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

We consider the focusing cubic half-wave equation on the real line $i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(ξ)=|ξ|\hat{u}(ξ), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$ 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.

No reviews yet.

No comments yet.