Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

  • Pavel Naumkin

    Universidad Michoacana, Morelia, Mexico
  • Nakao Hayashi

    Tokyo University of Science, Japan
  • Hidetake Uchida

    Tokyo University of Science, Japan

Abstract

We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form

where

, ; , as , , are real valued functions. Here the parameters , and are such that and . If and , , equation (A) appears in the classical pseudospin magnet model [9]. We prove that if and the norm is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate , where , . Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.

Cite this article

Pavel Naumkin, Nakao Hayashi, Hidetake Uchida, Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, pp. 501–513

DOI 10.2977/PRIMS/1195143611