Swift-Hohenberg equation with broken cubic-quintic nonlinearity.

Normal Form for Spatial Dynamics in the Swift-hohenberg Equation

The reversible Hopf bifurcation with 1:1 resonance holds the key to the presence of spatially localized steady states in many partial differential equations on the real line. Two different techniques for computing the normal form for this bifurcation are described and applied to the Swift-Hohenberg equation with cubic/quintic and quadratic/cubic nonlinearities.

Exact solutions of the cubic–quintic Swift–Hohenberg equation and their bifurcations

Modulation Instability of Optical Waves in the Cubic-Quintic Complex Ginzburg-Landau Equation with Fourth-Order Dispersion and Gain Terms

The modulation instability of the one-dimensional cubic-quintic complex Ginzburg-Landau equation with fouth-order dispersion and gain terms, a. k. a., the quintic complex Swift-Hohenberg equation, is investgated. The effects of the fourth-order terms to the modulational instability is studied. We numerically investigate the dynamics of the modulational instability in the presence of the fourtho...

Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation

Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by ce...

Variational approximations to homoclinic snaking in continuous and discrete systems.

Localized structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of length scales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behavior can be obtained using a variational method, for two systems. I...