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arXiv · 2412.00413

Global existence and large-time behavior of solutions to cubic nonlinear Schr\"odinger systems without coercive conserved quantity

Satoshi Masaki
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Abstract

In this article, we investigate the large-time behavior of small solutions to a system of one-dimensional cubic nonlinear Schr\"odinger equations with two components. In previous studies, a structural condition on the nonlinearity has been employed to guarantee the existence of a coercive, mass-type conserved quantity. We identify a new class of systems that do not satisfy such a condition and thus lack a coercive conserved quantity. Nonetheless, we establish the global existence and describe the large-time behavior for small solutions in this class. In this setting, the asymptotic profile is described in terms of solutions to the corresponding system of ordinary differential equations (ODEs). A key element of our analysis is the use of a quartic conserved quantity associated with the ODE system. Moreover, for a specific example within this class, we solve the ODE system explicitly, showing that the asymptotic behavior is expressed using Jacobi elliptic functions.

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Satoshi Masaki. 2024-11-30. Global existence and large-time behavior of solutions to cubic nonlinear Schr\"odinger systems without coercive conserved quantity. https://arxiv.org/abs/2412.00413

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