非线性Schrodinger方程孤立子的动力学特征

In this project, we study the Cauchy problems for the Gross-Pitaveskii equation and Davey-Stewarston system. We investigate the solitons of the nonlinear Schr?dinger equations with and without potentials derived from the above Cauchy problems, and research the variational structures and properties of the minimal energy solutions for the related nonlinear elliptic equation by the variational methods. In terms of the related nonlinear elliptic equations, we construct a series of constrained variational problems and cross-constrained variational problems. Then, comprehensively using the various modern compactness techniques, we solve these variational problems and analyze their variational characteristics. Combining the well-posedness of the Cauchy problems with various symmetric invariances(conservation laws), and applying harmonic analysis, spectral analysis and the above variational characteristic, we decompose (express) properly the global solutions and blow-up solutions of the Cauchy problems in terms of the solitons. According to these expressions, we realize the elaborate descriptions of asymptotic behavior of the global solutions in various scaled spaces, and the whole profile of the blow-ip solutions. Thus, we can obtain the dynamical properties of global solutions and blow-up solutions, and reveal the solition dynamics of the evolutional systems.

研究Gross-Pitaveskii方程与Davey-Stewarston系统的Cauchy 问题.探究由它们导出的各类带势的与不带势的非线性Schr?dinger 方程的孤立子, 利用变分法研究与之相关的非线性椭圆方程最小能量解的变分结构与性质. 针对相关的非线性椭圆方程, 构造一系列的强制变分问题、交叉强制变分问题. 综合利用各种现代紧性技术分析求解这些变分问题及其变分特征. 结合Cauchy 问题的适定性及各种对称不变性(守恒律), 运用调和分析、谱分析及上述变分特征, 以孤立子为主成份实现对Cauchy 问题整体解与爆破解的恰当分解与展开. 通过展开式, 实现对整体解在多种尺度空间中渐近行为的精细刻画, 以及爆破解爆破图景的全景描述. 从而得到整体解与爆破解的动力学性质, 揭示各种孤立子在发展系统中的动力学特征.

研究了带质量临界及能量临界非线性幂的Gross-Pitaveskii方程、分数阶非线性Schrodinger方程、Schrodinger-Poisson-Slater 方程、Davey-Stewartson系统、带非齐次非线性项Schrodinger方程以及带随机项的非线性波动系统的 Cauchy 问题。利用变分法研究与之相关的非线性椭圆方程最小能量解的变分结构与性质。针对相关的非线性椭圆方程, 构造一系列的变分问题, 求解其变分特征。结合 Cauchy 问题的适定性及各种对称不变性, 运用调和分析、谱分析及上述变分特征, 实现上述系统爆破解爆破图景的描述以及爆破动力学的研究。在项目执行过程中，我们已发表研究论文28篇，其中23篇被SCI收录。同时，培养博士研究生3名。