Qualitative analysis of three-wave interaction with periodic boundary condition

TL;DR

This paper analyzes three-wave resonant interactions with periodic boundary conditions, providing regularity results for positive energy cases and finite-time blow-up solutions for negative energy cases, along with classifying uniform solutions.

Contribution

It offers the first regularity theorem for positive energy three-wave interactions and classifies solutions for negative energy cases, including blow-up behavior.

Findings

Regularity theorem established for positive wave energy cases.

Finite-time blow-up solutions identified for negative wave energy cases.

Complete classification of spatially uniform solutions provided.

Abstract

First, for 3WRI with positive wave energy, we present a regularity theorem for all spatial dimension. Second, for 3WRI with negative wave energy, we present a class of solution in general spatial dimension that will blow up in finite time. Moreover, a complete classification of spatial uniform solution is given for this particular system.