Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

Abstract

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.1991 Mathematics subject classification: 58F39, 34C27, 34C35, 34C37, 35Q55. Background and definitionsOne way to investigate the dynamics of a pattern formation system modeled by a partial differential equation (PDE) of evolution type, with a single space variable, is via the traveling frame reduction. Introducing the traveling coordinate z = x -ct with wave speed c, we get a boundary value problem for a system of ordinary differential equations (ODE). The critical points of these ODEs correspond to the plane waves or the zero amplitude (trivial) wave of the original PDE. The connections, or transitions, between critical points carry important dynamical information. These connections are a subclass (i.e., uniformly translating structures) of the so-called "coherent structures" (e.g. Saarloos and Hohenberg [29,30]). A heteroclinic connection between the zero amplitude wave and a plane wave is called a front. We call a heteroclinic connection between two different plane waves a heteroclinic domain wall. We call a homoclinic connection from a plane wave to itself a homoclinic domain wall. A homoclinic orbit from the zero amplitude wave to itself is called a pulse. For example, in binary fluid convection (see below), these connections correspond to the "boundary layer" between the coexisting conductive state and periodic, confined convective states.Several

Existence and stability of standing hole solutions to complex Ginzburg-Landau equations

“…To do this, one can set φ(x) = r(x)e i x 0 ψ(s) ds , and then study trajectories in the (r, r ′ , ψ) phase space. This task has been done in a series of papers, of which Doelman and Doelman et al [8,9,10,11], Duan et al [13], Holmes [21], Jones et al [26], Kapitula and Kapitula et al [29,31,33], Marcq et al [38], and Van Saarloos et al [44] are a sample. In Section 2 we prove the following theorem regarding the persistence of the wave given by (1.2).…”

Front propagation in weakly subcritical pattern-forming systems

“…In order to present a unified discussion of both processes we focus here on the bistable Ginzburg-Landau equation [6][7][8][9][10][11][12][13] A t = µA + A xx + ia 1 |A| 2 A x + ia 2 A 2Ā…”

Bifurcations of plane wave (CW) solutions in the complex cubic–quintic Ginzburg–Landau equation

“…Among the multitude of other papers, we shall only refer to two sets of studies which will directly pertain to the work in this paper. The first of this series of papers [15,9,10,12,8] used dynamical systems techniques to prove that the cubic-quintic CGLE admits periodic and quasi-periodic traveling wave solutions. The second class of papers [20,1], primarily involving numerical simulations of the full cubic-quintic CGL PDE in the context of Nonlinear Optics, revealed various branches of plane wave solutions which are referred to as continuous wave (CW) solutions in the Optics literature.…”

“…Motivated by the above, we begin a fresh look at the traveling wave solutions of the cubic-quintic CGLE in this paper. Besides attempting to understand the complex numerical coherent structures in [1], one other goal is to build a bridge between the dynamical systems approach in [15,9,10,12,8] and the numerical one in [20,1]. Given the importance of the cubic-quintic CGLE as a canonical pattern-forming system, this is clearly important in and of itself.…”