Dynamics of solutions for nonlinear dispersive equation

利用长波模型的单调性前提，研究了长波模型孤立波的稳定性。证明了KP-II方程中线孤子的调制可以用Burgers方程组来描述，并证明了线孤子解的稳定性。证明了Fermi-Pasta-Ulam晶格的多孤立波在KdV极限下的稳定性，以及类似于Boussinesq系统的双向模型Benney-Luke方程的孤立波解。在与Tzvetkov的合作中，我们利用Miura变换证明了L^2(R_x×T_y)中KP-II方程的线孤子解的稳定性与零解的稳定性等价，并得到了L^2(R_x×T_y)中线孤子的稳定性。在与佩利诺夫斯基的联合工作中，这个想法被转移到了一维立方体NLS。

I study stability of solitary waves for long wave models using their monotonicity preoperty. I show that modulations of the line soliton for the KP-II equation can be described a system of the Burgers equation and prove the stability of line soliton solution. I also prove stability of multi solitary waves of the Fermi-Pasta-Ulam lattices in the KdV limit and solitary wave solutions of the Benney-Luke equation which is a kind of bidirectional models akin to the Boussinesq systems. In a joint work with Tzvetkov, we show that stability of line soliton solutions for the KP-II equation in L^2(R_x×T_y) is equivalent to the stability of the null solution by using the Miura transformation and obtain L^2(R_x×T_y) stability of line solitons. The idea was tranfered to 1D cubic NLS in a joint work with Pelinovsky.