Variational approximations to homoclinic snaking in continuous and discrete systems

Abstract

Localised 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 lengthscales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behaviour can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. Secondly, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.

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Cited by in corpus (15)

• Ultradiscrete kinks with supersonic speed in a layered crystal with realistic potentials
• Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity
• The Swift-Hohenberg equation with a nonlocal nonlinearity
• Snakes and ghosts in a parity-time-symmetric chain of dimers
• Variational approximation and the use of collective coordinates
• Unidirectional transport of wave packets through tilted discrete breathers in nonlinear lattices with asymmetric defects
• Homoclinic snaking in the discrete Swift-Hohenberg equation
• Snakes in square, honeycomb, and triangular lattices
• A discrete complex Ginzburg-Landau equation for a hydrodynamic active lattice
• Analysis of multistability in discrete quantum droplets and bubbles
• Snakes on Lieb lattice
• Gain-loss-driven travelling waves in PT-symmetric nonlinear metamaterials
• Nonlinear states of the conservative complex Swift-Hohenberg equation
• Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices
• Maxwell Fronts in the Discrete Nonlinear Schrödinger Equations with Competing Nonlinearities