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arXiv · 1611.08482

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

Patrick Gérard·Enno Lenzmann·Oana Pocovnicu·Pierre Raphaël
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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.

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Patrick Gérard, Enno Lenzmann, Oana Pocovnicu, Pierre Raphaël. 2016-11-25. A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line. https://arxiv.org/abs/1611.08482

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