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Notes on gap solitons for periodic discrete nonlinear Schrödinger equations
Author(s) -
Ding Liang,
Wei Jinlong
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5183
Subject(s) - mathematics , nonlinear system , class (philosophy) , schrödinger's cat , type (biology) , variational principle , function (biology) , variational method , mathematical analysis , mathematical physics , quantum mechanics , physics , ecology , artificial intelligence , evolutionary biology , computer science , biology
Consider a class of periodic discrete Schrödinger equations L u n − ω u n = σ f nu n, n ∈ Z . Under the condition of f nu n= χ nu n2 u n , by applying the linking theorem, Pankov obtained the existence of gap solitons. Pankov's result was then generalized to various type of nonlinear function f n , but most of studying works were concentrated on the case of f n ( t ) = O ( | t |s 1) as t →0 ( s 1 ⩾ 1 ) . In this paper, by ingeniously using Ekeland variational principle, for the nonlinearity which meets f n ( t ) = O ( | t |s 1) as t →0 ( s 1 ∈ (0,1)), we obtain the existence of gap solitons, which supplements the existing ones and gives a positive answer in part to the open problem proposed by Pankov.