Gauge transformation for the kinetic derivative nonlinear Schrödinger equation on the torus

2023/03/30 by Kishimoto, Nobu, Tsutsumi, Yoshio · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.17359

Abstract

We consider the kinetic derivative nonlinear Schrödinger equation, which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. In our previous work, we proved small-data global well-posedness of the Cauchy problem on the torus in Sobolev space H^s for s>1/2 by combining the Fourier restriction norm method with the parabolic smoothing effect, which is available in the periodic setting. In this article, we improve the regularity range to s>1/4 for the global well-posedness by constructing an effective gauge transformation. Moreover, we remove the smallness assumption by making use of the dissipative nature of the equation.

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• Global solution and asymptotic behavior for the kinetic derivative NLS on \mathbb R
• Local and global well-posedness for the kinetic derivative NLS on ℝ

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