On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear schrödinger equation

Nakao Hayashi, Pavel Naumkin

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

We study the asymptotic behavior for large time of small solutions to the Cauchy problem for the modified Benjamin-Ono equation: ut + (u3)x + Huxx = 0, where H is the Hilbert transformation, x,t ∈ R. We investigate the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Shrödinger equation and then apply techniques developed in [11] - [14] to the resulting cubic nonlocal nonlinear Schrödinger equation. Our method is simpler than that used in [10] because we can use the factorization of the free Schrödinger group. Our purpose in this paper is to show that solutions have the same L time decay rate as in the corresponding linear Benjamin-Ono equation and to prove the existence of modified scattering states, when the initial data are sufficiently small in the weighted Sobolev spaces H2,0 ∩ H1,1, where Hm,s = {φ ∈ S': ||φ||m,s = ||(1 + x2)s/2(1 - δx2)m/2φL2 < ∞}, m,s ∈ R. This is an improvement of the previous result, where we considered small initial data from the space H3,0 ∩ H1,2. Our method is based on a certain gauge transformation and an appropriate phase function.

Original languageEnglish
Pages (from-to)237-255
Number of pages19
JournalDiscrete and Continuous Dynamical Systems
Volume8
Issue number1
DOIs
Publication statusPublished - 2002 Jan
Externally publishedYes

Keywords

  • Asymptotic behavior of solutions
  • Modified B-O equation
  • Modified scattering states

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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