Traveling waves for the quartic focusing Half Wave equation in one space dimension

TL;DR

This paper demonstrates the existence of small-energy traveling wave solutions for the quartic focusing Half Wave equation in one dimension, contrasting with classical NLS behavior, and proves no traveling waves move at light speed.

Contribution

It establishes the existence of small-energy traveling waves and the non-existence of light-speed traveling waves for the quartic focusing Half Wave equation.

Findings

Existence of traveling waves with arbitrarily small $H^{1/2}$ norm.

Small data scattering is impossible for (HW).

Traveling waves at the speed of light do not exist.

Abstract

We consider the quartic focusing Half Wave equation (HW) in one space dimension. We show first that that there exist traveling wave solutions with arbitrary small $H^{\frac 12}(\R)$ norm. This fact shows that small data scattering is not possible for (HW) equation and that below the ground state energy there are solutions whose energy travels as a localised packet and which preserve this localisation in time. This behaviour for (HW) is in sharp contrast with classical NLS in any dimension and with fractional NLS with radial data. The second result addressed is the non existence of traveling waves moving at the speed of light. The main ingredients of the proof are commutator estimates and a careful study of spatial decay of traveling waves profile using the harmonic extension to the upper half space.