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

TL;DR

This paper demonstrates the global existence and large-time behavior of small solutions to certain cubic nonlinear Schrödinger systems lacking coercive conserved quantities, using a novel quartic conserved quantity and explicit solutions involving Jacobi elliptic functions.

Contribution

It introduces a new class of Schrödinger systems without coercive conserved quantities and establishes their global behavior using a quartic conserved quantity and explicit ODE solutions.

Findings

Global existence of small solutions is proven.

Asymptotic behavior described via ODE solutions.

Explicit solutions involve Jacobi elliptic functions.

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.