四阶非线性Schrödinger方程解的爆破图景

This project is devoted to blow-up solutions of the fourth-order nonlinear Schr?dinger equation, which models the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. By searching the inner relationship between the fourth-order nonlinear Schr?dinger equation and its corresponding nonlinear elliptic equation, proper functionals and constrained variational problems are constructed, and these variational problems and their variational characteristic are solved by the profile decomposition theory. The spectral properties of the linearized equation to the fourth-order nonlinear Schr?dinger equation is studied. Basing on the variational characteristic, the solution of the fourth-order nonlinear Schr?dinger equation is directly decomposed, and the existence of blow-up solutions is studied by the profile decomposition theory. Moreover, by applying the decomposition of solutions, spectral properties of the linearized equation and variational characteristic of the ground state to the fourth-order nonlinear Schr?dinger equation, dynamic behaviors of blow-up solutions of the fourth-order nonlinear Schr?dinger equation are studied, including limiting profile of minimal-mass blow-up solutions, sharp blow-up rate,the distribution and topological structure of blow-up points, mass concentration and rate of mass concentration etc.

本项目拟研究四阶非线性Schr?dinger方程的爆破解, 这类方程描述了强激光束通过具有Kerr非线性效应的大体积介质的传播. 我们探寻四阶非线性Schr?dinger方程与对应椭圆方程的内在联系, 构造恰当的泛函和约束变分问题, 利用Profile分解理论求解上述变分问题与相应变分特征, 并讨论四阶非线性Schr?dinger方程线性化方程对应算子的谱性质. 然后, 以基态变分特征为依托, 利用Profile分解理论对四阶非线性Schr?dinger方程的解进行分解, 讨论其爆破解的存在性. 进而综合利用四阶非线性Schr?dinger方程解的分解式、线性算子的谱性质以及基态变分特征, 讨论其爆破解的动力学性质, 包括最小质量爆破解的极限行为、最佳爆破速率、爆破点集的分布及其拓扑结构、质量集中性质以及集中速率等.

本项目研究了四阶非线性Schrodinger 方程和两类带无界势的非线性Schrodinger 方程：带导数项的非线性Schrodinger 方程和非奇次非线性Schrodinger方程。我们以Cauchy问题的适定性为基础，得到了上述方程的轨道稳定性和爆破解动力学性质。对于四阶非线性Schrodinger 方程，利用Profile 分解理论对其爆破解进行了以孤立为主成份的线性分解，讨论了其轨道稳定性的最佳条件。对于两类带无界势的非线性Schrodinger 方程，我们构造了多个Profile分解引理，得到了方程爆破解的爆破速率、集中性质以及极小质量爆破解的极限行为。在该项目执行过程中，我们共形成论文5篇，其中2篇已发表并被SCI收录。