Well-posedness for a generalized derivative nonlinear Schrödinger equation

Abstract

We study the Cauchy problem for a generalized derivative nonlinear Schrödinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space $H^1$.

v2: minor revision; 21 pages. The assumption of Theorem 1.7 is weakened slightly. To appear in Journal of Differential Equations

References in corpus (1)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data

Cited by in corpus (19)

• A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
• A note on the nonlinear Schrödinger equation in a general domain
• Small data well-posedness for derivative nonlinear Schrödinger equations
• Local structure of singular profiles for a Derivative Nonlinear Schrödinger Equation
• Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
• Potential well theory for the derivative nonlinear Schrödinger equation
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Orbital stability of solitary waves for generalized derivative nonlinear Schrödinger equations in the endpoint case
• Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case
• Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case
• On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$
• Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations
• Stability of the sum of two solitary waves for (gDNLS) in the energy space
• Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case
• Lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation
• On The Cauchy Problem For A Derivative Nonlinear Schr{ö}Dinger Equation With Nonvanishing Boundary Conditions
• On Uniqueness for Schrödinger maps with low regularity large data
• Construction of the multi-soliton trains for a generalized derivative nonlinear Schr\''odinger equations by a fixed point method
• Instability of degenerate solitons for nonlinear Schrödinger equations with derivative