Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. (English) Zbl 1178.35351

Summary: We discuss how to use critical point theory to study the existence of gap solitons for periodic discrete nonlinear Schrödinger equations. An open problem proposed by Professor Alexander Pankov is solved [A. Pankov, Nonlinearity 19, No. 1, 27–40 (2006; Zbl 1220.35163)].


35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
39A12 Discrete version of topics in analysis


Zbl 1220.35163
Full Text: DOI


[1] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations, and inverse scattering, (1991), Cambridge Univ. Press Cambridge · Zbl 0762.35001
[2] Aceves, A.B., Optical gap solutions: past, present, and future; theory and experiments, Chaos, 10, 584-589, (2000) · Zbl 0971.78014
[3] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037
[4] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[5] Bartsh, T.; Ding, Y.H., On a nonlinear Schrödinger equation with periodic potential, Math. ann., 313, 15-37, (1999)
[6] Bronski, J.C.; Segev, M.; Weinstein, M.I., Mathematical frontiers in optical solitons, Proc. natl. acad. sci. USA, 98, 12872-12873, (2001)
[7] de Sterke, C.M.; Sipe, J.E., Gap solitons, Progr. opt., 33, 203-260, (1994)
[8] Ding, Y.H.; Luan, S.X., Multiple solutions for a class of nonlinear Schrödinger equations, J. differential equations, 207, 423-457, (2004) · Zbl 1072.35166
[9] Flash, S.; Willis, C.R., Discrete breathers, Phys. rep., 295, 181-264, (1998)
[10] Gorbach, A.; Jonasson, M., Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. phys. J. D, 29, 77-93, (2004)
[11] Kevreides, P.G.; Rasmussen, K.Ø.; Bishop, A.R., The discrete nonlinear Schrödinger equation: A survey of recent results, Internat. J. modern phys. B, 15, 2883-2900, (2001)
[12] Landau, L.D.; Lifshitz, E.M., Quantum mechanics, (1979), Pergamon New York · Zbl 0081.22207
[13] Mills, D.L., Nonlinear optics, (1998), Springer Berlin · Zbl 0914.00011
[14] Pankov, A., Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19, 27-40, (2006) · Zbl 1220.35163
[15] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, (1986), Amer. Math. Soc. Providence, RI, New York · Zbl 0609.58002
[16] Sukhorukov, A.A.; Kivshar, Y.S., Generation and stability of discrete gap solitons, Optim. lett., 28, 2345-2348, (2003)
[17] Teschl, G., Jacobi operators and completely integrable nonlinear lattices, (2000), Amer. Math. Soc. Providence, RI, New York · Zbl 1056.39029
[18] Wang, M.L.; Zhou, Y.B., The periodic wave solutions for the klein – gordor – schrödinger equations, Phys. lett. A, 318, 84-92, (2003) · Zbl 1098.81770
[19] Willem, M.; Zou, W., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana univ. math. J., 52, 109-132, (2003) · Zbl 1030.35068
[20] Wu, X.F., Solitary wave and periodic wave solutions for the quintic discrete nonlinear Schrödinger equation, Chaos solitons fractals, 40, 1240-1248, (2009) · Zbl 1197.81130
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