Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

Abstract

We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation

“…To overcome the nonintegrability of the solutions, we worked in a periodic domain. The present report extends our result in Reference [27] by proposing a method that also works in L 2 (R).…”

“…Recently we have considered the reduction of a Klein-Gordon equation with external damping and drive into a damped driven discrete nonlinear Schrödinger equation [27]. To overcome the nonintegrability of the solutions, we worked in a periodic domain.…”

Justification of the Lugiato-Lefever model from a damped driven $ϕ^4$ equation

“…To overcome the nonintegrability of the solutions, we worked in a periodic domain. The present report extends our result in [19] by proposing a method that works also in L 2 (R).…”

“…To handle the non-integrability condition we can work on T, rather than on R (see [19] where a similar problem was considered in the discrete case). In this paper, we propose a different approach by introducing the decomposition,…”

“…Recently we consider the reduction of a Klein-Gordon equation with external damping and drive into a damped driven discrete nonlinear Schrödinger equation [19]. To overcome the nonintegrability of the solutions, we worked in a periodic domain.…”