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Modified Scattering for the One-Dimensional Schrödinger Equation with a Subcritical Dissipative Nonlinearity

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Abstract

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity \(\lambda |u|^\alpha u\), where \(0<\alpha <2\), and \(\lambda \) is a complex constant satisfying \(\text {Im} \lambda >\frac{\alpha |\text {Re} \lambda |}{2\sqrt{ \alpha +1}}\). For arbitrary large initial data, we present the uniform time decay estimates when \(4/3\le \alpha <2\), and the large time asymptotics of the solution when \(\frac{7+\sqrt{145}}{12}<\alpha <2\). The proof is based on the vector fields method and a semiclassical analysis method.

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Funding

This work is partially supported by National Natural Science Foundation of China 11931010, and Zhejiang Provincial Natural Science Foundation of China LDQ23A010001.

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  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310058, China

    Xuan Liu & Ting Zhang

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  1. Xuan Liu
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  2. Ting Zhang
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Contributions

XL completed the proof of the main theorem and wrote the manuscript; TZ introduced the question studied in this paper with constructive discussions.

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Correspondence to Ting Zhang.

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Liu, X., Zhang, T. Modified Scattering for the One-Dimensional Schrödinger Equation with a Subcritical Dissipative Nonlinearity. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10272-4

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