ON SOLUTIONS OF NONLINEAR DISPERSIVE EQUATIONS

1.P.I.Naumkin和我研究了具有立方非线性项的一维非线性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地依赖于数据的非线性和角度部分的结构。E.I.Kaiina和我研究了各种耗散非线性方程在正半直线上解的渐近行为。我们利用了解在边界上的反射比完全问题有更好的时间衰减的事实。P.I.Naumkin和我研究了具有二次非线性的二维非线性Schredinger方程的渐近和散射问题，其中涉及未知函数的导数。在这项工作中，我们使用一些解析函数空间来获得充分的时间衰减解，以获得期望的结果。P.I.Naumkin，H.Uchida和我研究了椭圆-双曲Davey-Stwertson系统，当数据在无穷远处指数衰减时，证明了解的解析光滑性。我们利用了非线性满足的规范不变量和与薛定谔算符可交换的算符。P.I.Naumkin和我研究了具有二次非线性项的一维非线性Schredinger方程解的渐近行为。我们使用标准形方法将方程转化为具有三次非线性的方程，并应用了(1)中所述的方法。E.I.Kaiina，P.I.Naumkin和我研究了具有临界非线性的复Landau-Ginzburg方程，利用解的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.