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

Abstract

We consider the focusing cubic half-wave equation on the real lineWe 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.

On small traveling waves to the mass critical fractional NLS

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We consider the mass critical fractional (NLS)We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [6] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.

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“…Related results for the travelling waves of the Gross Pitaevski equation are given in [7]. In [6], the sharp description of the tail of the travelling wave is an essential step for the derivation of the modulation equations associated to energy exchanges between two interacting solitary waves. The derivation of related modulation equations for 1 < s < 2 and the description of multiple bubbles interaction is a challenging problem due to the presence of additional high Galilean like oscillations, but Theorems 1.3-1.4 are the necessary starting point for such an investigation.…”

“…(1.2) s = 1: The half wave case is treated in details in [6] where the existence of travelling u(t, x) = Q s,β (x − βt) is proved with lim β↑1 Q s,β L 2 = 0.…”

“…where Q + is the ground state to the limiting non-local Szegő equation ∂ t u = Π + (u|u| 2 ), Π + u = 1 ξ>0û , see [5,16]. The existence of the critical speed β = 1 is the starting point for the construction of two bubbles interacting solitons with growing Sobolev norms, [16], [6].…”

Inelasticity of soliton collisions for the 5D energy critical wave equation

“…Such refined approximate solutions were introduced in several other situations related to blow up or soliton interactions, see e.g. [41,39,45,32,33,34,19,37,16]. In the case of the gKdV equation [34], since solitons decay exponentially in space, the method of separation of variables applies and correction terms have simple expressions in terms of solutions of elliptic problems.…”