Localized states in the generalized Swift-Hohenberg equation.

The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time. I. INTRODUCTION Ever since the observation that the subcritical complex Ginzburg-Landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states. The presence of these states has important consequences for other systems described by partial differential equations on the line since the Ginzburg-Landau equation describes the behavior of such systems near bifurcation from the trivial state of the system. Specifically, the complex Ginzburg-Landau equation describes the evolution of a long wavelength oscillatory instability, as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry. In contrast, near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real Ginzburg-Landau equation, and this equation possesses only unstable spatially localized states. It is of interest therefore to examine what happens to these unstable states at larger amplitude, where the real Ginzburg-Landau equation no longer provides an adequate description of the system. In this paper we show that the localized states can become stable at such amplitudes, and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions. We are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1:1 resonance, and use this theory to construct a large number of such states. The stability properties of these states are also determined, and the evolution of nonstationary localized states is studied by numerical integration in time. It is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation. The theory was originally developed in the context of water waves, where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ODE. The resulting localized states are called solitary waves, and in some cases turn out to be solitons. Kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area. Specifically, the ODE is viewed as a dynamical system in space, and localized states are sought as homoclinic orbits connecting the trivial state to itself. Whether such orbits are possible depends in part on the stability properties of the trivial state: eigenvalues with positive real part indicate that a nontrivial state can grow from x=, while eigenvalues with negative real part indicate that such a state may return, under appropriate conditions, back to the trivial state as x →. The spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system. In many cases, and in particular in the case considered here, the ODE is reversible. As a result the bifurcations that are encountered as a parameter is increased are nongeneric. In the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1:1 resonance. The unfolding of this resonance has been worked out in detail by Iooss and Peroueme 3, and can be used to understand the appearance of a variety of homoclinic orbits in this system, and hence of localized states with different spatial structure.