ON SOLUTIONS OF NONLINEAR DISPERSIVE EQUATIONS

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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.