ON SOLUTIONS OF NONLINEAR DISPERSIVE EQUATIONS

1. P.I. Naumkin和我研究了具有三次非线性的一维非线性Schredinger方程解的渐近行为。在这个项目中，我们证明了解的性质强烈依赖于非线性结构和数据的角部分。E.I. Kaikina和我研究了各种耗散非线性方程解在正半线上的渐近行为。我们利用了这样一个事实，即解在边界处通过反射有更好的时间衰减。P.I. Naumkin和我研究了包含未知函数导数的二次非线性二维Schredinger方程的渐近和散射问题。在此工作中，我们利用一些解析函数空间来得到解的足够的时间衰减，从而得到期望的结果。P.I. Naumkin， H.Uchida和我研究了椭圆-双曲Davey-Stwertson系统，并展示了当数据在无穷远处呈指数衰减时解的解析平滑性质。我们利用了非线性满足规范不变量和与Schredinger算子交换的算子。P.I. Naumkin和我研究了具有二次非线性的一维非线性Schredinger方程解的渐近行为。我们使用范式方法将方程转化为另一个具有三次非线性的方程，并应用(1)中所述的方法。E.I. Kaikina， P.I. Naumkin和我研究了具有临界非线性的复朗道-金兹堡方程，并利用解的第三次逼近和非线性变换证明了解的渐近性。

1. P.I. Naumkin and I studied asymptotic behavior of solutions to one dimensional nonlinear Schredinger equations with cubic nonlinearities. In this project we showed properties of solutions strongly depend on structures of nonlinearities and angular part of the data.2. E.I. Kaikina and I studied asymptotic behavior of solutions to various dissipative nonlinear equations on the positive half line. We made use of the facts that solutions have better time decay by refrection at the boundary compared to the problem in the full.3. P.I. Naumkin and I studied asymptotic and scattering problems of two dimensional nonlinear Schredinger equations with quadratic nonlinearities involving derivatives of the unknown function. In this work we used some analytic function space to get a sufficient time decay of solutions to get the desired result.4. P.I. Naumkin, H.Uchida and I studied the elliptic-hyperbolic Davey-Stwertson system and showed an analytical smoothing property of solutions when the data decay exponentially at infinity. We made use of the nonlinearity satisfies the gauge invariant and the operator which commute with the Schredinger operator.5. P.I. Naumkin and I studied asymptotic behavior of solutions to one dimensional nonlinear Schredinger equations with quadratic nonlinearities. We used the normal form method to translate the equation to another one which-has cubic nonlinearities and applied the previous method stated in (1).6. E.I. Kaikina, P.I. Naumkin and I studied complex Landau-Ginzburg equations with critical nonliaearities and showed asymptotics of solutions by using third approximation of solutions and nonlinear transformation.