Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves

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Abstract

We consider the fourth order nonlinear Schrödinger type equation (4NLS) which arises in context of the motion of vortex filament. The purposes of this paper are twofold. Firstly, we consider the initial value problem for (4NLS) under the periodic boundary condition. By refining the modified energy method used in our previous paper [23], we prove the unique existence of the global solution for (4NLS) in the energy space H2per (0,2L) with L > 0. Secondly, we study the stability property of periodic standing waves for (4NLS). Using the spectrum properties of the Schrödinger operators associated to the periodic standing wave developed by Angulo [1], we prove that standing wave of dnoidal type is orbitally stable under the time evolution by (4NLS). Fourth order nonlinear Schrödinger type equation, stability of standing waves.

Original languageEnglish
Pages (from-to)843-859
Number of pages17
JournalCommunications on Pure and Applied Analysis
Volume14
Issue number3
DOIs
Publication statusPublished - May 1 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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