Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

Abstract

The main result of the paper concerns the existence of nontrivial exponentially decaying solutions to periodic stationary discrete nonlinear Schrödinger equations with saturable nonlinearities, provided that zero belongs to a spectral gap of the linear part. The proof is based on the critical point theory in combination with periodic approximations of solutions. As a preliminary step, we prove also the existence of nontrivial periodic solutions with arbitrarily large periods.

Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

“…Only recently, due to implementation of variational techniques, the study of standing waves for non-translation invariant DNLS has been started [6][7][8][9][10][11]. More precisely, periodic DNLS, with power-like and saturable nonlinearities, is studied in [7][8][9], while papers [10,11] are devoted to the case of infinitely growing potential V ¼ {v n } and power-like nonlinearity. In all these results the nonlinearity is supposed to be either positive (self-focusing), or negative (defocusing) for all n 2 N.…”

Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities

“…If ω is below or above the spectrum of the difference operator L, the existence and nonexistence of solutions ware considered by many authors [7,15,18,32]. If ω lies in a finite gap (then the associated energy functional is strongly indefinite, thus it is much more difficult to obtain the existence results, see [8] for discussions on strongly indefinite problem), there are some authors who have obtained the existence or multiplicity of gap solitons of periodic DNLS equations with superlinear [5,16,17,26] or asymptotically linear [6,19,21,22,26,33] nonlinearities at infinity. For related problems, we refer the reader to [3,[9][10][11]25].…”

“…In the cases where (V 1 ) holds, and the nonlinearities g n (s) are asymptotically linear as |s| → ∞, there are some papers [6,19,21,22,26,33] concerning the existence of nontrivial solutions of (1.1).…”

“…(2) In [19], the author assumed that (V 1 ), (G 1 ) and (G 2 ) hold, g n (s) − V n s is bounded, G n (s) ≥ 0, ∀(n, s) ∈ Z × R, inf |s|≥r 0 H n (s) > 0 for any r 0 > 0 and V > inf σ (L + 1 ) − ω ( V := min n∈Z {V n }), and obtained a nontrivial gap soliton for (1.1) in l 2 . It is not hard to see that our Theorem 1.1 improves the result in [19]. We give an example here in which g n (s) − V n s is unbounded.…”

Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential

“…For example, Pankov and Zakharchenko used a variational method known as Nehari manifolds and a discrete version of the Lions concentration-compactness principle in [1] to establish existence results of nontrivial standing wave solutions for discrete nonlinear Schrödinger equation. Lately in [2], Pankov and Rothos employed Nehari manifolds approach and the Mountain Pass argument to demonstrate the existence of solutions in the discrete nonlinear Schrödinger equation with saturable nonlinearity. While for discrete Hamiltonian systems, Yu and Guo established a variational structure and introduced variational technique to the study of periodic solutions in [3][4][5].…”