Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

doi:10.1007/s11464-020-0857-6

Front. Math. China

2020, Vol. 15

Issue (5) : 923-957 https://doi.org/10.1007/s11464-020-0857-6

RESEARCH ARTICLE

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

Ze LI(

)

School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

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Abstract

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.

Keywords Nonlinear Schrödinger equation (NLS) solitons weak nonlinearity asymptotic stability

Corresponding Author(s): Ze LI

Issue Date: 19 November 2020

Cite this article:

Ze LI. Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case[J]. Front. Math. China, 2020, 15(5): 923-957.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0857-6
https://academic.hep.com.cn/fmc/EN/Y2020/V15/I5/923

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