On the Cauchy problem for the derivative nonlinear Schroedinger equation with periodic boundary condition

Abstract

It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm.

22 pages

Cited by in corpus (22)

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• Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation
• Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
• Orbital stability of solitary waves for derivative nonlinear Schrödinger equation
• A priori estimates for the derivative nonlinear Schrödinger equation
• Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation
• On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$
• On the stability of periodic waves for the cubic derivative NLS and the quintic NLS
• Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations
• Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle
• Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle
• A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below $H^{1/2}$
• Lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation
• Self-similar solutions to the derivative nonlinear Schrödinger equation
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with low regularity periodic initial data
• Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations
• Invariant Cantor manifolds of quasi-periodic solutions for the derivative nonlinear Schrodinger equation
• Well-posedness in energy space for the periodic modified Benjamin-Ono equation
• Ill-Posedness of the Third Order NLS Equation with Raman Scattering Term