Standing lattice solitons in the discrete NLS equation with saturation

Abstract

We consider standing lattice solitons for discrete nonlinear Schrödinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order differential equation arising as the next-order continuum approximation of the discrete NLSS and (ii) the advance-delay version of the discrete NLSS.In the context of (i), the fourth-order differential equation generally does not have solitary wave solutions due to small oscillatory tails. Nevertheless, we show that solitary waves solutions exist for specific values of governing parameter that form an infinite sequence. We present an asymptotic formula for the distance between two subsequent elements of the sequence in terms of the small parameter of lattice spacing. To derive this formula, we used two different analytical techniques: the semi-classical limit of oscillatory integrals and the beyond-all-order asymptotic expansions. Both produced the same result that is in excellent agreement with our numerical data.In the context of (ii), we also derive an asymptotic formula for values of lattice spacing for which approximate standing lattice solitons can be constructed. The asymptotic formula is in excellent agreement with the numerical approximations of transparent points. However, we show that the asymptotic formulas for the cases (i) and (ii) are essentially different and that the transparent points do not generally imply existence of continuous standing lattice solitons in the advance-delay version of the discrete NLSS.

Kuznetsov–Ma breather-like solutions in the Salerno model

“…The striking feature of the solutions presented in panels (b)-(e) is the formation of an oscillatory background (of small amplitude) in contrast with the KM breather of panel (a) where the localized wave sits atop a constant background. Such profiles featuring small amplitude wave trains are strongly reminiscent of nanoptera, and to the best of our knowledge are first reported for the Salerno model in the present work (see, also the recent works of [50,51] for Toda lattices and the DNLS with saturation). The other striking feature of our findings is that the KM breather (4) of the AL model (g = 1) seems not to be the only solution at that limit, as this has already been evident in panels (b)-(e) of Fig.…”

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

“…11-13. GSWs have also more recently been studied in discrete problems, including both difference and differential-difference equations. Systems that have been shown to demonstrate this behavior include discrete nonlinear Schrödinger lattices, [14][15][16][17] Fermi-Pasta-Ulan-Tsingou systems, 18,19 and Toda chains, 20,21 the discrete Klein-Gordon equation, 22 the Frenkel-Kontorova dislocation model, 3 and nonlinear chains of oscillators. 23 Given that there are many difference equations which produce GSWs, it seems natural to expect that there are also lattice equations which produce GSW solutions.…”

Solvability of some integro-differential equations with anomalous diffusion and transport

“…Let us discuss assertions (1) and (2). By means of parts ( 1) and ( 2) of Lemma 6 above, for a > 0, problems (4.1) and (4.12) admit unique solutions u(x) ∈ H 1 (R) and Evidently, the second term in the right side of (4.18) can be estimated from above in the L 2 (R) norm by…”

“…Non Fredholm operators arise also in the context of the wave systems with an infinite number of localized traveling waves (see [1]). Standing lattice solitons in the discrete NLS equation with saturation were studied in [2]. Weak solutions of the Dirichlet and Neumann problems with drift were considered in [18].…”