A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line

TL;DR

This paper constructs a global two-soliton solution for the focusing cubic half-wave equation on the real line, exhibiting a transient turbulent phase with norm growth followed by a saturated regime with persistent large norm.

Contribution

It introduces an explicit two-soliton solution demonstrating a transient turbulent phase and a subsequent stable large-norm regime for the cubic half-wave equation.

Findings

Demonstrates a finite-time growth of the H^1-norm during turbulence

Shows the solution stabilizes with a large H^1-norm indefinitely

Constructs solutions with arbitrarily small L^2-norm exhibiting complex dynamics

Abstract

We consider the focusing cubic half-wave equation on the real line $$i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(\xi)=|\xi|\hat{u}(\xi), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.