Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$

Nobu Kishimoto; Kiyeon Lee

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arXiv · 2508.08647

Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$

Nobu Kishimoto·Kiyeon Lee

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Abstract

In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t \to \infty$.

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Nobu Kishimoto, Kiyeon Lee. 2025-08-12. Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$. https://arxiv.org/abs/2508.08647

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