RESEARCH ARTICLE

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

  • Ze LI
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  • School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

Received date: 09 Jun 2020

Accepted date: 10 Aug 2020

Published date: 15 Oct 2020

Copyright

2020 Higher Education Press

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.

Cite this article

Ze LI . Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case[J]. Frontiers of Mathematics in China, 2020 , 15(5) : 923 -957 . DOI: 10.1007/s11464-020-0857-6

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