Accumulation of embedded solitons in systems with quadratic nonlinearity

Abstract

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multi-wave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model.An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter ε, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of WKB type yields an asymptotic formula for the distribution of discrete values of ε at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, −6/5, and a fairly good approximation for the numerical factor in front of the scaling term. In this work, we resolve these issues, by developing an asymptotic analysis and verifying the predictions against numerical results. As a result, we derive a universal asymptotic approximation (normal form) for models with quadratic nonlinearity that support embedded solitons. The normal form amounts to a system of two second-order equations, which is well known as a basic model for ordinary solitons in second-harmonic-generating systems. There was a rather common belief that all soliton solutions had 3 Accumulation of embedded solitons been found in this much-studied system. Nevertheless, in this work we are able to predict the existence of an infinite series of previously unknown embedded solitons in the model. This is done in an analytical form, by realizing that the embedded solitons feature a broad inner zone, the solution in which must be matched to exponentially decaying ones in outer zones. Further, a method (Bohr-Sommerfeld quantization rule) borrowed from quantum mechanics is applied to the solution in the inner zone. The most essential part of the eventual analytical result is an asymptotic distribution law for values of the single control parameter of the normal-form system at which the embedded solitons exist. Comparison with numerical results clearly shows that the analysis has produced an exact value of the respective scaling index.

Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

“…We note that these types of ESs in a nonlinear optical model very similar to (4.2) were observed numerically in [4,29] earlier. Similar types of ESs have also been studied in a model with quadratic nonlinearities in [3,13].…”

Solitons in nonintegrable systems

“…The issue closes with the paper by Malomed, Wagenknecht, Champneys, and Pearce, 15 who study embedded solitons in a coupled three-wave model with quadratic nonlinearities. Accumulations of embedded solitons near the fourfold zero eigenvalue bifurcation are traced in the normalform analysis with WKB approximations and numerical continuation software.…”

Embedded Solitons With X(2) Nonlinear Susceptibility

“…These give way to bright, dark as well as singular solitons. These embedded solitons have been studied in the past using a variety of analytical approaches [1][2][3][4][5][6][7][8][9][10][11]. It is now time to revisit the same arena using a wider variety of mathematical approaches.…”