Relations between properties of solutions and geometric symmetry of solutions for nonlinear wave and dispersive equations

研究结果如下：(1)证明了调和映象热流、Landau-Lifshitz方程和薛定谔映象的涡旋的稳定性和不稳定性。(2)证明了具有幂非线性项的非线性薛定谔柯西问题解的无条件唯一性。(3)得到了Lugiato-Lefever方程的定常解分支的线性稳定性和线性不稳定性的结果，该方程是具有阻尼和强迫的非线性薛定谔方程。

The research results are as follows.(1) We proved the stability and instability of vortices for the harmonic map heat-flows, the Landau-Lifshitz equations and the Schrodinger maps.(2) We proved the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger with powernonlinearity.(3) We obtained the results about the linear stability and the linear instability of stationary solution bifurcating from the constant stationary solution for the Lugiato-Lefever equation, which is the nonlinear Schrodinger equation with damping and forcing.