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Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials

Masaki Kawamoto, Hayato Miyazaki

Abstract

This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22].

Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials

Abstract

This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22].
Paper Structure (23 sections, 209 equations)

This paper contains 23 sections, 209 equations.

Table of Contents

  1. Introduction
  2. Setting and Backgounrd
  3. Main results
  4. Strategy on the proof of main results
  5. Strategy on the proof of Theorem \ref{['thm:1']}
  6. Strategy on the proof of Theorem \ref{['thm:iv2']}
  7. Organization of this paper
  8. Preliminaries
  9. Some tools
  10. Persistence of regularity of solutions
  11. Nonlinear estimates
  12. Construction of the modified wave operators
  13. Estimates for the external term
  14. Estimation for leading terms
  15. Proof of Theorem \ref{['thm:1']}
  16. ...and 8 more sections

Theorems & Definitions (21)

  • proof : Proof of Lemma \ref{['M-6:L1']}
  • proof : Proof of Lemma \ref{['M-6:P1']}
  • proof : Proof of Proposition \ref{['pro:ext1']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 11 more