Pattern Formation by Traveling Localized Modes in Two-Dimensional Dissipative Media with Lattice Potentials

Besse, Valentin; Leblond, Hervé; Mihalache, Dumitru; Malomed, Boris A.

doi:10.1007/978-3-319-21045-2_4

Cited by 1 publication

(1 citation statement)

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“…Related work on two-dimensional systems where solitary waves are relevant includes Besse et al [5,6], who summarise the behaviour of dipoles and vortices in the two-dimensional Ginzburg-Landau equation with cubic-quintic nonlinearities. In [12], Cisneros-Ake and Minzoni analyse lateral dispersion in discrete and continuous two-dimensional models using the Kadomtsev-Petviashvili equation.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of breather modes in a two-dimensional mechanical lattice

Wattis

Alzaidi

2020

Physica D: Nonlinear Phenomena

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We consider a two-dimensional square lattice in which each node is restricted to the plane of the lattice, but is permitted to move in both directions of the lattice. We assume nodes are connected to nearest neighbours along the lattice directions with nonlinear springs, and to diagonal neighbours with linear springs. We consider a generalised Klein-Gordon system, that is, where there is an onsite potential at each node in addition to the (nonlinear) nearest-neighbour interactions. We derive the equations of motion for the displacements from the Hamiltonian. We use asymptotic techniques to derive the form of small amplitude breather solutions, and find necessary conditions required for their existence. We find two types of mode, which we term 'optical' and 'acoustic', based on the analysis of other lattices which support dispersion relations with multiple branches. In addition to the usual inequality on the sign of the nonlinearity in order for the NLS to be of the focusing type, we obtain an additional ellipticity constraint, that is a restriction in the two-dimensional wavenumber space, required for the spatial differential operator to be elliptic. Highlights• we consider a 2D square lattice with in-plane motion of nodes • we use a weakly nonlinear asymptotic expansion to derive envelope equation • we find breather solutions of an associated 2D NLS

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Section: Introductionmentioning

confidence: 99%