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

2016/11/25 by Gérard, Patrick, Lenzmann, Enno, Pocovnicu, Oana +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.08482

Abstract

We consider the focusing cubic half-wave equation on the real line i ∂_t u + |D| u = |u|² u, \widehat|D|u(ξ)=|ξ|u(ξ), (t,x)∈ \Bbb R₊× \Bbb R. We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small L²-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its H¹-norm on a finite time interval, followed by (ii) a saturation regime in which the H¹-norm remains stationary large forever in time.

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