Ill-Posedness of the Third Order NLS Equation with Raman Scattering Term

Abstract

We consider the ill-posedness and well-posedness of the Cauchy problem for the third order NLS equation with Raman scattering term on the one dimensional torus. It is regarded as a mathematical model for the photonic crystal fiber oscillator. Regarding the ill-posedness, we show the nonexistence of solutions in the Sobolev space and the norm inflation of the data-solution map under slightly different conditions, respectively. We also prove the local unique existence of solutions in the analytic function space.

38 pages. The main results have been improved. This version has been accepted for publication in Mathematical Research Letters

References in corpus (4)

• On the Cauchy problem for the derivative nonlinear Schroedinger equation with periodic boundary condition
• Nonlinear wave interactions for the Benjamin-Ono equation
• Ill-posedness issues for nonlinear dispersive equations
• Nonuniqueness of weak solutions of the nonlinear Schroedinger equation