Non-polynomial conserved quantities for ODE systems and its application to the long-time behavior of solutions to cubic NLS systems

TL;DR

This paper explores the long-time behavior of small solutions to cubic NLS systems in one dimension by identifying new conserved quantities for associated ODEs, enabling analysis without traditional polynomial invariants.

Contribution

It introduces a novel class of conserved quantities for ODE reductions of cubic NLS systems, extending understanding of solution boundedness and asymptotics without relying on polynomial invariants.

Findings

Established global boundedness of small solutions

Identified new conserved quantities for ODE systems

Analyzed asymptotic behavior of solutions

Abstract

In this paper, we investigate the asymptotic behavior of small solutions to the initial value problem for a system of cubic nonlinear Schrodinger equations (NLS) in one spatial dimension. We identify a new class of NLS systems for which the global boundedness and asymptotics of small solutions can be established, even in the absence of any effective conserved quantity. The key to this analysis lies in utilizing conserved quantities for the reduced ordinary differential equation (ODE) systems derived from the original NLS systems. In a previous study, the first author investigated conserved quantities expressed as quartic polynomials. In contrast, the conserved quantities considered in the present paper are of a different type and are not necessarily polynomial.