Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

Abstract

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity {i∂t(u-∂x2u)+∂x2u-a∂x4u=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where a>15,λ∈ R. We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the L2-boundedness of the pseudodifferential operators.

Original languageEnglish
Article number2
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume73
Issue number1
DOIs
Publication statusPublished - 2022 Feb

Keywords

  • Cubic nonlinear Schrödinger equation
  • Decay estimates
  • Modified scattering
  • Nonlocal Schrödinger equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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