bibliography[1] Ablowitz, M.J.; Clarkson, P.A.; Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991).
[2] Ablowitz, M.J.; Prinari, B.; Trubach, A.D.; Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, Cambridge, 2004).
[3] Aftalion, A.; Vortices in Bose–Einstein Condensates, Progress in Nonlinear Differential Equations and Their Applications 67 (Birkhäuser, Boston, 2006).
[4] Aftalion, A.; Helfler, B.; “On mathematical models for Bose–Einstein condensates in optical lattices”, Rev. Math. Phys. 21 (2009), 229–278.
[5] Agrawal, G.P.; Kivshar, Yu.S.; Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
[6] Agueev, D.; Pelinovsky, D.; “Modeling of wave resonances in low-contrast photonic crystals”, SIAM J. Appl. Math. 65 (2005), 1101–1129.
[7] Akhmediev, N.; Ankiewicz, A.; Solitons: Nonlinear Pulses and Beams (Kluwer Academic, Dordrecht, 1997).
[8] Al Khawaja, U.; “A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations”, J. Math. Phys. 51 (2010), 053506 (11 pp.).
[9] Alama, S.; Li, Y.Y.; “Existence of solutions for semilinear elliptic equations with indefinite linear part”, J. Diff. Eqs. 96 (1992), 89–115.
[10] Alfimov, G.L.; Brazhnyi, V.A.; Konotop, V.V.; “On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation”, Physica D 194 (2004), 127–150.
[11] Angulo Pava, J.; Nonlinear Dispersive Equations (Existence and Stability of Solitary and Periodic Travelling Wave Solutions) (AMS, Providence, RI, 2009).
[12] Azizov, T.Ya.; Iohvidov, I.S.; Elements of the Theory of Linear Operators in Spaces with Indefinite Metric (Nauka, Moscow, 1986) [in Russian].
[13] Bambusi, D.; Sacchetti, A.; “Exponential times in the one-dimensional Gross–Pitaevskii equation with multiple well potential”, Comm. Math. Phys. 275 (2007), 1–36.
[14] Barashenkov, I.V.; Oxtoby, O.F.; Pelinovsky, D.E.; “Translationally invariant discrete kinks from one-dimensional maps”, Phys. Rev. E 72 (2005), 035602(R) (4 pp.).
[15] Barashenkov, I.V.; van Heerden, T.C.; “Exceptional discretizations of the sine-Gordon equation”, Phys. Rev. E 77 (2008), 036601 (9 pp.).
[16] Bao, W.; Du, Q.; “Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow”, SIAM J. Sci. Comput. 25 (2004), 1674–1697.
[17] Belmonte-Beitia, J.; Pérez-García, V.M.; Vekslerchik, V.; Torres, P. J.; “Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities”, Discr. Cont. Dyn. Syst. B 9 (2008), 221–233.
[18] Belmonte-Beitia, J.; Pelinovsky, D.E.; “Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient”, Applic. Anal. 89 (2010), 1335–1350.
[19] Belmonte-Beitia, J.; Cuevas, J.; “Solitons for the cubic–quintic nonlinear Schrödinger equation with time- and space-modulated coefficients”, J. Phys. A: Math. Theor. 42 (2009), 165201 (11 pp.).
[20] Bloch, F.; “Über die Quantenmechanik der Electronen in Kristallgittern”, Z. Phys. 52 (1928), 555–600.
[21] Bluman, G.W.; Kumei, S.; Symmetries and Differential Equations (Springer, New York, 1989).
[22] Brugarino, T.; Sciacca, M.; “Integrability of an inhomogeneous nonlinear Schrödinger equation in Bose–Einstein condensates and fiber optics”, J. Math. Phys. 51 (2010), 093503 (18 pp.).
[23] Buffoni, B.; Sere, E.; “A global condition for quasi-random behaviour in a class of conservative systems”, Commun. Pure Appl. Math. 49 (1996), 285–305.
[24] Busch, K.; Schneider, G.; Tkeshelashvili, L.; Uecker, H.; “Justification of the nonlinear Schrödinger equation in spatially periodic media”, Z. Angew. Math. Phys. 57 (2006), 905–939.
[25] Buslaev, V.S.; Perelman, G.S.; “Scattering for the nonlinear Schrödinger equation: states close to a soliton”, St. Petersburg Math. J. 4 (1993), 1111–1142.
[26] Buslaev, V.S.; Perelman, G.S.; “On the stability of solitary waves for nonlinear Schrödinger equations”, Amer. Math. Soc. Transl. 164 (1995), 75–98.
[27] Buslaev, V.S; Sulem, C.; “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 419–475.
[28] Calvo, D.C.; Akylas, T.R.; “On the formation of bound states by interacting nonlocal solitary waves”, Physica D 101 (1997), 270–288.
[29] Carretero-Gonzáles, R.; Talley, J.D.; Chong, C.; Malomed, B.A.; “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation”, Physica D 216 (2006), 77–89.
[30] Casenawe, T.; Semilinear Schrödinger Equations, Courant Lecture Notes (Courant Institute, New York, 2003).
[31] Champneys, A.R.; Yang, J.; “A scalar nonlocal bifurcation of solitary waves for coupled nonlinear Schrödinger systems”, Nonlinearity 15 (2002), 2165–2192.
[32] Chang, S.M.; Gustafson, S.; Nakanishi, K.; Tsai, T.P.; “Spectra of linearized operators for NLS solitary waves”, SIAM J. Math. Anal. 39 (2007), 1070–1111.
[33] Chugunova, M.; Pelinovsky, D.; “Count of unstable eigenvalues in the generalized eigenvalue problem”, J. Math. Phys. 51 (2010), 052901 (19 pp.).
[34] Chugunova, M.; Pelinovsky, D.; “Two-pulse solutions in the fifth-order KdV equation: rigorous theory and numerical approximations”, Discr. Cont. Dyn. Syst. B 8 (2007), 773–800.
[35] Chugunova, M.; Pelinovsky, D.; “Block-diagonalization of the symmetric first-order coupled-mode system”, SIAM J. Appl. Dyn. Syst. 5 (2006), 66–83.
[36] Chugunova, M.; Pelinovsky, D.; “Spectrum of a non-self-adjoint operator associated with the periodic heat equation”, J. Math. Anal. Appl. 342 (2008), 970–988.
[37] Chugunova, M.; Pelinovsky, D.; “On quadratic eigenvalue problems arising in stability of discrete vortices”, Lin. Alg. Appl. 431 (2009), 962–973.
[38] Colin, Th.; “Rigorous derivation of the nonlinear Schrödinger equation and Davey–Stewartston systems with quadratic hyperbolic systems”, Asymptot. Anal. 31 (2002), 69–91.
[39] Colin, M.; Lannes, D.; “Short pulse approximations in dispersive media”, SIAM J. Math. Anal. 41 (2009), 708–732.
[40] Comech, A.; Pelinovsky, D.; “Purely nonlinear instability of standing waves with minimal energy”, Commun. Pure Appl. Math. 56 (2003), 1565–1607.
[41] Cuccagna, S.; “On asymptotic stability in energy space of ground states of NLS in 1D”, J. Diff. Eqs. 245 (2008), 653–691.
[42] Cuccagna, S.; Pelinovsky, D.; “Bifurcations from the end points of the essential spectrum in the linearized NLS problem”, J. Math. Phys. 46 (2005) 053520 (15 pp.).
[43] Cuccagna, S; Pelinovsky, D.; Vougalter, V.; “Spectra of positive and negative energies in the linearized NLS problem”, Commun. Pure Appl. Math. 58 (2005), 1–29.
[44] Cuccagna, S.; Tarulli, M.; “On asymptotic stability of standing waves of discrete Schrödinger equation in ℤ”, SIAM J. Math. Anal. 41 (2009), 861–885.
[45] Demanet, L.; Schlag, W.; “Numerical verification of a gap condition for a linearized NLS equation”, Nonlinearity 19 (2006), 829–852.
[46] Derks, G.; Gottwald, G.A.; “A robust numerical method to study oscillatory instability of gap solitary waves”, SIAM J. Appl. Dyn. Syst. 4 (2005), 140–158.
[47] Dimassi, M.; Sjöstrand, J.; Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Notes 268 (Cambridge University Press, Cambridge, 1999).
[48] Dmitriev, S.V.; Kevrekidis, P.G.; Yoshikawa, N.; “Discrete Klein–Gordon models with static kinks free of the Peierls–Nabarro potential”, J. Phys. A: Math. Gen. 38 (2005), 7617–7627.
[49] Dmitriev, S.V.; Kevrekidis, P.G.; Sukhorukov, A.A.; Yoshikawa, N.; Takeno, S.; “Discrete nonlinear Schrödinger equations free of the Peierls–Nabarro potential”, Phys. Lett. A 356 (2006), 324–332.
[50] Dohnal, D.; Pelinovsky, D.; Schneider, G.; “Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential”, J. Nonlin. Sci. 19 (2009), 95–131.
[51] Dohnal, D.; Uecker, H.; “Coupled-mode equations and gap solitons for the 2D Gross–Pitaevskii equation with a non-separable periodic potential”, Physica D 238 (2009), 860–879.
[52] Eastham, M.S.; The Spectral Theory of Periodic Differential Equations (Scottish Academic Press, Edinburgh, 1973).
[53] Eckmann, J.-P.; Schneider, G.; “Nonlinear stability of modulated fronts for the Swift–Hohenberg equation”, Comm. Math. Phys. 225 (2002), 361–397.
[54] Esteban, M.J.; Séré, É.; “Stationary states of the nonlinear Dirac equation: a variational approach”, Comm. Math. Phys. 171 (1995), 323–350.
[55] Evans, L.C.; Partial Differential Equations (AMS, Providence, RI, 1998).
[56] Feckan, M.; Topological Degree Approach to Bifurcation Problems (Springer, Heidelberg, 2008).
[58] Floer, A.; Weinstein, A.; “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential”, J. Funct. Anal. 69 (1986), 397–408.
[59] Floquet, G.; “Sur les équations Différentielles linéaires à coefficients périodique”, Ann. École Norm. Sup. 12 (1883), 47–88.
[60] Friesecke, G.; Pego, R.L.; “Solitary waves on FPU lattices”, Nonlinearity 12 (1999), 1601–1627; 15 (2002), 1343–1359; 17 (2004), 207–227; 17 (2004), 229–251.
[61] Gagnon, L.; Winternitz, P.; “Symmetry classes of variable coefficients nonlinear Schrödinger equations”, J. Phys. A: Math. Gen. 26 (1993), 7061–7076.
[62] Gang, Z.; Sigal, I.M.; “Asymptotic stability of nonlinear Schrödinger equations with potential”, Rev. Math. Phys. 17 (2005), 1143–1207.
[63] Gang, Z.; Sigal, I.M.; “Relaxation of solitons in nonlinear Schrödinger equations with potential”, Adv. Math. 216 (2007), 443–490.
[64] García-Ripoll, J.J.; Pérez-García, V.M.; “Optimizing Schrödinger functionals using Sobolev gradients: applications to quantum mechanics and nonlinear optics”, SIAM J. Sci. Comput. 23 (2001), 1315–1333.
[65] Georgieva, A.; Kriecherbauer, T.; Venakides, S.; “Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium”, SIAM J. Appl. Math. 60 (1999), 272–294.
[66] Giannoulis, J.; Mielke, A.; “The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities”, Nonlinearity 17 (2004), 551–565.
[67] Giannoulis, J.; Mielke, A.; Sparber, C.; “Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential”, J. Diff. Eqs. 245 (2008), 939–963.
[68] Giannoulis, J.; Herrmann, M.; Mielke, A.; “Continuum descriptions for the dynamics in discrete lattices: derivation and Justification”, in Analysis, Modeling and Simulation of Multiscale Problems (Springer, Berlin, 2006), pp. 435–466.
[69] Glazman, I.M.; Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Israel Program for Scienti.c Translation, Jerusalem, 1965).
[70] Gohberg, I.G.; Krein, M.G.; Introduction to the Theory of Linear Non-self-adjoint Operators, Translations of Mathematical Monographs 18 (AMS, Providence, RI, 1969).
[71] Golubitsky, M.; Schaeffer, D.G.; Singularities and Groups in Bifurcation Theory, Volume 1 (Springer, Berlin, 1985).
[72] Goodman, R.H.; Weinstein, M.I.; Holmes, P.J.; “Nonlinear propagation of light in one-dimensional periodic structures”, J. Nonlin. Sci. 11 (2001), 123–168.
[73] Gorshkov, K.A.; Ostrovsky, L.A.; “Interactions of solitons in nonintegrable systems: Direct perturbation method and applications”, Physica D 3 (1981), 428–438.
[74] Grillakis, M.; Shatah, J.; Strauss, W.; “Stability theory of solitary waves in the presence of symmetry. I”, J. Funct. Anal. 74 (1987), 160–197.
[75] Grillakis, M.; Shatah, J.; Strauss, W.; “Stability theory of solitary waves in the presence of symmetry. II”, J. Funct. Anal. 94 (1990), 308–348.
[76] Grillakis, M.; “Linearized instability for nonlinear Schrödinger and Klein–Gordon equations”, Commun. Pure Appl. Math. 41 (1988), 747–774.
[77] Grillakis, M.; “Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system”, Commun. Pure Appl. Math. 43 (1990), 299–333.
[78] Grimshaw, R.; “Weakly nonlocal solitary waves in a singularly perturbed nonlinear Schrödinger equation”, Stud. Appl. Math. 94 (1995), 257–270.
[79] Groves, M.D.; “Solitary-wave solutions to a class of fifth-order model equations”, Nonlinearity 11 (1998), 341–353.
[80] Groves, M.D.; Schneider, G.; “Modulating pulse solutions for a class of nonlinear wave equations”, Comm. Math. Phys. 219 (2001), 489–522.
[81] Groves, M.D.; Schneider, G.; “Modulating pulse solutions for quasilinear wave equations”, J. Diff. Eqs. 219 (2005), 221–258.
[82] Groves, M.D.; Schneider, G.; “Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales”, Comm. Math. Phys. 278 (2008), 567–625.
[83] Gross, E.P.; “Hydrodynamics of a superfluid concentrate”, J. Math. Phys. 4 (1963), 195–207.
[84] Gurski, K.F.; Kollar, R.; Pego, R.L.; “Slow damping of internal waves in a stably stratified fluid”, Proc. R. Soc. Lond. 460 (2004), 977–994.
[85] Haragus, M.; Kapitula, T.; “On the spectra of periodic waves for infinite-dimensional Hamiltonian systems”, Physica D 237 (2008), 2649–2671.
[86] Helffer, B.; Semi-classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics 1336 (Springer, New York, 1988).
[87] Heinz, H.P.; Küpper, T.; Stuart, C.A.; “Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation”, J. Diff. Eqs. 100 (1992), 341–354.
[88] Hernández-Heredero, R.; Levi, D.; “The discrete nonlinear Schrödinger equation and its Lie symmetry reductions”, J. Nonlin. Math. Phys. 10 (2003), 77–94.
[89] Hislop, P.D.; Sigal, I.M.; Introduction to Spectral Theory with Applications to Schrödinger Operators (Springer, New York, 1996).
[90] Ilan, B.; Weinstein, M. I.; “Band-edge solitons, nonlinear Schrödinger/Gross–Pitaevskii equations and effective media”, Multiscale Model. Simul. 8 (2010), 1055–1101.
[91] Iohvidov, I.S.; Krein, M.G.; Langer, H.; Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric (Mathematische Forschung, Berlin, 1982).
[92] Iooss, G.; “Travelling waves in the Fermi–Pasta–Ulam lattice”, Nonlinearity 13 (2000), 849–866.
[93] Iooss, G.; Adelmeyer, M.; Topics in Bifurcation Theory and Applications (World Scientific, Singapore, 1998).
[94] Iooss, G.; Kirchgassner, K.; “Travelling waves in a chain of coupled nonlinear oscillators”, Comm. Math. Phys. 211 (2000), 439–464.
[95] Iooss, G.; Pelinovsky, D.; “Normal form for travelling kinks in discrete Klein–Gordon lattices”, Physica D 216 (2006), 327–345.
[96] James, G.; Sire, Y.; “Travelling breathers with exponentially small tails in a chain of nonlinear oscillators”, Comm. Math. Phys. 257 (2005), 51–85.
[97] James, G.; Sire, Y.; “Numerical computation of travelling breathers in Klein–Gordon chains”, Physica D 204 (2005), 15–40.
[98] JeanJean, L.; Tanaka, K.; “A remark on least energy solutions in ℝN”, Proc. Amer. Math. Soc. 131 (2002), 2399–2408.
[99] Jones, C.K.R.T.; “An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation”, J. Diff. Eqs. 71 (1988), 34–62.
[100] Jones, C.K.R.T.; “Instability of standing waves for nonlinear Schrödinger-type equations”, Ergod. Theor. Dynam. Sys. 8 (1988), 119–138.
[101] Kapitula, T.; “Stability of waves in perturbed Hamiltonian systems”, Physica D 156 (2001), 186–200.
[102] Kapitula, T.; Kevrekidis, P.G.; “Stability of waves in discrete systems”, Nonlinearity 14 (2001), 533–566.
[103] Kapitula, T.; Kevrekidis, P.; “Bose–Einstein condensates in the presence of a magnetic trap and optical lattice: two-mode approximation”, Nonlinearity 18 (2005), 2491–2512.
[104] Kapitula, T.; Kevrekidis, P.; Chen, Z.; “Three is a crowd: solitary waves in photorefractive media with three potential wells”, SIAM J. Appl. Dyn. Syst. 5 (2006), 598–633.
[105] Kapitula, T.; Kevrekidis, P.; Sandstede, B.; “Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems”, Physica D 195 (2004), 263–282; Addendum: Physica D 201 (2005), 199–201.
[106] Kapitula, T.; Law, K.J.H.; Kevrekidis, P.G.; “Interaction of excited states in twospecies Bose–Einstein condensates: a case study”, SIAM J. Appl. Dyn. Syst. 9 (2010), 34–61.
[107] Kapitula, T.; Sandstede, B.; “Edge bifurcations for near integrable systems via Evans function techniques”, SIAM J. Math. Anal. 33 (2002), 1117–1143.
[108] Kato, T.; Perturbation Theory for Linear Operators (Springer, New York, 1976).
[109] Kaup, D.J.; “Perturbation theory for solitons in optical fibers”, Phys. Rev. A 42 (1990), 5689–5694.
[110] Kevrekidis, P.G.; The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, Springer Tracts in Modern Physics 232 (Springer, New York, 2009).
[111] Kevrekidis, P.G.; “On a class of discretizations of Hamiltonian nonlinear partial differential equations”, Physica D 183 (2003), 68–86.
[112] Kevrekidis, P.G.; Pelinovsky, D.E.; Stefanov, A.; “Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation in one dimension”, SIAM J. Math. Anal. 41 (2009), 2010–2030.
[113] Kirrmann, P.; Schneider, G.; Mielke, A.; “The validity of modulation equations for extended systems with cubic nonlinearities”, Proc. Roy. Soc. Edinburgh A 122 (1992), 85–91.
[114] Khare, A.; Dmitriev, S.V.; Saxena, A.; “Exact moving and stationary solutions of a generalized discrete nonlinear Schrödinger equation”, J. Phys. A: Math. Gen. 40 (2007), 11301–11317.
[115] Klaus, M.; Pelinovsky, D.; Rothos, V.M.; “Evans function for Lax operators with algebraically decaying potentials”, J. Nonlin. Sci. 16 (2006), 1–44.
[116] Klaus, M.; Shaw, K.; “On the eigenvalues of Zakharov–Shabat systems”, SIAM J. Math. Anal. 34 (2003), 759–773.
[117] Kohn, W.; “Analytic properties of Bloch waves and Wannier functions”, Phys. Rev. 115 (1959), 809–821.
[118] Kollar, R.; Pego, R.L.; “Spectral stability of vortices in two-dimensional Bose–Einstein condensates via the Evans function and Krein signature”, Appl. Math. Res. Express 2011 (2011), in press (46 pp.).
[119] Kollar, R.; “Homotopy method for nonlinear eigenvalue pencils with applications”, SIAM J. Math. Anal. 43 (2011), 612–633.
[120] Koukouloyannis, V.; Kevrekidis, P.G.; “On the stability of multibreathers in Klein–Gordon chains”, Nonlinearity 22 (2009), 2269–2285.
[121] Kominis, Y.; Hizanidis, K.; “Power dependent soliton location and stability in complex photonic structures”, Opt. Express 16 (2008), 12124–12138.
[122] Krieger, J.; Schlag, W.; “Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension”, J. Amer. Math. Soc. 19 (2006), 815–920.
[123] Kruskal, M.D.; Segur, H.; “Asymptotics beyond all orders in a model of crystal growth”, Stud. Appl. Math. 85 (1991), 129–181.
[124] Küpper, T.; Stuart, C.A.; “Necessary and sufficient conditions for gap-bifurcation”, Nonlin. Anal. 18 (1992), 893–903.
[125] Lakoba, T.; Yang, J.; “A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity”, J. Comput. Phys. 226 (2007), 1668–1692.
[126] Lakoba, T.I.; Yang, J.; “A mode elimination technique to improve convergence of iteration methods for finding solitary waves”, J. Comput. Phys. 226 (2007), 1693–1709.
[127] Lakoba, T.I.; “Conjugate gradient method for finding fundamental solitary waves”, Physica D 238 (2009), 2308–2330.
[128] Lafortune, S.; Lega, J.; “Spectral stability of local deformations of an elastic rod: Hamiltonian formalism”, SIAM J. Math. Anal. 36 (2005), 1726–1741.
[129] Lannes, D.; “Dispersive effects for nonlinear geometrical optics with rectification”, Asymptot. Anal. 18 (1998), 111–146.
[130] Lieb, E.H.; Loss, M.; Analysis, 2nd edn (AMS, Providence, RI, 2001).
[131] Lieb, E.H.; Seiringer, R.; Solovej, J.P.; Yngvason, J.; The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars 34 (Birkhäuser, Basel, 2005).
[132] Lin, Z.; “Instability of nonlinear dispersive solitary waves”, J. Funct. Anal. 255 (2008), 1191–1224.
[133] Linares, F.; Ponce, G.; Introduction to Nonlinear Dispersive Equations (Springer, New York, 2009).
[134] Lions, P.-L.; “The concentration compactness principle in the calculus of variations. The locally compact case”, Ann. Inst. Henri Poincaré 1 (1984), 223–283.
[135] Lukas, M.; Pelinovsky, D.E.; Kevrekidis, P.G.; “Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices”, Physica D 237 (2008), 339–350.
[136] MacKay, R.S.; Aubry, S.; “Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators”, Nonlinearity 7 (1994), 1623–1643.
[137] MacKay, R.S.; “Slow manifolds”, in Energy Localization and Transfer, eds. T., DauxoisA., Litvak-Hinenzon, R.S., MacKay, and A., Spanoudaki (World Scientific, Singapore, 2004), pp. 149–192.
[138] Mallet-Paret, J.; “The Fredholm alternative for functional differential equations of mixed type”, J. Dyn. Diff. Eqs. 11 (1999), 1–47.
[139] Melvin, T.R.O.; Champneys, A.R.; Kevrekidis, P.G.; Cuevas, J.; “Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: existence, stability and dynamics”, Physica D 237 (2008), 551–567.
[140] Melvin, T.R.O.; Champneys, A.R.; Pelinovsky, D.E.; “Discrete travelling solitons in the Salerno model”, SIAM J. Appl. Dyn. Syst. 8 (2009), 689–709.
[141] Mizumachi, T.; “Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential”, J. Math. Kyoto Univ. 48 (2008), 471–497.
[142] Morgante, A.M.; Johansson, M.; Kopidakis, G.; Aubry, S.; “Standing wave instabilities in a chain of nonlinear coupled oscillators”, Physica D 162 (2002), 53–94.
[143] Newell, A.C.; Moloney, J.V.; Nonlinear Optics (Westview Press, Boulder, CO, 2003).
[144] Olver, P.J.; Applications of Lie Groups to Differential Equations (Springer, New York, 1993).
[145] Oxtoby, O.F.; Barashenkov, I.V.; “Moving solitons in the discrete nonlinear Schödinger equation”, Phys. Rev. E 76 (2007), 036603 (18 pp.).
[146] Oxtoby, O.F.; Pelinovsky, D.E.; Barashenkov, I.V.; “Travelling kinks in discrete phi-4 models”, Nonlinearity 19 (2006), 217–235.
[147] Pankov, A.; Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lattices (Imperial College Press, London, 2005).
[148] Pankov, A.; “Periodic nonlinear Schrödinger equation with application to photonic crystals”, Milan J. Math. 73 (2005), 259–287.
[149] Pankov, A.; “Gap solitons in periodic discrete nonlinear Schrödinger equations”, Nonlinearity 19 (2006), 27–40.
[150] Pankov, A.; “Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach”, Discr. Cont. Dyn. Syst. 19 (2007), 419–430.
[151] Pego, R.L.; Warchall, H.A.; “Spectrally stable encapsulated vortices for nonlinear Schrödinger equations”, J. Nonlin. Sci. 12 (2002), 347–394.
[152] Pelinovsky, D.E.; “Asymptotic reductions of the Gross–Pitaevskii equation”, in Emergent Nonlinear Phenomena in Bose–Einstein Condensates eds. P.G., KevrekidisD.J., Franzeskakis, and R., Carretero-Gonzalez (Springer, New York, 2008), pp. 377–398.
[153] Pelinovsky, D.E.; “Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations”, Proc. Roy. Soc. Lond. A, 461 (2005), 783–812.
[154] Pelinovsky, D.E.; “Translationally invariant nonlinear Schrödinger lattices”, Nonlinearity 19 (2006), 2695–2716.
[155] Pelinovsky, D.E.; Kevrekidis, P.G.; Frantzeskakis, D.J.; “Stability of discrete solitons in nonlinear Schrödinger lattices”, Physica D 212 (2005), 1–19.
[156] Pelinovsky, D.E.; Kevrekidis, P.G.; Frantzeskakis, D.J.; “Persistence and stability of discrete vortices in nonlinear Schrödinger lattices”, Physica D 212 (2005), 20–53.
[157] Pelinovsky, D.E.; Melvin, T.R.O.; Champneys, A.R.; “One-parameter localized traveling waves in nonlinear Schrödinger lattices”, Physica D 236 (2007), 22–43.
[158] Pelinovsky, D.E.: Rothos, V.M.; “Bifurcations of travelling breathers in the discrete NLS equations”, Physica D 202 (2005), 16–36.
[159] Pelinovsky, D.; Sakovich, A.; “Internal modes of discrete solitons near the anti–continuum limit of the dNLS equation”, Physica D 240 (2011), 265–281.
[160] Pelinovsky, D.E.; Stepanyants, Yu.A.; “Convergence of Petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations”, SIAM J. Numer. Anal. 42 (2004), 1110–1127.
[161] Pelinovsky, D.E.; Sukhorukov, A.A.; Kivshar, Yu.S.; “Bifurcations and stability of gap solitons in periodic potentials”, Phys. Rev. E 70 (2004), 036618 (17 pp.).
[162] Pelinovsky, D.; Schneider, G.; “Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential”, Applic. Anal. 86 (2007), 1017–1036.
[163] Pelinovsky, D.; Schneider, G.; “Moving gap solitons in periodic potentials”, Math. Meth. Appl. Sci. 31 (2008), 1739–1760.
[164] Pelinovsky, D.; Schneider, G.; “Bounds on the tight-binding approximation for the Gross–Pitaevskii equation with a periodic potential”, J. Diff. Eqs. 248 (2010), 837–849.
[165] Pelinovsky, D.; Schneider, G.; MacKay, R.; “Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential”, Comm. Math. Phys. 284 (2008), 803–831.
[166] Pelinovsky, D.E.; Yang, J.; “Instabilities of multihump vector solitons in coupled nonlinear Schrödinger equations”, Stud. Appl. Math. 115 (2005), 109–137.
[167] Perelman, G.; “On the formation of singularities in solutions of the critical nonlinear Schrödinger equation”, Ann. Henri Poincaré 2 (2001), 605–673.
[168] Petviashvili, V.I.; “Equation of an extraordinary soliton”, Sov. J. Plasma Phys. 2 (1976), 257–258.
[169] Pillet, C.A.; Wayne, C.E.; “Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations”, J. Diff. Eqs. 141 (1997), 310–326.
[170] Pitaevskii, L.P.; “Vortex lines in an imperfect Bose gas”, Sov. Phys. JETP 13 (1961), 451–454.
[171] Pitaevskii, L.; Stringari, S.; Bose–Einstein Condensation (Oxford University Press, Oxford, 2003).
[172] Pomeau, Y.; Ramani, A.; Grammaticos, B.; “Structural stability of the Korteweg–de Vries solitons under a singular perturbation”, Physica D 31 (1988), 127–134.
[173] Pontryagin, L.S.; “Hermitian operators in spaces with indefinite metric”, Izv. Akad. Nauk SSSR Ser. Mat. 8 (1944), 243–280.
[174] Porter, M.A.; Chugunova, M.; Pelinovsky, D.E.; “Feshbach resonance management of Bose–Einstein condensates in optical lattices”, Phys. Rev. E 74 (2006), 036610 (8 pp.).
[175] Qin, W.-X.; Xiao, X.; “Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices”, Nonlinearity 20 (2007), 2305–2317.
[176] Reed, M.; Simon, B.; Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press, New York, 1978).
[177] Salerno, M.; “Quantum deformations of the discrete nonlinear Schrödinger equation”, Phys. Rev. A 46 (1992), 6856–6859.
[178] Sánchez, A.; Bishop, A.R.; “Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations”, SIAM Rev. 40 (1998), 579–615.
[179] Sandstede, B.; “Stability of multiple-pulse solutions”, Trans. Amer. Math. Soc. 350 (1998), 429–472.
[180] Scheel, A.; Van Vleck, E.S.; “Lattice differential equations embedded into reaction–Diffusion systems”, Proc. Roy. Soc. Edinburgh A 139 (2009), 193–207.
[181] Schlag, W.; “Stable manifolds for an orbitally unstable NLS”, Ann. Math. 169 (2009), 139–227.
[182] Schneider, G.; “Validity and limitation of the Newell–Whitehead equation”, Math. Nachr. 176 (1995), 249–263.
[183] Schneider, G.; “The validity of generalized Ginzburg–Landau equations”, Math. Meth. Appl. Sci. 19 (1996), 717–736.
[184] Schneider, G.; “Justification of modulated equations for hyperbolic systems via normal forms”, Nonlin. Diff. Eqs. Appl. 5 (1995), 69–82.
[185] Schneider, G.; Uecker, H.; “Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems”, Asymptot. Anal. 28 (2001), 163–180.
[186] Schneider, G.; Wayne, C.E.; “The long-wave limit for the water wave problem. I. The case of zero surface tension”, Commun. Pure Appl. Math. 53 (2000), 1475–1535.
[187] Schneider, G.; Wayne, C.E.; “The rigorous approximation of long-wavelength capillary–gravity waves”, Arch. Ration. Mech. Anal. 162 (2002), 247–285.
[188] Serkin, V.N.; Hasegawa, A.; Belyaeva, T.L.; “Nonautonomous solitons in external potentials”, Phys. Rev. Lett. 98 (2007), 074102 (4 pp.).
[189] Shatah, J.; Strauss, W.; “Instability of nonlinear bound states”, Comm. Math. Phys. 100 (1985), 173–190.
[190] Shi, H.; Zhang, H.; “Existence of gap solitons in periodic discrete nonlinear Schrödinger equations”, J. Math. Anal. Appl. 361 (2010), 411–419.
[191] Sivan, Y.; Fibich, G.; Efremidis, N.K.; Bar-Ad, S.; “Analytic theory of narrow lattice solitons”, Nonlinearity 21 (2008), 509–536.
[192] Skorobogatiy, M.; Yang, J.; Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, 2009).
[193] Soffer, A.; Weinstein, M.I.; “Multichannel nonlinear scattering theory for nonintegrable equations”, Comm. Math. Phys. 133 (1990), 119–146.
[194] Soffer, A.; Weinstein, M.I.; “Multichannel nonlinear scattering theory for nonintegrable equations II: The case of anisotropic potentials and data”, J. Diff. Eqs. 98 (1992), 376–390.
[195] Soffer, A.; Weinstein, M.I.; “Selection of the ground state for nonlinear Schrödinger equations”, Rev. Math. Phys. 16 (2004), 977–1071.
[196] Speight, J.M.; “Topological discrete kinks”, Nonlinearity 12 (1999), 1373–1387.
[197] de Sterke, C.M.; Sipe, J.E.; “Gap solitons”, Prog. Opt. 33 (1994), 203–259.
[198] Stuart, C.A.; “Bifurcations into spectral gaps”, Bull. Belg. Math. Soc. Simon Stevin, 1995, suppl. (59 pp.).
[199] Sulem, C.; Sulem, P.L.; The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse (Springer, New York, 1999).
[200] Sylvester, J.J.; “A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares”, Philos. Mag. IV (1852), 138–142.
[201] Tao, T.; Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106 (AMS, Providence, RI, 2006).
[202] Tao, T.; “Why are solitons stable?”, Bull. Amer. Math. Soc. 46 (2009), 1–33.
[203] Tovbis, A.; Tsuchiya, M.; Jaffe, C.; “Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Henon map as an example”, Chaos 8 (1998), 665–681.
[204] Tovbis, A.; “On approximation of stable and unstable manifolds and the Stokes phenomenon”, Contemp. Math. 255 (2000), 199–228.
[205] Tovbis, A.; “Breaking homoclinic connections for a singularly perturbed differential equation and the Stokes phenomenon”, Stud. Appl. Math. 104 (2000), 353–386.
[206] Tovbis, A.; Pelinovsky, D.; “Exact conditions for existence of homoclinic orbits in the fifth-order KdV model”, Nonlinearity 19 (2006), 2277–2312.
[207] Vakhitov, M.G.; Kolokolov, A.A.; “Stationary solutions of the wave equation in a medium with nonlinearity saturation”, Radiophys. Quantum Electron. 16 (1973), 783–789.
[208] Vougalter, V.; “On the negative index theorem for the linearized nonlinear Schrödinger problem”, Canad. Math. Bull. 53 (2010), 737–745.
[209] Vougalter, V.; Pelinovsky, D.; “Eigenvalues of zero energy in the linearized NLS problem”, J. Math. Phys. 47 (2006), 062701 (13 pp.).
[210] Weder, R.; “The Wk,p-continuity of the Schrödinger wave operators on the line”, Comm. Math. Phys. 208 (1999), 507–520.
[211] Weinstein, M.I.; “Liapunov stability of ground states of nonlinear dispersive evolution equations”, Commun. Pure Appl. Math. 39 (1986), 51–68.
[212] Weinstein, M.; “Excitation thresholds for nonlinear localized modes on lattices”, Nonlinearity 12 (1999), 673–691.
[213] Yang, J.; “Classification of the solitary waves in coupled nonlinear Schrödinger equations”, Physica D 108 (1997), 92–112.
[214] Yang, J.; Akylas, T.R.; “Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation”, Stud. Appl. Math. 111 (2003), 359–375.
[215] Yang, J.; Pelinovsky, D.E.; “Stable vortex and dipole vector solitons in a saturable nonlinear medium”, Phys. Rev. E 67 (2003), 016608 (12 pp.).
[216] Yang, J.; Lakoba, T.; “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations”, Stud. Appl. Math. 118 (2007), 153–197.
[217] Yang, J.; Lakoba, T.; “Accelerated imaginary-time evolution methods for computations of solitary waves”, Stud. Appl. Math. 120 (2008), 265–292.
[218] Yau, H.T.; Tsai, T.P.; “Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and radiation dominated solutions”, Commun. Pure Appl. Math. 55 (2002), 1–64.
[219] Yau, H.T.; Tsai, T.P.; “Stable directions for excited states of nonlinear Schrödinger equations”, Comm. Part. Diff. Eqs. 27 (2002), 2363–2402.
[220] Yau, H.T.; Tsai, T.P.; “Relaxation of excited states in nonlinear Schrödinger equations”, Int. Math. Res. Not. 2002 (2002), 1629–1673.
[221] Zelik, S.; Mielke, A.; “Multi-pulse evolution and space-time chaos in dissipative systems”, Mem. Amer. Math. Soc. 198 (2009), 1–97.
[222] Zhang, G.; “Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials”, J. Math. Phys. 50 (2009), 013505 (10 pp.).
[223] Zhou, Z.; Yu, J.; Chen, Y.; “On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity”, Nonlinearity 23 (2010), 1727–1740.