Nonexistence of small, smooth, time-periodic, spatially periodic solutions for nonlinear Schrödinger equations

Ambrose, David; Wright, J.

doi:10.1090/qam/1519

Authors: David M. Ambrose and J. Douglas Wright
Journal: Quart. Appl. Math. 77 (2019), 579-590
MSC (2010): Primary 35Q41; Secondary 35B10, 35A01
DOI: https://doi.org/10.1090/qam/1519
Published electronically: September 6, 2018
MathSciNet review: 3962583
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Abstract: We study the question of nonexistence of small spatially periodic, time-periodic solutions for cubic nonlinear Schrödinger equations. We prove that in certain regions of the period-amplitude plane, time-periodic solutions do not exist. To be more precise, we prove that for almost any value in a bounded set of possible temporal periods, there is an amplitude threshold, below which any initial value is not the initial value for a time-periodic solution. The proof requires a certain level of Sobolev regularity on solutions. The methods used are not based on any special structure of the nonlinear Schrödinger equation, and can be applied more generally.

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