Famous results by Rademacher, Kolmogorov and Paley and Zygmund state that random series on the torus enjoy better Lp bounds that the deterministic bounds. We present a natural extension of these harmonic analysis results to a partial-differential-equations setting. Specifically, we consider the one-dimensional nonlinear harmonic oscillator i∂tu + Δu − |x|2u = |u|r−1u, and exhibit examples for which the solutions are better behaved for randomly chosen initial data than would be predicted by the deterministic theory. In particular, on a deterministic point of view, the nonlinear harmonic oscillator equation is well posed in L2(ℝ) if and only if r ≤ 5. However, we shall prove that, for all nonlinearities |u|r−1u, r > 1, not only is the equation well posed for a large set of initial data whose Sobolev regularity is below L2, but also the flows enjoy very nice large-time probabilistic behaviour.

These results are joint work with Laurent Thomann and Nikolay Tzvetkov.

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

We review some recent results concerning tug-of-war games and their relation to some well-known partial differential equations (PDEs). In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero. Since the equations being studied are nonlinear and are not in divergence form, we will make extensive use of the concept of viscosity solutions.

We are interested in the positive doubly periodic solutions, which are even in each variable, of a stationary nonlinear Schrödinger equation in ℝ2, with a small parameter. For any pair of periods (2a, 2b), we construct a branch of solutions that concentrate uniformly to the ground-state solution of the equation.

If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L∈(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∈(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

We examine the regularity of the solution of a quasilinear system involving the curl of vector fields. This system arises in the mathematical theory of superconductivity. The C2+α regularity was obtained by Bates and Pan under the condition that Ω is simply connected and has no holes, and that the normal component of the curl of the boundary data vanishes. The aim of this paper is to remove these technical restrictions on the topology of the domain and on the boundary data. By carefully studying the related quasilinear Neumann problem, we obtain the C2+α regularity without assuming these technical conditions.

The self-adjointness of Δ2 + κ|x|−4 (κ>κ0 = κ0(N)) in L2(ℝN) is established as an application of the perturbation theorem in terms of Re(Au, Bεu), u ∈ D(A), for two non-negative self-adjoint operators A, B in a Hilbert space, where the family {Bε}ε>0 is the Yosida approximation of B. A key to the proof lies in a new inequality for the functions ν ∈ L2(ℝN) with |x|2Δν ∈ L2(ℝN) derived by using two real parameters.

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

This paper is concerned with the limit of the vanishing ratio of the electron mass to the ion mass in the hydrodynamic models for plasmas in critical Besov spaces. We give a new construction of approximation solutions and show that periodic initial-value problems of certain scaled hydrodynamic models have smooth solutions in a (finite) time interval where the Euler solution is known to exist. Furthermore, it is justified that, as the electron mass tends to zero, the smooth solutions converge rigorously to solutions of the incompressible Euler equations, and the definite convergence orders are also obtained.