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Global Behavior of Solutions to Generalized Gross-Pitaevskii Equation

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Abstract

This paper is concerned with time global behavior of solutions to nonlinear Schrödinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solution approaches to the non-vanishing state, then the equation admits a global solution which scatters to the non-vanishing element for both time directions.

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Notes

  1. The property that \(|f| \leqslant |g|\) a.e. implies \(\left\Vert f\right\Vert \leqslant \left\Vert g\right\Vert\).

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Authors and Affiliations

  1. Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka, Toyonaka, 560-8531, Japan

    Satoshi Masaki

  2. Teacher Training Courses, Faculty of Education, Kagawa University, Kagawa, Takamatsu, 760-8522, Japan

    Hayato Miyazaki

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Masaki, S., Miyazaki, H. Global Behavior of Solutions to Generalized Gross-Pitaevskii Equation. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00609-8

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