Local structure of singular profiles for a Derivative Nonlinear Schrödinger Equation

Abstract

The Derivative Nonlinear Schrödinger equation is an $L^2$-critical nonlinear dispersive equation model for Alfvén waves in space plasmas. Recent numerical studies on an $L^2$-supercritical extension of this equation provide evidence of finite time singularities. Near the singular point, the solution is described by a universal profile that solves a nonlinear elliptic eigenvalue problem depending only on the strength of the nonlinearity. In the present work, we describe the deformation of the profile and its parameters near criticality, combining asymptotic analysis and numerical simulations.

34 pages, 14 figures

References in corpus (3)

• Well-posedness for a generalized derivative nonlinear Schrödinger equation
• Global Existence for the Derivative Nonlinear Schrodinger Equation by the Method of Inverse Scattering
• Existence of global solutions to the derivative NLS equation with the inverse scattering transform method

Cited by in corpus (5)

• Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation
• 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
• The derivative NLS equation: global existence with solitons
• Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in $H^{\frac{1}{2}}$