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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.
The direct method based on the definition of conserved currents of a system of differential equations is applied to compute the space of conservation laws of the (1+1)-dimensional wave equation in the light-cone coordinates. Then Noethers theorem yie
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$.
In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open pr
We consider the half-wave maps equation $$ partial_t vec{S} = vec{S} wedge | abla| vec{S}, $$ where $vec{S}= vec{S}(t,x)$ takes values on the two-dimensional unit sphere $mathbb{S}^2$ and $x in mathbb{R}$ (real line case) or $x in mathbb{T}$ (periodi
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segment