begin{equation} \left. \begin{gathered} Q_{I} (t) = \frac{\xi_2 (\xi_3 - \xi_1) - \xi_1 (\xi_3-\xi_2)\,\mbox{sn}^2(u(t),k)}{(\xi_3-\xi_1)-(\xi_3-\xi_2) \,\mbox{sn}^2(u(t),k)},\quad u(t) = \frac{\gamma t}{2} + \mbox{sn}^{-1}(l,k) ,\\ \gamma = \sqrt{a (\xi_4-\xi_2)(\xi_3-\xi_1)}, \quad k = \sqrt{\frac{(\xi_3-\xi_2)(\xi_4-\xi_1)}{(\xi_4-\xi_2)(\xi_3-\xi_1)}}, \quad l = \sqrt{\frac{\xi_2 (\xi_3 - \xi_1)}{\xi_1 (\xi_3-\xi_2)}}, \end{gathered} \right\} \end{equation}

where $\mbox {sn}$ and $\mbox {sn}^{-1}$ are Jacobi elliptic functions defined here by (see e.g. Byrd & Friedman (1971, p. 18))

(4.9)\begin{equation} x = \mbox{sn}^{-1}(y,k) = \int_0^y \frac{{\rm d} \xi}{\sqrt{(1-\xi^2)(1-k^2 \xi^2)}}, \quad y = \mbox{sn} (x,k). \end{equation}

Period is

(4.10)\begin{equation} \tau = \frac{4}{\gamma} \,\mbox{K}(k), \quad \mbox{K}(k) = \int_0^{{\rm \pi}/2} \frac{{\rm d} \theta}{\sqrt{1-k^2 \sin^2 \theta}}, \end{equation}

where $\mbox {K}$ is the complete elliptic integral of the first kind. (Different conventions exist for the arguments of elliptic functions; we follow here the notations of Byrd & Friedman (1971). By contrast, in Wolfram Mathematica the entry for $\mbox {sn} (x, k)$ as defined in (4.9) is JacobiSN[x,m], where $m=k^2$; similarly for $\mbox {K}(k)$ defined in (4.10) for which the entry is EllipticK[m].)

Note that at the initial time

(4.11)\begin{equation} \dot Q_I (0) = \frac{\gamma \xi_1 \xi_2}{l (\xi_2 - \xi_1)} \,\mbox{cn} (\mbox{sn}^{-1}(l,k), k) \, \mbox{dn} (\mbox{sn}^{-1}(l,k), k), \end{equation}

where at fixed modulus $k$: $\mbox {cn}(x)= \sqrt {1-\mbox {sn}^2(x)}$ and $\mbox {dn}(x)= \sqrt {1-k^2 \,\mbox {sn}^2(x)}$ (Byrd & Friedman 1971, p. 19). Therefore, at fixed $k$:

(4.12)\begin{equation} \mbox{cn} (\mbox{sn}^{-1}(y)) \, \mbox{dn} (\mbox{sn}^{-1}(y)) = \sqrt{1-y^2}\sqrt{1-k^2y^2}. \end{equation}

Then

(4.13)\begin{equation} \dot Q_I (0) = \frac{\gamma \xi_1 \xi_2}{l (\xi_2 - \xi_1)} \sqrt{1-l^2}\sqrt{1-k^2l^2} \geq 0. \end{equation}

Formula (254.00) in Byrd & Friedman (1971, p. 112) has been used to get (4.8), plotted in figure 2(a). Similar solutions were found by Inoue (1975), Boyd & Turner (1978), Shemer & Stiassnie (1985), Chen & Snyder (1989), Cappellini & Trillo (1991) and Stiassnie & Shemer (2005).

Figure 2. Unsteady solutions $q(t)$ in a quartic potential (4.6) with $a=1$ (a) or $a=-1$ (b). (a) Periodic solution (4.8) with $\xi _1=-1.2$, $\xi _2=-1$, $\xi _3=2$, $\xi _4=2.4$ (solid line); breather solution (4.14) with $\xi _{12}=-1$, $\xi _3=2$, $\xi _4=2.4$ (dotted line); pump solution (4.16) with $\xi _{12}=-1$, $\xi _{34}=2$ (dashed line). (b) Periodic solution (4.21) with $\xi _1=-1$, $\xi _2=2$, $\xi _3=2.1$, $\xi _4=2.2$ (solid line); breather solution (4.23) with $\xi _{1}=-1$, $\xi _{23}=2$, $\xi _4=2.2$ (dotted line); rational breather solution (4.25) with $\xi _{1}=-1$, $\xi _{234}=2$ (dashed line).

4.2.2. Breather solution

If $f$ admits a double real root $\xi _{12}$ and two distinct real roots $\xi _3$ and $\xi _4$ such that $\xi _{12} < 0 \leq \xi _3 < \xi _4$ (figure 1b), then (2.18) with (2.5) has the pair of breather solutions $\{ Q_{I\!I}(t), Q_{I\!I} (- t) \}$ with

(4.14)\begin{equation} \left. \begin{gathered} Q_{I\!I} (t) =\xi_{12} \frac{v^4(t) + 4 n^2 v^3(t) - 2 s v^2(t) + 4n^2 r v(t) + r^2}{ v^4(t) - 2 (8 \xi_{12}^2 n^2 + r) v^2(t) + r^2 } ,\\ v(t) = (\xi_{12} (\xi_3 + \xi_4) - 2 (\xi_3 \xi_4 - n \sqrt{\xi_3 \xi_4})) \exp(\sqrt{a}n t), \\ n = \sqrt{(\xi_3-\xi_{12})(\xi_4-\xi_{12})}, \quad r = \xi_{12}^2 (\xi_4 - \xi_3)^2, \\ s = 4 \xi_{12} (\xi_{12}^2 + \xi_3 \xi_4) (\xi_3 + \xi_4) - \xi_{12}^2 (3 \xi_3 + \xi_4) (\xi_3 + 3 \xi_4). \end{gathered} \right\} \end{equation}

At the initial time: $\dot Q_{I\!I} (0)= - \sqrt {a \xi _3 \xi _4} \xi _{12} \geq 0$. Note that from (4.1): $\xi _{12} = - q_1 = - q_2$.

Derived with the assistance of a computer algebra system and plotted in figure 2(a), solution (4.14) has not been found in the literature in this general form. However, if $\xi _3=0$, we get

(4.15)\begin{equation} Q_{I\!I} (t) =\xi_{12} - \frac{2n^2}{2 \xi_{12} - \xi_4 (1+ \cosh (\sqrt{a}n t))}, \quad n=\sqrt{-\xi_{12}(\xi_4-\xi_{12})}, \end{equation}

corresponding to expression (22a) in Inoue (1975). The ‘pump-depletion’ solutions (A6) and (A7) in Cappellini & Trillo (1991) are similar to (4.14), but correspond in their formulation to specific values of the roots $\xi _i$. Finally, connection between (4.14) and the ‘discrete Akhmediev breathers’ (5.4) and (5.15) in Andrade & Stuhlmeier (2023b) remains to be clarified since their solutions are given in implicit form, contrary to (4.14).

4.2.3. Pump solution

If $f$ admits two double real roots $\xi _{12}$ and $\xi _{34}$ such that $\xi _{12} < 0 < \xi _{34}$ (figure 1c), then (2.18) with (2.5) has the pair of pump solutions $\{ Q_{I\!I\!I} (t), \, Q_{I\!I\!I} (- t) \}$ with

(4.16)\begin{equation} Q_{I\!I\!I} (t) = \xi_{34} \frac{\mathrm{e}^{\mu t} - 1}{ \mathrm{e}^{\mu t} - m }, \quad \mu = \sqrt{a}(\xi_{34} - \xi_{12}) , \quad m = \frac{\xi_{34}}{\xi_{12}}. \end{equation}

At the initial time: $\dot Q_{I\!I\!I} (0) = - \sqrt {a} \xi _{12} \xi _{34} > 0$. The ‘shock-like’ solutions of Inoue (1975) and the ‘multibreather’ solution of Andrade & Stuhlmeier (2023a,b) are equivalent to (4.16), plotted in figure 2(a).

From (4.1): $\xi _{12} = -q_1 = -q_2 = -q_{12}$ and $\xi _{34} = q_3 = q_4 = q_{34}$; from (4.6):

(4.17)\begin{equation} f(q)=a(q - \xi_{12})^2(q - \xi_{34})^2 = (16T^2 - A^2) (q + q_{12})^2(q - q_{34})^2. \end{equation}

Identification with (2.18) yields, excluding the case $A=0$, the following two compatibility conditions (Andrade & Stuhlmeier 2023a):

(4.18)\begin{equation} \cos p_0 = \frac{A}{4T} \in [-1,1], \quad \Delta \omega + (B_1+B_2-A)q_{12} - (B_3+B_4-A)q_{34} = 0. \end{equation}

Finally, if $q_{12} = q_{34} = q_0>0$, the second condition in (4.18) gives $\Delta \omega + (\Delta B) q_0 = 0$, and (4.16) with $-\xi _{12} = \xi _{34} = q_0$ becomes simply

(4.19)\begin{equation} Q_{I\!I\!I} (t) = q_0 \tanh (\sqrt{a} q_0 t). \end{equation}

4.3. The case $a < 0$

4.3.1. Periodic solution (four real roots)

If $f$ admits four distinct real roots $\xi _1,\ldots, \xi _4$ such that (figure 1d)

(4.20)\begin{equation} \xi_1 \leq 0 < \xi_2 < \xi_3 \leq \xi_4 \quad \mbox{or} \quad \xi_1 < 0 \leq \xi_2 < \xi_3 \leq \xi_4, \end{equation}

then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{I\!V} (t), \, Q_{I\!V} (- t) \}$ with

(4.21)\begin{align} \left. \begin{gathered} Q_{I\!V} (t) = \frac{\xi_1 (\xi_4 - \xi_2) + \xi_4 (\xi_2-\xi_1)\,\mbox{sn}^2(u(t),k)}{(\xi_4-\xi_2)+(\xi_2-\xi_1) \,\mbox{sn}^2(u(t),k)},\quad u(t) = \frac{\gamma t}{2} + \mbox{sn}^{-1}(l, k) ,\\ \gamma = \sqrt{-a (\xi_4-\xi_2)(\xi_3-\xi_1)}, \quad k = \sqrt{\frac{(\xi_4-\xi_3)(\xi_2-\xi_1)}{(\xi_4-\xi_2)(\xi_3-\xi_1)}}, \quad l = \sqrt{-\frac{\xi_1 (\xi_4 - \xi_2)}{\xi_4 (\xi_2-\xi_1)}}. \end{gathered} \right\} \end{align}

Period is $\tau = 4 \gamma ^{-1}\mbox {K}(k)$. At the initial time:

(4.22)\begin{equation} \dot Q_{I\!V} (0) = \frac{\gamma l (\xi_2- \xi_1 ) \xi_4^2}{(\xi_4-\xi_1)(\xi_4 - \xi_2)} \sqrt{1-l^2}\sqrt{1-k^2l^2} \geq 0. \end{equation}

Formula (252.00) in Byrd & Friedman (1971, p. 103) has been used to get (4.21), plotted in figure 2(b). A similar solution is given by Chen (1989).

4.3.2. Breather solution

If $f$ admits a double real root $\xi _{23}$ and two distinct real roots $\xi _1$ and $\xi _4$ such that $\xi _{1} \leq 0 < \xi _{23} < \xi _4$ (figure 1e), then (2.18) with (2.5) has the pair of breather solutions $\{ Q_{V} (t), Q_{V} (- t) \}$ with

(4.23)\begin{equation} \left. \begin{gathered} Q_{V} (t) =\xi_{23} \frac{v^4(t) + 4 n^2 v^3(t) - 2 s v^2(t) + 4n^2 rv(t) + r^2}{ v^4(t) + 2 (8 \xi_{23}^2n^2 - r) v^2(t) +r^2 } ,\\ \displaystyle v(t) =- (n^2 + 2 n \sqrt{- \xi_1 \xi_4} + \xi_{23}^2 - \xi_1 \xi_4) \exp(\sqrt{-a}n t), \\ n = \sqrt{(\xi_{23}-\xi_1)(\xi_4-\xi_{23})}, \quad r = \xi_{23}^2 (\xi_4 - \xi_1)^2, \\ s = 4 \xi_{23} (\xi_{23}^2 + \xi_1 \xi_4) (\xi_1 + \xi_4) - \xi_{23}^2 (3 \xi_1 + \xi_4) (\xi_1 + 3 \xi_4). \end{gathered} \right\} \end{equation}

At the initial time: $\dot Q_{V} (0)= \sqrt {a \xi _1 \xi _4} \xi _{23} \geq 0$. Plotted in figure 2(b), solution (4.23) has not been found in the literature. If $\xi _1=0$, we get

(4.24)\begin{equation} Q_{V} (t) =\xi_{23} + \frac{2n^2}{2 \xi_{23} - \xi_4 (1+ \cosh (\sqrt{-a}n t))}, \quad n=\sqrt{\xi_{23}(\xi_4-\xi_{23})}. \end{equation}

4.3.3. Rational breather solution

If $f$ admits a triple real root $\xi _{234}$ and a single real root $\xi _1$ such that $\xi _{1} \leq 0 < \xi _{234}$ (figure 1f), then (2.18) with (2.5) has the pair of rational breather solutions $\{ Q_{V\!I}(t), \, Q_{V\!I} (- t) \}$ with

(4.25)\begin{equation} Q_{V\!I} (t) = \frac{\xi_{234} \mu t (\mu t + 4\;m )}{ \mu t( \mu t + 4 m) + 4 (1 + m^2)}, \quad \mu = \sqrt{-a} (\xi_{234} - \xi_1),\quad m = \sqrt{-\xi_1/\xi_{234}}. \end{equation}

At the initial time: $\dot Q_{V\!I} (0)= \sqrt {a \xi _1 \xi _{234}^3} \geq 0$. Plotted in figure 2(b), a solution equivalent to (4.25) is given by Turner (1980).

Finally, if $\xi _1=0$, (4.25) may be written as

(4.26)\begin{equation} \displaystyle Q_{V\!I} (t) = \xi_{234} \tilde Q(\tilde t), \quad \tilde Q (\tilde t) = \tilde t^2/(\tilde t^2 + 4), \quad \tilde t = \sqrt{-a} \xi_{234} t. \end{equation}

4.3.4. Periodic solution (two real roots and two complex conjugate roots)

Finally, if $f$ admits two distinct real roots $\xi _1$ and $\xi _2$ such that $\xi _1 \leq 0 < \xi _2$ or $\xi _1 < 0 \leq \xi _2$, and two complex conjugate roots $\xi _3$ and $\xi _4 = \xi _3^*$ (figure 1g), then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{V\!I\!I} (t), \, Q_{V\!I\!I} (- t) \}$ with

(4.27)\begin{align} \left. \begin{gathered} Q_{V\!I\!I} (t) = \frac{(\beta_1 \xi_2 + \beta_2 \xi_1) - (\beta_1 \xi_2 - \beta_2 \xi_1)\,\mbox{cn}(w(t),k)}{(\beta_1 + \beta_2) - (\beta_1 - \beta_2) \,\mbox{cn}(w(t),k)},\quad w(t) = \gamma t + \mbox{cn}^{-1}(l, k) ,\\ \gamma = \sqrt{- a \beta_1 \beta_2}, \quad k = \sqrt{\frac{(\xi_2-\xi_1)^2 - (\beta_2-\beta_1)^2}{4 \beta_1 \beta_2}}, \quad l = \frac{\beta_1 \xi_2 + \beta_2 \xi_1}{\beta_1 \xi_2 - \beta_2 \xi_1},\\ \beta_1 = \sqrt{(\xi_1 - x_3)^2 + y_3^2}, \quad \beta_2 = \sqrt{(\xi_2 - x_3)^2 + y_3^2},\quad x_3 = \mathrm{Re} (\xi_3), \quad y_3 = \mathrm{Im} (\xi_3). \end{gathered} \right\} \end{align}

Period is $\tau = 4 \gamma ^{-1} \mbox {K}(k)$. At the initial time:

(4.28)\begin{equation} \dot Q_{V\!I\!I} (0) =-\frac{a(\beta_1 \xi_2 - \beta_2 \xi_1)^2}{2\gamma(\xi_2-\xi_1)} \sqrt{1-l^2} \sqrt{1+k^2(l^2-1)} \geq 0, \end{equation}

since at fixed $k$, $\mbox {sn} (\mbox {cn}^{-1}(y))\,\mbox {dn} (\mbox {cn}^{-1}(y)) = \sqrt {1-y^2}\sqrt {1+k^2(y^2-1)}$.

Equation (259.00) in Byrd & Friedman (1971, p. 133) has been used to get (4.27). A similar solution is given by Turner (1980).

5. Deep-water gravity waves

5.1. The truncated quartet model

Deep-water irrotational gravity waves propagating in an inviscid incompressible fluid are governed in spectral space at third order in amplitude by the Zakharov equation (Zakharov 1966, 1968; Krasitskii 1990, 1994):

(5.1)\begin{align} \mathrm{i} \frac{\partial {\mathcal{B}} (\boldsymbol{k})}{\partial t} = \omega(\boldsymbol{k}) {\mathcal{B}} (\boldsymbol{k}) + \int_{\mathbb{R}^{6}} T (\boldsymbol{k},\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}) {\mathcal{B}}^*(\,\boldsymbol{p}) {\mathcal{B}}(\boldsymbol{q}) {\mathcal{B}}(\boldsymbol{r}) \delta(\boldsymbol{k}+\boldsymbol{p}-\boldsymbol{q}-\boldsymbol{r}) \,\mathrm{d} \boldsymbol{p} \,\mathrm{d} \boldsymbol{q} \,\mathrm{d} \boldsymbol{r}, \end{align}

where $\boldsymbol {k} \in \mathbb {R}^{2}$, $\omega (\boldsymbol {k})=\sqrt {g|\boldsymbol {k}|}$, $\delta (\boldsymbol {k})$ is Dirac delta function and the real-valued function $T(\boldsymbol {k},\boldsymbol {p},\boldsymbol {q},\boldsymbol {r})$ is Krasitskii's kernel given in Appendix C. At leading order, ${\mathcal {B}}(\boldsymbol {k},t)$ is related to the free-surface elevation $z=\eta (\boldsymbol {x},t)$, $\boldsymbol {x} \in \mathbb {R}^{2}$, by (we follow Janssen's (2004, p. 132) convention for the Fourier transform, so that expressions given here for $T$ (and $T_{ij}$) must be divided by $(2{\rm \pi} )^2$ to recover those given in Krasitskii (1994))

(5.2)\begin{equation} \eta (\boldsymbol{x},t)= \int_{\mathbb{R}^2} \hat \eta(\boldsymbol{k},t) \mathrm{e}^{\mathrm{i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}}\mathrm{d} \boldsymbol{k}, \quad \hat \eta(\boldsymbol{k},t)=\sqrt{\frac{|\boldsymbol{k}|}{2\omega(\boldsymbol{k})}} ( {\mathcal{B}}(\boldsymbol{k},t)+ {\mathcal{B}}^*(-\boldsymbol{k},t) ). \end{equation}

According to Zakharov (1966, 1968), an exact solution of (5.1) is the Stokes wave:

(5.3)\begin{align} {\mathcal{B}}(\boldsymbol{k},t) = b(t) \delta (\boldsymbol{k} - \boldsymbol{k}_0), \quad b(t)=b_0 \mathrm{e}^{-\mathrm{i} \varOmega_0 t}, \quad \varOmega_0 = \omega(\boldsymbol{k}_0) + T(\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0) |b_0|^2. \end{align}

If we now consider a linear combination of waves ${\mathcal {B}}(\boldsymbol {k},t) = \sum _{i=1}^N b_i(t) \delta (\boldsymbol {k} - \boldsymbol {k}_i)$ with $N>1$, the Zakharov equation (5.1) yields a system of ordinary differential equations which is not closed, as noticed by Okamura (1985); it leads indeed to the generation of higher harmonics on time scales of order $(|T||b|^2)^{-1}$. The mathematical validity of such an ansatz is therefore an open question. (This was pointed to me out by an anonymous reviewer even for $N=2$; see also discussion in Badulin et al. (1995). Zakharov (1967) already noticed that the bichromatic wave (3.3) is an approximate solution.)

If the terms corresponding to higher harmonics are neglected, one gets a truncated model consisting of a closed system of ordinary differential equations considered in various textbooks and review articles (e.g. Yuen & Lake 1982; Craik 1985; Shemer & Stiassnie 1991; Janssen 2004; Kartashova 2010) and implicitly or explicitly used in a number of articles, including those by Saffman & Yuen (1980), Caponi, Saffman & Yuen (1982), Okamura (1984, 1985), Shemer & Stiassnie (1985), Hogan et al. (1988), Badulin et al. (1995), Stiassnie & Shemer (2005), Leblanc (2009) and Andrade & Stuhlmeier (2023a,b):

(5.4)\begin{equation} \mathrm{i} \frac{\mathrm{d} b_i}{\mathrm{d} t} = \omega_i b_i + \sum_{j,m,n=1}^N \hat T_{ijmn} b_j^* b_m b_n , \quad i=1,\ldots, N, \end{equation}

where $\omega _i =\sqrt {g|\boldsymbol {k}_i|}$ and $\hat T_{ijmn} = T(\boldsymbol {k}_i,\boldsymbol {k}_j,\boldsymbol {k}_m,\boldsymbol {k}_n)$ if $\boldsymbol {k}_i+\boldsymbol {k}_j=\boldsymbol {k}_m+\boldsymbol {k}_n$ and $0$ otherwise. In the case of four waves satisfying (1.3), (5.4) reduces to (1.1) where $T \equiv T(\boldsymbol {k}_1,\boldsymbol {k}_2,\boldsymbol {k}_3,\boldsymbol {k}_4)$ and $T_{ij} = T(\boldsymbol {k}_i,\boldsymbol {k}_j,\boldsymbol {k}_i,\boldsymbol {k}_j)$. The ‘free-surface elevation’ of this truncated low-order model corresponding to an ‘isolated’ quartet would be

(5.5)\begin{equation} \eta_{quartet} (\boldsymbol{x},t) = \sum_{i=1}^4 \sqrt{\frac{|\boldsymbol{k}_i|}{2\omega_i}} \left( b_i(t) \mathrm{e}^{\mathrm{i} \boldsymbol{k}_i \boldsymbol{\cdot} \boldsymbol{x}} + b_i^*(t) \mathrm{e}^{-\mathrm{i} \boldsymbol{k}_i \boldsymbol{\cdot} \boldsymbol{x}} \right)\!. \end{equation}

Although the formal validity of such a model with respect to actual solutions of the Zakharov equation (5.1) is an open question, numerical and experimental results support its usefulness (Liu & Liao 2014; Liu et al. 2015; Liao et al. 2016). If the quartet is resonant, Benney's equations (1.1) obtained with the method of multiple scales are recovered.

5.2. Bidirectional standing waves and their stability

A particular case of interaction (1.3) concerns bidirectional standing waves for which

(5.6)\begin{equation} (\boldsymbol{k}_1,\boldsymbol{k}_2,\boldsymbol{k}_3,\boldsymbol{k}_4) = (\boldsymbol{k}_1,-\boldsymbol{k}_1,\boldsymbol{k}_3,-\boldsymbol{k}_3), \quad \boldsymbol{k}_1 \not= \boldsymbol{k}_3. \end{equation}

For simplicity, we consider the case where initially $q_1=q_2=q_3=q_4 \equiv q_0 >0$. From criterion 3.1 (§ 3.3), the wave quartet is steady providing that $q_0=-\Delta \omega / \Delta B >0$, where $\Delta B$ is given in (3.11). For deep water, we have (see Appendix C)

(5.7)\begin{equation} T_{11}=T_{22}=k_1^3, \quad T_{33}=T_{44}=k_3^3, \quad T_{12}=-k_1^3, \quad T_{34}=-k_3^3, \end{equation}

so that $\Delta B = - 2(k_1^3-k_3^3)$ and $-\Delta \omega / \Delta B = (\omega _1/k_1^3) \rho (k_3/k_1)$, where the function $\rho (\kappa ) = (1-\sqrt {\kappa })/(1-\kappa ^3)$ is strictly positive for any $\kappa >0$ (the discontinuity at $\kappa =1$ may be removed since $\rho (\kappa ) \to 1/6$ when $\kappa \to 1$). Therefore $q_0 = (\omega _1-\omega _3)/(k_1^3-k_3^3)$ is defined and strictly positive if $k_1 \not = k_3$. If $k_1=k_3$, then both $\Delta \omega =0$ and $\Delta B=0$ so that, from criterion 3.1, $q_0>0$ may be chosen arbitrarily.

Now, from (3.6):

(5.8)\begin{equation} \left. \begin{gathered} \varOmega_1 = \varOmega_2 = \omega_1 + (2 \bar{T} - k_1^3)q_0, \quad \varOmega_3 = \varOmega_4 = \omega_3 + (2 \bar{T} - k_3^3)q_0,\\ \bar{T} = T(\boldsymbol{k}_1,\boldsymbol{k}_3,\boldsymbol{k}_1,\boldsymbol{k}_3) + T(\boldsymbol{k}_1,-\boldsymbol{k}_3,\boldsymbol{k}_1,-\boldsymbol{k}_3) + T(\boldsymbol{k}_1,-\boldsymbol{k}_1,\boldsymbol{k}_3,-\boldsymbol{k}_3) \cos p_0, \end{gathered} \right\} \end{equation}

where $\bar T$ may be evaluated explicitly thanks to the expressions given in Appendix C. Choosing $\varphi _1 = \varphi _ 2 = 0$ and $\varphi _3 = \varphi _ 4 = -p_0/2$, it may be shown from (5.5) that the free-surface elevation is, at leading order,

(5.9)\begin{align} \left. \begin{gathered} \eta_{quartet}(\boldsymbol{x},t) = 2 {\mathcal{A}}_1\cos (\boldsymbol{k}_1 \boldsymbol{\cdot} \boldsymbol{x}) \cos(\varOmega_1 t) + 2 {\mathcal{A}}_3 \cos (\boldsymbol{k}_3 \boldsymbol{\cdot} \boldsymbol{x}) \cos\left(\varOmega_3 t + \frac{p_0}{2} \right),\\ {\mathcal{A}}_i = \sqrt{ \frac{2k_i q_0}{\omega_i}}, \quad i=1,3, \quad q_0 = \frac{\omega_1-\omega_3}{k_1^3-k_3^3},\quad p_0 = 0 \mbox{ or } {\rm \pi}. \end{gathered} \right\} \end{align}

Free surface (5.9) is periodic if $\varOmega _1$ and $\varOmega _3$ are commensurate. If $k_1 = k_3 \equiv k_0$ and $p_0=0$, we recover the bidirectional standing waves studied by Okamura (1985), since in that case $\varOmega _1= \varOmega _3 = \omega _0 + (2 \bar T - k_0^3) q_0$, where $q_0>0$.

Let us now apply the instability criterion 3.1 (§ 3.3). Using homogeneity property (C4), we can define, without lost of generality, a function $\chi (\kappa,\psi )$ such that

(5.10)\begin{equation} \left. \begin{gathered} \frac{A(\boldsymbol{k}_1, - \boldsymbol{k}_1, \boldsymbol{k}_3 , -\boldsymbol{k}_3 )}{T(\boldsymbol{k}_1, - \boldsymbol{k}_1, \boldsymbol{k}_3 , -\boldsymbol{k}_3 )} = \frac{A(\boldsymbol{i}, - \boldsymbol{i}, \boldsymbol{\kappa} , -\boldsymbol{\kappa} )}{T(\boldsymbol{i}, - \boldsymbol{i}, \boldsymbol{\kappa} , -\boldsymbol{\kappa} )} \equiv 4 \chi (\kappa,\psi),\\ \boldsymbol{i} = (1,0), \quad \boldsymbol{\kappa} = (\kappa \cos \psi, \kappa \sin \psi),\quad \kappa = k_3/k_1. \end{gathered} \right\} \end{equation}

Since $\psi = \mbox {angle} (\boldsymbol {i},\boldsymbol {\kappa }) = \mbox {angle} (\boldsymbol {k}_1,\boldsymbol {k}_3)$, it is sufficient to consider $\psi \in [0,{\rm \pi} /2]$. Function $\chi (\kappa,\psi )$ is plotted on figure 3; since $\chi (\kappa,\psi ) > 0$, there is no instability according to (3.12) for $p_0 = {\rm \pi}$. On the contrary, if $p_0=0$, standing waves are unstable in regions where $\chi (\kappa,\psi ) > 1$, that is, when $0<\kappa <1/\kappa _c$ or $\kappa > \kappa _c$, where $\kappa _c \equiv \kappa _c(\psi )>1$ is such that $\chi (\kappa _c(\psi ),\psi )=1$. It is found numerically that $1.730<\kappa _c(\psi ) <1.861$, $\forall \psi \in [0,{\rm \pi} /2]$. Therefore, our results support the following statement:

Proposition On deep water, bidirectional standing waves (5.9) with (5.8) and $p_0=0$ are exponentially unstable when $k_3 / k_1$ (or $k_1/k_3$) $\geq 1.861$.

Figure 3. Graph of $\chi (\kappa,\psi )$ defined in (5.10) for $\psi = 0$ (solid line); $\psi = {\rm \pi}/8$ (dotted line); $\psi = {\rm \pi}/2$ (dashed line). Steady bidirectional standing waves (5.9) with $p_0=0$ are exponentially unstable if $\chi (\kappa,\psi ) >1$, where $\kappa =k_3/k_1$ and $\psi = \mbox {angle} (\boldsymbol {k}_1,\boldsymbol {k}_3)$.

Theoretical predictions have been compared with numerical integration of (1.1) and results are in excellent agreement. A representative example is plotted in figure 4. Of course, the unsteady perturbed solutions that appear on this plot may be expressed with one of the explicit formulas given in § 4. To this aim, we first determine in each case the function $f$ in (2.20). From the data given in the caption of figure 4, we get $a<0$ so that $f$ may be factorized as (4.6), where the four roots $\xi _i$, $i=1,\ldots,4$, may be explicitly evaluated thanks to the expressions given in Appendix A since in both cases $q_1=q_2=q_3=q_4=q_0(1+\varepsilon )$; their numerical values are reported in table 1. Therefore, because of the Manley–Rowe relations (2.4), the unsteady periodic solutions plotted in figure 4 are $|b_1(t)|^2 =q_1 + q(t)$, where the functions $q(t)$ are given by (4.21) for $p_0=0$ (dashed line) and by (4.27) for $p_0={\rm \pi}$ (solid line). Their respective periods are given in table 1. Their graphs cannot be distinguished from their numerical counterparts, as the relative errors between numerical and exact solutions is of the order of $0.1\,\%$ at the final time of the computations.

Figure 4. Numerical integration of system (1.1) with $g=1$, $\boldsymbol {k}_1=-\boldsymbol {k}_2=(1,0)$, $\boldsymbol {k}_3=-\boldsymbol {k}_4=(0,2)$, $q_i=q_0(1+\varepsilon )$ with $q_0=(\omega _1-\omega _3)/(k_1^3-k_3^3)$ and $\varphi _1 = \varphi _2=0$, $\varphi _3= \varphi _4=-p_0/2$. Dotted line ($\varepsilon =0$, $p_0={\rm \pi}$): steady bidirectional standing waves (5.9). Dashed line ($\varepsilon =0.01$, $p_0=0$): unstable disturbance. Solid line ($\varepsilon =0.01$, $p_0={\rm \pi}$): stable disturbance.

Table 1. Truncated values of the roots $\xi _i$ of the quartic function (4.6) with $a=-25.57$ corresponding to the parameters of the unsteady periodic solutions represented in figure 4 for $\varepsilon =0.01$. Their respective exact solutions $q(t)$ are (4.21) for $p_0=0$ and (4.27) for $p_0={\rm \pi}$; their crest–trough amplitude is $\xi _2 - \xi _1$ and their period is $\tau$.

Finally, the free-surface elevation of these bifurcated solutions may be written as

(5.11)\begin{align} \left. \begin{gathered} \eta_{quartet}(\boldsymbol{x},t) = 2 {\mathcal{A}}_1 \eta_1(t) \cos (\boldsymbol{k}_1 \boldsymbol{\cdot} \boldsymbol{x}) \cos p_1(t) + 2 {\mathcal{A}}_3 \eta_3(t) \cos (\boldsymbol{k}_3 \boldsymbol{\cdot} \boldsymbol{x}) \cos p_3(t),\\ \eta_1(t) = \sqrt{1+\varepsilon + q(t)/q_0}, \quad \eta_3(t) = \sqrt{1+\varepsilon - q(t)/q_0}, \end{gathered} \right\} \end{align}

where ${\mathcal {A}}_i$ and $q_0$ are defined in (5.9) and where the phases $p_i(t)$ are integrated numerically from (2.3). Results are plotted in figure 5: they show in the unstable case (figure 5b) five large-scale beats over ten periods $\tau$ of the corresponding function $q(t)$. This is due to the fact that the amplitudes $|b_i(t)|$ involve the square root of $q(t)$; see (2.4).

Figure 5. Free-surface elevation (5.11) at $\boldsymbol {x} = (0,0)$ for $\varepsilon =0.01$. Stable disturbance $p_0={\rm \pi}$ (a); unstable disturbance $p_0=0$ (b). Data are those of figure 4 and table 1.

5.3. Steady wave quartets and their stability

We now turn to the general case (1.3), on or off resonance. Consider first Kartashova's resonant quartet (Lvov, Nazarenko & Pokorni 2006; Kartashova 2010, p. 78):

(5.12)\begin{equation} \boldsymbol{K}_1= (495, 90), \quad \boldsymbol{K}_2 = (64, 128), \quad \boldsymbol{K}_3 = (359, 118), \quad \boldsymbol{K}_4 = (200, 100), \end{equation}

useful for symbolic computation. Since $\Delta \omega = 0$ and $\Delta B \not = 0$ in that case, criterion 3.1 (§ 3.3) is of no use to construct a steady solution; we need another strategy. Testing $q_1=q_2\equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$, the compatibility condition (3.5) with $\Delta \omega =0$ becomes

(5.13)\begin{equation} \frac{q_{34}}{q_{12}} = \frac{4T - (B_1+B_2)\cos p_0}{4T - (B_3+B_4)\cos p_0} \approx \left\{ \begin{array}{@{}ll} 0.8546, & \mbox{if} \ p_0=0,\\ 0.5857, & \mbox{if} \ p_0= {\rm \pi}. \end{array} \right. \end{equation}

Since $q_{34}/q_{12}>0$ in both cases, steady states exist. It may be checked that the instability condition (3.9) becomes also (3.12) when $q_1=q_2\equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$. Since $A/T \approx -7.430< -4$ for Kartashova's resonant quartet (5.12) or for any rescaled non-trivial resonant quartet $(\lambda \boldsymbol {K}_1,\lambda \boldsymbol {K}_2,\lambda \boldsymbol {K}_3,\lambda \boldsymbol {K}_4)$ according to (C4), we conclude immediately that the corresponding steady wave quartet is stable if $p_0=0$ and unstable if $p_0={\rm \pi}$. This method may be employed for any other resonant quartet provided that the ratio $q_{34}/q_{12}$ is positive. Therefore in summary, we have proved:

Criterion 5.1 Steady resonant wave quartets (3.6) with $q_1=q_2 \equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$ exist if

(5.14)\begin{equation} \frac{q_{34}}{q_{12}} = \frac{4T - (B_1+B_2)\cos p_0}{4T - (B_3+B_4)\cos p_0} > 0 \quad (\,p_0= 0 \mbox{ or } {\rm \pi}). \end{equation}

They are exponentially unstable if $(A/T) \cos p_0 > 4$.

(A companion criterion is given in Appendix D for $q_1=q_3$ and $q_2=q_4$.)

Unstable disturbed solutions can lie on the same resonant surface (figure 6a), or not. For instance, if we consider the following non-resonant (but nearly resonant) quartet:

(5.15)\begin{equation} \boldsymbol{k}_1= (500, 90), \quad \boldsymbol{k}_2 = (60, 130), \quad \boldsymbol{k}_3 = (360, 120), \quad \boldsymbol{k}_4 = (200, 100), \end{equation}

the corresponding unsteady solution is plotted in figure 6(b), showing again instability.

Figure 6. Wave action of steady Kartashova's resonant quartet $(\lambda \boldsymbol {K}_1,\lambda \boldsymbol {K}_2,\lambda \boldsymbol {K}_3,\lambda \boldsymbol {K}_4)$ given by (5.12), $\lambda =1/300$ and $p_0={\rm \pi}$ (dotted lines) versus unstable solutions according to criterion 5.1 (solid lines). Initial wave actions are in each case $q_{12}\approx 1.380$ and $q_{34} \approx 0.8084$. (a) Resonant solution with perturbed initial phase mismatch $p_0=1.05{\rm \pi}$ and same wave vectors. (b) Non-resonant solution with same phase mismatch $p_0={\rm \pi}$ and perturbed wave vectors $(\lambda \boldsymbol {k}_1,\lambda \boldsymbol {k}_2,\lambda \boldsymbol {k}_3,\lambda \boldsymbol {k}_4)$ given by (5.15) and $\lambda =1/300$. Solutions are $|b_1(t)|^2 = q_1+q(t)$ with $q(t)$ given by (a) (4.27) and (b) (4.21).

Let us now construct a steady solution for the non-resonant quartet (5.15). We first test criterion 3.1 (§ 3.3), but since in that case $\Delta \omega / \Delta B > 0$, a steady solution with $q_1 = q_2 = q_3 =q_4$ does not exist; we need one more time another strategy. We are, however, lucky with (5.15) because all the $B_i$ defined in (2.9) have the same sign: negative. Thus we can set $q_i = \alpha /B_i$ expecting $\alpha <0$ for existence. From (3.5) we get

(5.16)\begin{equation} \alpha =- \frac{\sqrt{B_1 B_2 B_3 B_4}\Delta \omega}{2T\Delta B} \cos p_0 \quad (\,p_0 = 0 \mbox{ or } {\rm \pi}). \end{equation}

Since $T>0$ for (5.15), we conclude that the only possibility to have $\alpha <0$ is $p_0=0$. Concerning stability, condition (3.9) with $q_i=\alpha /B_i$ becomes

(5.17)\begin{equation} \frac{A}{2T} \sqrt{B_1 B_2 B_3 B_4} \cos p_0 > \frac{(\Delta B)^2}{4} +(B_1 + B_2)(B_3 + B_4) - B_1 B_2 - B_3 B_4. \end{equation}

Applying this condition for quartet (5.15) and $p_0=0$ gives stability.

The method presented here may be applied to any non-resonant quartet providing that all the $B_i$ have the same sign:

Criterion 5.2 Steady non-resonant wave quartets (3.6) with $B_i>0$ (respectively $<0$) and $q_i=\alpha / B_i$, $i=1,\ldots,4$, where $\alpha$ is defined in (5.16), exist if $\alpha >0$ (respectively $<0$). They are exponentially unstable if (5.17) holds.

Other strategies might be used to construct steady non-resonant quartets. For instance we can choose $q_1=q_2$ and $q_3=q_4$, or $q_1=q_3$ and $q_2=q_4$. In these cases, compatibility condition (3.5) would yield affine relations between $q_i$, but conditions for exponential instability resulting from (3.9) would involve $q_i$, contrary to the instability conditions given in criteria 1 to 4. We shall not pursue this issue here.

6. The X-quartet: pump and beats

After steady and periodic solutions, let us now turn to the ‘exceptional cases’ mentioned by Bretherton (1964). We start with the pump solution (4.16). Although an example has already been given by Andrade & Stuhlmeier (2023a), we purpose here another specimen based on the following orthogonal resonant quartet, the ‘X-quartet’:

(6.1)\begin{equation} \boldsymbol{k}_1^ 0 = (k_0,0), \quad \boldsymbol{k}_2^0 = (-k_0,0), \quad \boldsymbol{k}_3^0 = (0,k_0), \quad \boldsymbol{k}_4^0 = (0,-k_0), \quad k_0>0, \end{equation}

with initially $q_1=q_2=q_3=q_4\equiv q_0 >0$. We have, from Appendix C:

(6.2)\begin{equation} \frac{T}{k_0^3}=- \frac{3}{4}, \quad \left[ \frac{T_{ij}}{k_0^3} \right] = \left[ \begin{array}{@{}cccc@{}} 1 & -1 & \nu & \nu \\ -1 & 1 & \nu & \nu \\ \nu & \nu & 1 & -1 \\ \nu & \nu & -1 & 1 \end{array} \right] , \quad \nu = \frac{4 \sqrt{2}-5}{28}, \end{equation}

so that, from (2.9) and (2.21):

(6.3)\begin{align} \frac{A}{k_0^3} =-\frac{4}{7} (1+ 2 \sqrt{2}), \quad \frac{A}{4T} = \frac{4}{21}(1+ 2 \sqrt{2}) \in [-1,1], \quad \frac{a}{k_0^6} = \frac{1}{49} (297 - 64 \sqrt{2})> 0. \end{align}

Furthermore, since $\Delta \omega = 0$ and $\Delta B =0$ in that case, both conditions in (4.18) are fulfilled for the existence of the pump solution (4.19). It remains to set $p_0=P_0$, where

(6.4)\begin{equation} P_0 = \arccos \left( \frac{A}{4T} \right) = \arccos \left( \frac{4}{21}(1+ 2 \sqrt{2}) \right)\!. \end{equation}

Once $q(t)$ is replaced by (4.19), it may be checked, after squaring (2.7), that $p(t)=P_0$. Furthermore, the individual phases may be integrated since, from (2.3), we have simply $d p_i / dt= -\omega _0 + 2k_0^3 q_0$, $i=1,\ldots,4$. Therefore, the solution of (1.1) is the ‘X-pump’:

(6.5)\begin{equation} b_1(t)=b_2(t)= \sqrt{q_0+q(t)}\mathrm{e}^{-\mathrm{i} \varOmega_0 t} ,\quad b_3(t)=b_4(t)= \sqrt{q_0-q(t)}\mathrm{e}^{-\mathrm{i} (\varOmega_0 t + \frac{1}{2} P_0)}, \end{equation}

where $q(t) = q_0 \tanh (\sqrt a q_0 t)$, $\varOmega _0 = \omega _0 -2 k_0^3 q_0$ and $\omega _0 = \sqrt {g k_0}$. Free surface is

(6.6)\begin{align} \left. \begin{gathered} \eta_{quartet}(\boldsymbol{x},t) = 2{\mathcal{A}}_0 \eta_+(t) \cos (k_0x) \cos(\varOmega_0 t) + 2{\mathcal{A}}_0 \eta_-(t) \cos (k_0y) \cos\left(\varOmega_0 t + {\frac{P_0}{2}} \right),\\ {\mathcal{A}}_0 = \sqrt{\frac{2k_0q_0}{\omega_0}}, \quad \eta_\pm(t) = \sqrt{1 \pm \tanh(\sqrt a q_0 t)}, \quad \varOmega_0 = \omega_0 -2 k_0^3 q_0, \quad q_0 > 0, \end{gathered} \right\} \end{align}

and where $a$ and $P_0$ are given respectively in (6.3) and (6.4). Figure 7 illustrates the fact that the energy of components $(\boldsymbol {k}_3^0,\boldsymbol {k}_4^0)$ is pumped and totally transferred to $(\boldsymbol {k}_1^0,\boldsymbol {k}_2^0)$, that is, from the $y$ standing wave to the $x$ standing wave.

Figure 7. Snapshots of the free-surface elevation (6.6) of the X-pump in the region $k_0x \in [-4{\rm \pi},4{\rm \pi} ]$, $k_0y \in [-4{\rm \pi},4{\rm \pi} ]$ ($g=1$): (a) $\varOmega _0 t = -2$, (b) $\varOmega _0 t =-1$, (c) $\varOmega _0 t = 0$, (d) $\varOmega _0 t =1$, (e) $\varOmega _0 t =2$.

One might expect that this pump solution could be reproduced experimentally; however, both standing waves constituting the complete solution (6.6) are linearly unstable with respect to infinitesimal disturbances constituted by the other couple of waves according to instability condition (3.8) applied for instance to the bichromatic wave $(\boldsymbol {k}_1^0,\boldsymbol {k}_2^0)$ with $q_1=q_2$ disturbed by $(\boldsymbol {k}_3^0,\boldsymbol {k}_4^0)$ yielding $16T^2 > 4(1+4\nu )^2 k_0^6$, true from (6.2). The converse being true, both instabilities may feed each other.

This is illustrated in figure 8 in which is represented the X-pump (dashed line) and a perturbed solution where a relative error of $0.1\,\%$ has been made on the initial phase mismatch $p_0$: the perturbed solution is periodic (solid line). In fact, the roots of (4.6) are, from (A2) with $B=0$ and $T$ and $A$ given by (6.2) and (6.3):

(6.7)\begin{equation} \frac{\xi_i}{q_0} \in \left\{{\pm} \sqrt{\frac{21(1+\cos p_0)}{25+8\sqrt 2} }, \pm \sqrt{\frac{21(1-\cos p_0)}{17-8\sqrt 2}} \right\}\!. \end{equation}

They are plotted in figure 9(a). They coincide by pairs for $p_0=P_0$ and are distinct for any $p_0 \notin \{ 0, P_0, {\rm \pi}\}$, two being positive, two negative. The solution is in that case periodic and completely determined by (4.8) together with $a$ in (6.3) and $\xi _i$ (6.7); their period is (4.10) (figure 9b), and their crest–trough amplitude at fixed $p_0$ is: $2 \min |\xi _i|$, $i=1,\ldots,4$. These are the ‘X-beats’ in which energy is periodically exchanged between components $(\boldsymbol {k}_1^0, \boldsymbol {k}_2^0)$ and $(\boldsymbol {k}_3^0, \boldsymbol {k}_4^0)$. Note finally that $0$ is a double root if $p_0=0$ or ${\rm \pi}$, corresponding to steady standing waves discussed in § 5.2.

Figure 8. Solutions of (2.18) for (6.1) with $q_1=q_2=q_3=q_4\equiv q_0$. X-pump with $p_0=P_0$ (dashed line); X-beats with $p_0=1.0001P_0$ (solid line). $P_0$ is given by (6.4).

Figure 9. (a) Dimensionless roots $\tilde \xi _i = |\xi _i /q_0|$ given by (6.7) of the quartic (4.6) for the X-quartet (6.1). The dashed line corresponds to $p_0=P_0$ (6.4) of the X-pump solution (6.6). (b) If $p_0 \notin \{ 0, P_0, {\rm \pi}\}$, solutions are periodic X-beats given by (4.8) with dimensionless period $\tilde \tau = k_0^3 q_0 \tau$ given by (4.10).

7. Breathers

7.1. The $\varPsi$-breathers

Looking for breathers is a difficult task because two or three roots of the quartic function (4.6) have to coincide in a non-symmetric way, as compared with the X-pump solution (4.19) for which the roots were equal by pairs and opposite.

Fortunately, one branch of such breathers bifurcates from the X-pump solution studied previously. Indeed, consider the one-parameter family of resonant quartet:

(7.1)\begin{equation} \left. \begin{gathered} \boldsymbol{k}_1^{\varepsilon}= (\alpha_+, 0), \quad \boldsymbol{k}_2^{\varepsilon} = (-\alpha_-, 0), \quad \boldsymbol{k}_3^{\varepsilon} = (\beta, \gamma), \quad \boldsymbol{k}_4^{\varepsilon} = (\beta, -\gamma),\\ \alpha_\pm= k_0 \left(1 \pm \tfrac{1}{2} \varepsilon + \tfrac{1}{4} \varepsilon^2\right)^2, \quad \beta = k_0 \varepsilon\left(1 + \tfrac{1}{4} \varepsilon^2\right), \quad \gamma = k_0 \left(1 - \tfrac{1}{16} \varepsilon^4\right), \end{gathered} \right\} \end{equation}

where $\varepsilon \geq 0$, with $\varepsilon \not =2$ to avoid $\boldsymbol {k}_3^{\varepsilon }=\boldsymbol {k}_4^{\varepsilon }$, is the main parameter ($k_0 >0$ is just a scaling parameter). Quartet (7.1) is a member of the ‘tridents’ (D2) proposed by Lvov et al. (2006), with $(m,n) = (1,\varepsilon /2)$. The X-quartet (6.1) is matched for $\varepsilon =0$.

With (7.1), it is found numerically that $T<0$, $A<0$ and $\Delta B>0$ for any positive $\varepsilon \not =2$. Furthermore, $4T-A>0$ (then $a<0$) for any $\varepsilon \not =2$ such that $\varepsilon _1 < \varepsilon < \varepsilon _2$, and $4T-A<0$ (then $a>0$) otherwise, where $\varepsilon _1 \approx 0.4672$ and $\varepsilon _2 \approx 8.562$. Therefore, we may expect to find breathers of the kind (4.14) for $0<\varepsilon < \varepsilon _1$ (because $a>0$), and those of the kind (4.23) or (4.25) for $\varepsilon _1 < \varepsilon < \varepsilon _2$, $\varepsilon \not =2$ (because $a<0$). The case $\varepsilon = \varepsilon _1$ (then $a=0$) will be considered later.

To construct breathers (4.14) bifurcating from the X-pump (6.6), we consider $0 < \varepsilon < \varepsilon _1$ (so that $a>0$) and choose again initially $q_1=q_2=q_3=q_4 \equiv q_0>0$. To do so, two roots have to coincide, so we impose arbitrarily the following constraint for the negative roots: $\xi _{12} = \xi _1 = \xi _2 = - q_0<0$, while $0 \leq \xi _3 < \xi _4$. Since the initial wave actions are equal, the four roots are given explicitly by (A2), here with

(7.2)\begin{equation} \Delta \omega=0, \quad \Delta B >0, \quad T<0, \quad 4T\pm A <0, \end{equation}

since $0 < \varepsilon < \varepsilon _1$. Roots are therefore real and such that $\xi _-^{(-)} < 0 < \xi _-^{(+)}$. Since the two negative roots must coincide with $-q_0$, we get a first compatibility condition:

(7.3)\begin{equation} \sqrt{(\Delta B)^2 + 16T(4T-A)(1-\cos p_0)} = 2(A-4T) - \Delta B. \end{equation}

Positiveness of the right-hand side is ensured when $0<\varepsilon \leq \varepsilon _3$, with $\varepsilon _3 \approx 0.3705$. In that range, on squaring the above equality, we get $\cos p_0 = (A- \Delta B)/(4T) \geq 0$. Furthermore, it may be checked numerically that the constraint $\cos p_0 \leq 1$ is ensured providing that $0<\varepsilon \leq \varepsilon _4$, with $\varepsilon _4 \approx 0.3213$. Therefore, in that range, we may choose

(7.4)\begin{equation} p_0 = \arccos \left( \frac{A-\Delta B}{4T} \right)\!. \end{equation}

With (7.4), the four roots are therefore $\xi _{12}= \xi _\pm ^{(\pm )}$, $\xi _{3}= \xi _+^{(-)}$, $\xi _4= \xi _-^{(+)}$:

(7.5)\begin{equation} \xi_{12}=-q_0, \quad \xi_3 = \left( 1 - \frac{\Delta B}{A-4T} \right) q_0, \quad \xi_4 = \left( 1 - \frac{\Delta B}{A+4T} \right) q_0. \end{equation}

Note that $p_0=0$, $\xi _3=0$ and $\xi _4/q_0 = 8T/(A+4T)$ when $\varepsilon = \varepsilon _4 \approx 0.3213$, which corresponds to the end point of this branch of solutions.

Together with $a=(4T+A)(4T-A)$, roots (7.5) completely determine the solution (4.14), referred to as the ‘$\varPsi$-breathers’ represented in figure 10 for various values of $\varepsilon$. Note that $\Delta B = 0$ for $\varepsilon =0$, so that (7.4) matches with (6.4) obtained for the X-pump solution. Expansion near $\varepsilon = 0$ gives, up to the second order:

(7.6)\begin{equation} \left. \begin{gathered} \frac{\xi_{12}}{q_0} =-1,\quad \frac{\xi_3}{q_0} = 1 - \frac{5}{46}(17 + 8 \sqrt{2}) \varepsilon^2 , \quad \frac{\xi_4}{q_0} = 1 + \frac{5}{142}(25- 8 \sqrt{2}) \varepsilon^2 ,\\ \frac{a}{k_0^6} = \frac{1}{49}(297 -64 \sqrt{2}) - \frac{8}{7}(17- 8 \sqrt{2})\varepsilon +\frac{1}{9604}(35\,017 - 86\,052 \sqrt{2}) \varepsilon^2 . \end{gathered} \right\} \end{equation}

It is interesting for the $\varPsi$-breathers to reconstruct the free-surface elevation. Since the relative $q(t)$ is given explicitly by (4.14), the relative phase $p(t)$ defined in (2.2) may be deduced from (2.7) and plotted in figure 10. Then, the four individual moduli $|b_i(t)|$ are deduced from the Manley–Rowe equation (2.4), while numerical integration of (2.3) is necessary to obtain the individual phases $p_i(t)=\arg b_i(t)$ in order reconstruct the free-surface elevation given by (5.5). This is achieved in figure 11 for the $\varPsi$-breathers defined by (7.1), (7.4) and (7.5) for different values of the parameter $\varepsilon$ that quantifies the departure from the X-pump solution (6.6). Is is clear from figure 11 that the sole considerations of $q(t)$ and $p(t)$ represented in figure 10 cannot explain the local behaviour of $\eta _{quartet}(0,t)$ and that phase mixing must be invoked to explain such destructive/constructive interferences (the ${\rm \pi} /2$ relative phase jump is also worth noting). Whether this kind of behaviour may be related or not to the occurrence of extreme events is left for future investigation.

Figure 10. The $\varPsi$-breathers corresponding to quartets (7.1) with $k_0=1$, $q_0=0.01$ and $p_0$ given by (7.4), for $\varepsilon = 0.2$ (dotted line), $\varepsilon =0.1$ (dot-dashed line), $\varepsilon =0.05$ (dashed line) and $\varepsilon =0.01$ (solid line); the grey line corresponds to the X-pump (4.19) for which $\varepsilon =0$. (a) Relative action $q(t)$. (b) Relative phase $p(t)$.

Figure 11. Free-surface elevation (5.5) at $\boldsymbol {x} = (0,0)$ for the $\varPsi$-breathers (7.1) with (7.4) for $\varepsilon = 0.01$ (a), $\varepsilon = 0.05$ (b), $\varepsilon = 0.1$ (c) and $\varepsilon = 0.2$ (d). Computations have been carried out backward and forward from $t=0$, with $g=1$, $k_0=1$ and $q_0=0.01$.

Finally, if $\varepsilon = \varepsilon _1$ or $\varepsilon = \varepsilon _2$, then $A=4T$ and $a=0$, so that $f$ in (2.18) becomes cubic and reads as (B1), where here $b=-2A (\Delta B) q_0 >0$. It may be deduced from the considerations of Appendix B in that case that if we constrain the positive roots to coincide, i.e. $\xi _{23} \equiv \xi _2 = \xi _3 = q_0$, then we get the following compatibility conditions:

(7.7)\begin{equation} \cos p_0 = 1 + \frac{\Delta B}{A} \in [-1,1], \quad \xi_1 =- \left( 1 + \frac{\Delta B}{2 A} \right)q_0 \leq 0. \end{equation}

For both $\varepsilon = \varepsilon _1$ and $\varepsilon = \varepsilon _2$ we have the same numerical value for $\Delta B / A \approx -0.8487$; both conditions above are fulfilled and the corresponding solution is breather (B4).

7.2. Looking for rational breathers

Since $a<0$ when $\varepsilon \not = 2$ lies in the range $\varepsilon _1 < \varepsilon < \varepsilon _2$, we expect to find rational breathers (4.25). Suppose that the triple root which characterizes this solution is positive, $\xi _2=\xi _3=\xi _4 \equiv \xi _{234}>0$, so that $\xi _1 \leq 0$. Now, contrary to the case $a>0$ for which the multiple roots must coincide (in absolute value) with the initial wave actions because of (4.1), roots satisfy in the present case the following inequalities:

(7.8)\begin{equation} - q_0 \leq \xi_1 \leq 0 < \xi_{234} \leq q_0. \end{equation}

However, no rational breather has been found for the resonant trident (7.1).

Thus, constraints imposed by the coincidence of three roots of the quartic function $f$ make the search for rational breathers even more difficult than for other breathers. For reasons that will be made clear later, still for the quartet (7.1), we consider the point in parameter space where $A=12T$. It is found numerically that it corresponds to $\varepsilon = \varepsilon _0$, where $\varepsilon _0 \approx 0.7475$. Now, from this point of bifurcation, we consider the following family of non-resonant quartet:

(7.9)\begin{equation} \boldsymbol{k}_1 = \boldsymbol{k}_1^{\varepsilon_0} + \boldsymbol{\kappa}, \quad \boldsymbol{k}_2 = \boldsymbol{k}_2^{\varepsilon_0} - \boldsymbol{\kappa}, \quad \boldsymbol{k}_3 = \boldsymbol{k}_3^{\varepsilon_0}, \quad \boldsymbol{k}_4 = \boldsymbol{k}_4^{\varepsilon_0}, \quad \boldsymbol{\kappa} = (k_0 \kappa_x,k_0 \kappa_y), \end{equation}

where the wave vectors $\boldsymbol {k}_i^{\varepsilon }$ are defined in (7.1) and where $(\kappa _x, \kappa _y)$ are real numbers.

Now, if we impose on the roots in (7.8) the constraints $\xi _1=0$ and $\xi _{234}=q_0>0$, and if we assume that $A=12T$ and $\cos p_0=-1$, then it may be deduced from (A2) that $q_0$ must satisfy the following compatibility condition:

(7.10)\begin{equation} q_0 =- \frac{\Delta \omega}{\Delta B + 16 T} > 0, \end{equation}

where all parameters are to be evaluated on the non-resonant manifold (7.9). In figure 12 is represented in the $(\kappa _x,\kappa _y)$ plane the curve $(\varGamma )$ defined by $A=12T$ that bifurcates from the origin $O$ corresponding to $\boldsymbol {\kappa }=0$. This curve passes through points $P_1$, $P_2$ and $P_3$ and forms a loop which crosses light-grey regions in which condition (7.10) is violated, so that rational breathers of the kind considered here exist on any point of $(\varGamma )$, except in these shadowed regions.

Figure 12. Curve $(\varGamma )$ defined by $A=12T$ in the $(\kappa _x,\kappa _y)$ plane defined in (7.9). The origin $O$ corresponds to $(\kappa _x,\kappa _y)=(0,0)$ for which (7.9) matches (7.1). The oval (dashed line) crossing $(\varGamma )$ in $O$ is the resonant curve $\Delta \omega =0$. The dashed curve is defined by $\Delta B + 16 T=0$. The light-grey region delimited by these two curves is defined by $\Delta \omega /(\Delta B + 16 T) > 0$. The dark-grey regions are defined by $a=16T^2 - A^2>0$. The coordinates of points $P_1$, $P_2$ and $P_3$ are reported in table 2.

Table 2. Coordinates and values of $a$, $q_0$ and $|\Delta \omega |$ for $P_1$, $P_2$ and $P_3$ of figure 12.

Truncated values of $a= -128 T^2$ (since $A=12T$) and $q_0$ in (7.10) are reported in table 2. Since $\xi _1=0$ and $\xi _{234}=q_0$, the corresponding rational breathers are (4.26). The values of the frequency mismatch show that for $|\Delta \omega |/(k_0^3q_0)$ to stay of order 1, one must stay on curve $(\varGamma )$ between $O$ and approximately $P_1$ to keep physical insight.

8. Summary and perspectives

The complete solutions of the cubic nonlinear equations (1.1) first derived by Armstrong et al. (1962) and Benney (1962) have been presented in order to determine, paraphrasing Phillips (1960), the ‘elementary interactions’ inside an isolated quartet of waves satisfying (1.3). Following the works of Bretherton (1964), Inoue (1975), Turner (1980), Shemer & Stiassnie (1985), Cappellini & Trillo (1991), Stiassnie & Shemer (2005) and Andrade & Stuhlmeier (2023a,b), the system has been reduced in $\S \,2$ to a single ordinary differential equation – the Bretherton equation (2.18) – governing the evolution of an auxiliary variable: the relative action $q(t)$ vanishing initially and related to the modulus $|b_i(t)|$ by the Manley–Rowe relations (2.4).

To summarize, the nature of a given non-degenerate wave quartet with wave vectors $\boldsymbol {k}_i$ satisfying (1.3) and prescribed initial amplitudes $b_i(0) \not = 0$, $i=1,\ldots,4$, may be determined through the following steps:

1. (a) Compute the initial wave actions $q_i=|b_i(0)|^2$, the initial phase mismatch $p_0$ from (2.12), the linear frequencies $\omega _i = \omega (\boldsymbol {k}_i)$ from the dispersion law of the medium under consideration and the frequency mismatch $\Delta \omega$ from (1.4).

2. (b) Compute the nonlinear coupling coefficients $T_{ij}$ and $T$ as given in Appendix C for deep-water gravity waves, and the coefficients $A$, $B$, $C$ and $a$, $b$, $c$, $d$, $e$ from (2.9), (2.19) and (2.21). Form the polynomial function $f$ in (2.20).

3. (c) Nature of the quartet.

   1. (i) If $p_0=0$ or $p_0={\rm \pi}$, and if condition (3.5) holds, then the wave quartet is steady (theorem 3.1). In that case, the quartet is exponentially unstable if inequality (3.9) holds (theorem 3.2), algebraically unstable if equality holds, otherwise the quartet is nonlinearly stable in the sense of Lyapunov (Lagrange theorem).

   2. (ii) If $p_0 \not = 0$ and $p_0 \not = {\rm \pi}$, or if condition (3.5) is violated, then the wave quartet is unsteady. In that case, we have to compute the roots $\xi _i$ of $f$ and determine the corresponding solution $q(t)$ according to the classification of either § 4 if $a \not = 0$, or Appendix B if $a=0$.

As anticipated by Bretherton (1964), periodic solutions $q(t)$ are the rule (the full field is, however, generally not periodic, as discussed by Stiassnie & Shemer (2005)), steady or unsteady non-periodic solutions are the exceptions. Indeed, steady solutions are constrained by the conditions recalled above, while unsteady non-periodic solutions (breathers or pump) are constrained by the equality of two or three roots $\xi _i$ of the function $f$.

The reverse problem which consists of finding a priori steady or non-periodic solutions is much more delicate, because of the complexity of the coefficients $T_{ij}$ and $T$ of the dispersive medium under consideration. In the steady case, one has to solve the algebraic compatibility relation (3.5), while in the unsteady case one has to elaborate strategies to make two or three roots $\xi _i$ of $f$ coincide. This reverse problem was the object of § 5 (steady solutions), and §§ 6 and 7 (pump and breather solutions).

By steady solutions, we mean those for which the relative wave action $q(t)$ vanishes identically, the corresponding waves of the quartet having therefore constant amplitudes $|b_i|^2 = q_i$. In addition to theorems 3.1 and 3.2 giving general conditions for existence and stability of non-degenerate steady quartets, four criteria have been deduced for particular choices of the initial amplitudes $q_i$. As pointed out in § 3.4, the various stability criteria derived through the present paper are also formally valid in the nonlinear regime because of a famous theorem by Lagrange, stability being defined in the sense of Lyapunov. This formal result is, however, of restricted utility for reasons explained at the end of § 3.4, essentially because our study is restricted to a single isolated quartet. However, it might be useful for experimental purposes to observe the steady stable solutions under controlled conditions.

Most of the unsteady solutions of the Bretherton equation (2.18) discussed in § 4 and Appendix B have already been published in the literature, but spread in different fields of physics including essentially plasma physics, nonlinear optics and hydrodynamics, with more or less permeability between the various fields. Chronologically, these include the works by Inoue (1975), Boyd & Turner (1978), Turner (1980), Shemer & Stiassnie (1985), Chen & Snyder (1989), Chen (1989), Cappellini & Trillo (1991), Stiassnie & Shemer (2005) and recently Andrade & Stuhlmeier (2023a,b). Some solutions were known only in particular cases, such as the breathers (4.14) and (4.23), and, to our point of view, a complete and unified classification was lacking.

Various examples of isolated deep-water gravity wave quartets have then been elaborated, such as among others, the X-pump and X-beats (§ 6) which are bifurcated solutions from the bidirectional standing wave built on the X-quartet (6.1), and the $\varPsi$-breathers (§ 7) built from the trident quartets (7.1) of Lvov et al. (2006). A number of these solutions have been found to bifurcate to periodic large-amplitude solutions (the X-beats for instance) in a way similar to the Fermi–Pasta–Ulam recurrence for the modulational instability (Yuen & Lake 1982; Shemer & Stiassnie 1985, 1991; Trillo & Wabnitz 1991; Janssen 2004; Leblanc 2009). This behaviour is also illustrated in the phase portraits in figure 2 of Andrade & Stuhlmeier (2023a): in this figure, the dots correspond to steady solutions and are connected by a heteroclinic orbit in phase space which corresponds to a breather; one sees that any departures from these saddle points or from heteroclinic orbits bifurcate to closed orbits corresponding to periodic large-amplitude solutions. Similar observations were made by Cappellini & Trillo (1991) and Andrade & Stuhlmeier (2023b) for the three-wave cubic interaction.

What was surprising to me is that stability or instability of non-degenerate quartets has to be understood as structural since no external disturbance is necessary for a solution to be unstable, in contrast for instance to the decay instability or to the modulational instability of Stokes wave or bichromatic wavetrains, where disturbances assimilated to interacting waves with initially infinitesimal amplitudes have to be superimposed to the initial states. Here, in the unstable case, any small departure from the initial conditions leads to a solution that diverges initially from the steady state.

I conclude this paper with possible related issues:

1. (a) Compute the response of the Zakharov equation (5.1) to initial data of the form ${\mathcal {B}}(\boldsymbol {k},0) = \sum _{i=1}^4 b_i(0) \delta (\boldsymbol {k} - \boldsymbol {k}_i)$ and compare with solutions of the truncated model (1.1), on or off resonance. Alternatively, is the Stokes wave the unique exact solution of (5.1) in the form ${\mathcal {B}}(\boldsymbol {k},t) = \sum _{i=1}^N b_i(t) \delta (\boldsymbol {k} - \boldsymbol {k}_i)$, $N\geq 1$?

2. (b) Can isolated quartet models such as the bidirectional standing wave (5.9) and the X-pump (6.6), the $\varPsi$-breather or their bifurcated periodic states be reproduced numerically or experimentally, as for steady states (Liu et al. 2015)?

3. (c) Is there a connection between the large variations observed in figure 11 and the occurrence of rogue waves (Onorato et al. 2013)?

4. (d) Would instabilities grow super-exponentially if a linear forcing term were added to (1.1) or (1.6), as predicted from cubic forced nonlinear Schrödinger and Davey–Stewartson equations (Leblanc 2007, 2008)?