一类离散广义非线性 Schröd inger系统周期解的存在性

谭伟明,覃学文

(梧州学院数理系,中国 梧州 543002)

一类离散广义非线性 Schröd inger系统周期解的存在性

谭伟明,覃学文＊①

(梧州学院数理系,中国 梧州 543002)

把一些文献讨论的离散广义非线性 Schrödinger方程推广到了n维空间,应用临界点理论,得到了一类离散广义非线性 Schrödinger系统存在多个非零周期解的充分条件.

Schrödinger系统 ;临界点 ;周期解

非线性 Schrödinger方程是现代物理和数学理论研究中的一个基本方程,对这个方程的研究在推动现代物理和数学的发展起着非常重要的作用.从数学家的观点来看,非线性 Schrödinger方程也具有极大的吸引力,数学家们对这个方程关注和探讨的问题也是多方面的.近十多年来,许多学者对非线性 Schrödinger方程及其应用作了深入的研究,取得了一定的成果[1-9].

文 [4]从离散非线性 Schrödinger方程

1 预备知识和变分结构

2 主要结果

下面证明系统(6)还有其它的非零T-周期解,为此只需证明泛函J(X)在ET上还有其它的非零临界点.应用山路引理证明.

可证明泛函J满足环绕定理的条件.由于已证明泛函J满足 P-S条件,因此只需证明泛函J满足环绕定理的条件(1)和(2).

即转化为(1)的情形,由上述讨论可知,泛函 -J至少有 2个临界点,从而泛函J至少有 2个非零临界点,于是系统(6)至少存在 2个非零T-周期解.

[1]BEREZ IN F A,SHUB IN M A.The Schrödinger equation[M].Netherlands:KluwerAcademic Publishers,2002.

[2]ABLOW ITZM J,PR INAR IB,TRUBATCH A D.Diserete and continuous nonlinear Schrödinger systems[M].New York:Cambridge University Press,2004.

[3]MULLER-KI RSTEN H JW.Introduction to quantum mechanics:Schrödinger equation and path integral[M].Singapore:World Scientific Printers(S)Pte Ltd,2006.

[4]PANKOV A.Gap solitons in periodic discrete nonlinear Schrödinger equation[J].Nonlinearty,2006,19:27-40.

[5]ALEXANDRU D IONESCU,CARLOS E KEN IG.Uniqueness properties of solutions of Schrödinger equations[J].Journal of FunctionalAnalysis,2006,232:90-136.

[6]ALOMAR IA K,NOORAN IM SM,NAZAR R.Explicit series solutionsof some linear and nonlinear Schrödinger equations via the homotopy analysismethod[J].Communications in Nonlinear Science and Numerical Simulation,2009,14:1 196-1 207.

[7]G IOVANNA CERAM I,R ICCARDO MOLLE.Positive solutions for some Schrödinger equations having partially periodic potentials[J].Journal ofMathematicalAnalysis and Applications,2009,359:15-27.

[8]NAKAO HAYASH I,PAVEL INAUMK IN.Asymptotics of odd solutionsof quadratic nonlinear Schrödinger equations[J].Journal ofMathematicalAnalysis and Applications,2009,359:146-158.

[9]KOPYLOVA E A.On the asymptotic stability of solitarywaves in the discrete Schrödinger equation coupled to a nonlinear oscillator[J].NonlinearAnalysis,2009,71:3 031-3 046.

Existence Periodic Solution of the Generalized D iscrete Nonlinear Schrödinger System

TAN W ei-m ing,Q IN Xue-wen
(School ofMathematics and Physics,Wuzhou University,Wuzhou 543002,China)

The generalized discrete nonlinear Schrödinger equation discussed in some literature are extented ton-d imensional space.Using critical point theory,some sufficient conditions are obtained for the existence periodic solution of the generalized discrete nonlinear Schrödinger system.

Schrödinger systems;critical point theory;periodic solution

O413.1

A

1000-2537(2010)04-0046-07

2010-05-17

梧州学院科研基金资助项目 (2009B012);广西教育厅科研基金资助项目(2008MS121);广西自然科学基金资助项目 (桂科自 0991279)

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