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<p>In this paper, we study the large-time behavior of small solutions to the standard form of the systems of 1D cubic nonlinear Schrödinger equations consisting of two components and possessing a coercive mass-like conserved quantity. The cubic nonlinearity is known to be critical in one space dimension in view of the large-time behavior. By employing the result by Katayama-Sakoda (Partial Differ Equ Appl 1, 3(Paper No. 12):41 (2020). <a href="https://doi.org/10.1007/s42985-020-00012-4">https://doi.org/10.1007/s42985-020-00012-4</a>), one can obtain the large-time behavior of the solution if we can integrate the corresponding ODE system. We introduce an integration scheme suited to the system. The key idea is to rewrite the ODE system, which is cubic, as a quadratic system of quadratic quantities of the original unknown. By using this technique, we described the large-time behavior of solutions in terms of elementary functions and the Jacobi elliptic functions for several examples of standard systems.</p>

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Partial classification of the large-time behavior of solutions to cubic nonlinear Schrödinger systems

  • 期刊论文

  • 发表日期：

    2025年6月7日

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  • Satoshi Masaki

摘要

In this paper, we study the large-time behavior of small solutions to the standard form of the systems of 1D cubic nonlinear Schrödinger equations consisting of two components and possessing a coercive mass-like conserved quantity. The cubic nonlinearity is known to be critical in one space dimension in view of the large-time behavior. By employing the result by Katayama-Sakoda (Partial Differ Equ Appl 1, 3(Paper No. 12):41 (2020). https://doi.org/10.1007/s42985-020-00012-4), one can obtain the large-time behavior of the solution if we can integrate the corresponding ODE system. We introduce an integration scheme suited to the system. The key idea is to rewrite the ODE system, which is cubic, as a quadratic system of quadratic quantities of the original unknown. By using this technique, we described the large-time behavior of solutions in terms of elementary functions and the Jacobi elliptic functions for several examples of standard systems.