分数阶非线性Schrödinger方程的爆破动力学

In this project, we study the Cauchy problem of the fractional nonlinear Schrödinger equation, which is one of the fundament models of the fractional quantum mechanics. Firstly, constructing various constrained variational problems and homogeneous variational problems, we get the variational characteristic of solitons of ground states for the fractional nonlinear Schrödinger equation by the profile decomposition theory. Secondly, basing on the symmetric invariance of the fractional nonlinear Schrödinger equation, we use the solitons of the ground states and the solutions of the fractional linear Schrödinger equation as basic elements to give the nonlinear profile decomposition of solutions for the above Cauchy problem by the profile decomposition theory. Thus, we use the nonlinear profile decomposition to study the existence of blow-up solutions, and the dynamic behaviors of blow-up solutions, including blow-up rate, strong concentration and weak concentration, the distribution and topological structure of blow-up points and limiting profile of blow-up solutions etc. Finally, using the profile decomposition theory, we search the criteria of the existence of stable standing waves for the fractional nonlinear Schrödinger equation involving critical nonlinearities, and the global dynamics of global solutions including asymptotic stability etc.

本项目研究分数阶非线性Schrödinger方程的Cauchy问题, 它是分数阶量子力学的基础数学模型. 首先, 构造多种齐次变分问题和约束变分问题, 利用Profile分解理论求解上述变分问题, 讨论分数阶非线性Schrödinger方程对应各种基态孤立子的变分特征. 其次, 根据方程的对称不变性, 利用Profile分解理论对分数阶非线性Schrödinger方程的解做以基态孤立子和线性方程的解为基本元素的非线性图景分解与展开. 探索其爆破解存在的充分条件以及爆破解的爆破速率、强集中与弱集中现象、爆破点的空间分布与拓扑结构、极限图景等动力学性质. 最后, 利用Profile分解理论研究具有临界指数幂分数阶非线性Schrödinger方程驻波稳定性的判别条件、渐近稳定等动力学性质.

研究了带质量临界及质量超临界的分数阶非线性Schrodinger方程、Davey-Stewartson系统、带反调和势的非线性Schrodinger方程以及浅水波方程的 Cauchy 问题。构造多种齐次变分问题和约束变分问题, 利用Profile分解理论求解上述变分问题, 讨论上述非线性波动系统对应各种基态孤立子的变分特征。得到了系统爆破解存在的充分条件以及爆破解的爆破速率、爆破解的集中现象以及极限图景、驻波的轨道稳定性与强不稳定性、解散射与爆破的判别准则等动力学性质。在项目执行过程中，我们已发表研究论文12篇，其中9篇被SCI收录。同时，培养硕士研究生5名。