具有有界非周期位势的离散非线性薛定谔方程的同宿解

Discrete nonlinear Schrodinger (DNLS) equation is one of the most basic difference equation models in the field of nonlinear science. It can effectively describe discrete soliton phenomena in many branches of the natural sciences, such as the Bose-Einstein condensate (experimental implementation in 1995 and Nobel Prize in Physics in 2001). The existence problem of discrete solitons in the DNLS equations can be transformed into the existence problem of homoclinic solutions of the equations. The existing existence results of the non-trivial homoclinic solutions of the DNLS equations are mainly concentrated in the case of periodic or unbounded potentials, while the situation of bounded and non-periodic potentials remains to be solved. Based on the previous works on homoclinic solutions of the DNLS equations, by using the variational methods, we will study the existence and multiple solutions of homoclinic solutions of the DNLS equations with bounded and non-periodic potentials and characterize the behavior of homoclinic solutions. The research of this project can not only develop and improve the variational theory of difference equations, but also apply the results to the study of discrete solitons in the DNLS equations, which has important theoretical significance and application prospects.

离散非线性薛定谔（DNLS）方程是非线性科学领域的最基本的差分方程模型之一，可以有效地描述自然科学中很多分支领域出现的离散孤子现象，如Bose-Einstein凝聚态(1995年实验实现，2001年获诺贝尔物理学奖)。DNLS方程离散孤子的存在性问题可以转化为该方程同宿解的存在性问题。现有的关于DNLS方程非平凡同宿解的存在性结果主要集中在具有周期位势或者无界位势的情况，而对于具有有界非周期位势的情况仍有待解决。本项目将在前期DNLS方程同宿解研究工作的基础上，运用变分方法研究具有有界非周期位势的DNLS方程同宿解的存在性与多解性，并刻画同宿解的性态。本项目的研究不仅可以发展和完善差分方程的变分理论，而且能够将所得结果应用于DNLS方程离散孤子的研究，具有重要的理论意义和应用前景。