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

Research output: Contribution to journalArticle

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
Externally publishedYes

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Orbital Stability
Initial value problems
Standing Wave
Initial Value Problem
Fourth Order
Torus
Periodic Wave
Vortex Filament
Energy Method
Periodic Boundary Conditions
Global Solution
Refining
Vortex flow
Boundary conditions
Motion
Operator
Energy

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "We consider the fourth order nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger 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 Schr{\"o}dinger type equation, stability of standing waves.",
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