One-parameter localized traveling waves in nonlinear Schrödinger lattices

Abstract

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance-delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.

Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics

“…However we would expect the same qualitative behaviour of the core amplitude if the solution was continued to the edge of the spectral band) as it approaches the fundamental spectral band due to the related eigenfunction of the fundamental band, Figure 6. See [32], [31] for analysis and results around this band. All other branches terminate at a spectral band with non-zero amplitude.…”

“…Recent progress has been obtained for example by Oxtoby et al [31] by looking at the limit of small amplitude waves and computation of the so-called Stokes constant corresponding to the solutions originating at the fundamental spectral band in Figure 3. We also mention in related cubic DNLS models the work of Pelinovsky et al where a rigorous Melnikov theory is developed in a special case [32].…”

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

“…Nevertheless, in recent studies it has been shown that special nonlinearities can lead to so-called transparent points at which the on-site and off-site solitons have the same energy, see [28] and references therein. Thus, there can be values of parameters at which exact travelling solitons that do not shed radiation can be shown to occur [28,25,26]. Albeit, theses solitons cannot occur for arbitrarily small wave speeds, due to additional resonances with phonons [24,23].…”

Discrete Traveling Solitons in the Salerno Model

“…These include the AL lattice [10] and the lattice with translationally invariant stationary solutions derived in [17]. The Salerno model has a two-parameter family of exact traveling solitary wave solutions at α = 1 corresponding to the AL lattice; however, these solutions do not persist for α = 1 away from the integrable limit [10]; this result has been confirmed using a Melnikov method in [18]. However, further numerical results in [18] suggest that the Salerno model can still support traveling solutions for some α = 1 far from the integrable limit.…”

“…However, posing the DNLS in a traveling frame gives rise to a differential advance-delay equation which is notoriously hard to analyze. Recent progress in this area has been made by developing a Melnikov method around existing solution families [18] or by using a pseudospectral method to transform the advance-delay equation into a large system of algebraic equations [2,14]. Alternatively, looking for small-amplitude (but nonzero wavespeed) solutions bifurcating from the rest state involves computation of the so-called Stokes constants [22] in the method of beyond all orders asymptotics, which measures the splitting of the stable and unstable manifolds.…”