# Proof We start the proof with a lemma which

Let (u,v,P,η)(u,v,P,η)be an x-periodic solution to the irrotational water wave problem (1), (2), (3), (4), (5) and (6), with the property that JIB-04 η and the horizontal component u at the surface are both symmetric about x=c(t)x=c(t)at any instant of time. Then the solution defines a traveling wave.

Proof.

AcknowledgementsThe financial support of the ERC Advanced Grant “Nonlinear studies of water flows with vorticity” (NWFV 267116) is acknowledged.

35A05; 35B40; 35B45

Keywords

Fractional Ginzburg–Landau equation; Global smooth solution; Global attractor; Hausdorff dimension; Fractal dimension

1. Introduction

The Ginzburg–Landau equation [14] and [15] is one of the most-studied nonlinear equations in physics. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose–Einstein condensation to liquid crystals and strings in field theory. The Ginzburg–Landau equation with fractional derivatives was suggested in [41] and studied in [39] and [40], where precipitation is used to describe processes in media with fractal dispersion or long-range interaction.