Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

2024/07/25 by Kondo, Toshiki, Okamoto, Mamoru
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.17782

Abstract

We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity u^k ∂_x u, where k is a natural number. We prove the norm inflation in a subspace of the Sobolev space H^s(\mathbbT) for any s ∈ ℝ. In particular, the Cauchy problem is ill-posed in H^s(\mathbbT) for any s ∈ ℝ.

Related

• A remark on norm inflation for nonlinear Schrödinger equations
• Norm inflation for the derivative nonlinear Schrödinger equation
• On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle
• Ill-posedness of quintic fourth order Schrödinger equation
• Norm inflation for the viscous nonlinear wave equation