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

TL;DR

The paper studies the long-time behavior of small solutions to a 1D two-component cubic NLS system by leveraging conserved quantities of the associated reduced ODE system. It introduces non-polynomial conserved quantities tied to the eigenstructure of the linear part, enabling global existence and uniform bounds even when no polynomial invariant exists. Under Assumptions A1 or A2 (rank-3 regime) the reduced ODE is globally well-posed and its invariants drive the PDE asymptotics, yielding precise leading-order behavior $u_j(t,x) \approx (it)^{-1/2} e^{i|x|^2/(2t)} \varphi_j(\log t, x/t)$ with $\varphi_j$ solving the ODE. The authors also provide explicit standard forms and concrete examples, showing that non-polynomial conserved quantities suffice to control long-time dynamics and extending the reach of asymptotic analyses for multi-component NLS systems.

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.

Theorems & Definitions (18)

• Proposition 1.1: MaSeUr2
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• Proposition 2.2
• proof