A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

Abstract

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if ${\Vert u_0\Vert }_{L^2}^2<4\pi$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schrödinger equation", Analysis & PDE 8 (2015), no. 5, 1101--1112. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_0$ satisfies that ${\Vert u_0\Vert }_{L^2}^2=4\pi$ and the momentum $P(u_0)$ is negative.

arXiv commentTo appear in Analysis & PDE. We changed the title. Namely, this paper is a revised version of "Global Well-Posedness on a generalized derivative nonlinear Schrödinger equation"

References in corpus (11)

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• Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$
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• Multi-solitary waves for the nonlinear Klein-Gordon equation
• Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation
• Orbital stability of solitary waves for derivative nonlinear Schrödinger equation
• Stability of the traveling waves for the derivative Schrödinger equation in the energy space
• Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case
• Stability of the sum of two solitary waves for (gDNLS) in the energy space
• Solitary waves for nonlinear Schrödinger equation with derivative

Cited by in corpus (18)

• Modulational instability of periodic standing waves in the derivative NLS equation
• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• Global Existence for the Derivative Nonlinear Schrödinger Equation with Arbitrary Spectral Singularities
• Traveling waves for a nonlinear Schrödinger system with quadratic interaction
• Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation
• Potential well theory for the derivative nonlinear Schrödinger equation
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
• Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case
• Stability of the sum of two solitary waves for (gDNLS) in the energy space
• Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations
• Solitary waves for nonlinear Schrödinger equation with derivative
• Global Well-Posesedness for the Derivative Nonlinear Schrodinger Equation
• The stability of degenerate solitons for derivative nonlinear Schrodinger equations
• The derivative NLS equation: global existence with solitons
• Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case
• Instability of degenerate solitons for nonlinear Schrödinger equations with derivative
• Remarks on blow-up criteria for the derivative nonlinear Schrodinger equation under the optimal threshold setting