Moving lattice kinks and pulses: An inverse method

Abstract

We develop a general mapping from given kink or pulse shaped traveling-wave solutions including their velocity to the equations of motion on one-dimensional lattices which support these solutions. We apply this mapping-by definition an inverse method-to acoustic solitons in chains with nonlinear intersite interactions, nonlinear Klein-Gordon chains, reaction-diffusion equations, and discrete nonlinear Schrödinger systems. Potential functions can be found in a unique way provided the pulse shape is reflection symmetric and pulse and kink shapes are at least C2 functions. For kinks we discuss the relation of our results to the problem of a Peierls-Nabarro potential and continuous symmetries. We then generalize our method to higher dimensional lattices for reaction-diffusion systems. We find that increasing also the number of components easily allows for moving solutions.

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

“…Flach et al studied moving pulses of the generalized discrete nonlinear Schrödinger equation [42] i dψ n dt + α(ψ n+1 − 2ψ n + ψ n−1 ) + F (|ψ n | 2 )(ψ n+1 + ψ n−1 ) +G(|ψ n | 2 )ψ n = 0 .…”

High-speed kinks in a generalized discreteϕ4model

“…Our approach to deriving analytical solutions will be, at least in the beginning of this section somewhat similar in spirit to the inverse method of [35]. That is, we will postulate the desired solution and will identify the model parameters for which this solution will be a valid one, as we explain in more detail below.…”

Embedded soliton dynamics in the asymmetric array of Josephson junctions

“…If the inductive coupling between the junctions is small, the free fluxon motion becomes impossible. However, the possibility of radiationless propagation of solitary waves in discrete media for selected values of velocity has been reported in the number of papers [10][11][12][13][14][15][16][17][18][19][20]. Also, free propagation of the bound states of several discrete topological solitons has been reported by Peyrard and Kruskal more than 30 years ago [8].…”