Long time asymptotic behavior of the focusing nonlinear Schrödinger equation

Michael Borghese, Robert Jenkins, Kenneth D.T.R. McLaughlin

Research output: Contribution to journalArticlepeer-review

66 Scopus citations


We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the ∂‾ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution ψ(x,t) in any fixed space-time cone C(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt with x0∈[x1,x2],v∈[v1,v2]} up to an (optimal) residual error of order O(t−3/4). In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton–soliton and soliton–radiation interactions as one moves through the cone. Our results require that the initial data possess one L2(R) moment and (weak) derivative and that it not generate any spectral singularities.

Original languageEnglish (US)
Pages (from-to)887-920
Number of pages34
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Issue number4
StatePublished - Jul 2018


  • Focusing
  • Integrable systems
  • Long time asymptotics
  • Nonlinear Schrödinger
  • Riemann–Hilbert
  • Soliton resolution

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics


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