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arXiv · 2606.02088

Modified scattering type asymptotic behavior for a quadratic nonlinear Schrödinger system under the mass-resonance condition in two dimensions

Satoshi Masaki·Kota Uriya
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Abstract

We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with arbitrary initial data, where the amplitude is expressed explicitly in terms of Jacobi elliptic functions and the phase is given by elliptic integrals of the third kind. As a consequence, we obtain a fully explicit characterization of the asymptotic profile for the original PDE. In particular, the long-time behavior is described by a modified-scattering-type profile whose profile evolves according to the above integrable structure. While elliptic-function-type asymptotics were previously constructed for special solutions in final-state problems, the present work provides the first complete characterization, in two space dimensions, of asymptotic dynamics associated with arbitrarily small initial data through elliptic-function-type profiles.

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Satoshi Masaki, Kota Uriya. 2026-06-01. Modified scattering type asymptotic behavior for a quadratic nonlinear Schrödinger system under the mass-resonance condition in two dimensions. https://arxiv.org/abs/2606.02088

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