‘‘Robust’’ bistable solitons of the highly nonlinear Schrödinger equation

Abstract

We found that for some highly nonlinear Schrodinger equations (as contrasted to the cubic equation) the criteria of stability of solitary waves against small and large perturbations do not coincide, which results in the existence of "weak" and "robust" solitons, respectively. We have shown that bistable solitons, predicted earlier by Kaplan fPhys. Rev. Lett. 55, 1291Lett. 55, (1985j, are robust for some particular nonlinearities and, therefore, physically feasible. We have also suggested a general criterion for robustness of solitons.Solitons, by definition, are stable solitary solutions of nonlinear wave equations. These equations and their solitons result from many problems in the theory of elementary particles, nonlinear optics and electrodynamics, plasma physics, hydrodynamics, biology, etc. It is well known, for example, that the nondegenerate solitary-wave solutions of the cubic nonlinear Schrodinger equation (which has many applications in nonlinear optics) are stable against both small and large perturbations; in particular, two such singular solitary waves survive their collision, with their individual energies and momenta conserved after the collision, i.e. , they are solitons. Since the cubic nonlinear equation is the most famous of the nonlinear Schrodingerlike equations studied so far, the distinction between these two types of stability has never been, to the best of our knowledge, clearly drawn in the literature. Very often for such (and other) nonlinear equations, the conditions of stability from small-perturbation analysis are automatically regarded as universal criteria for soliton existence.However, this issue becomes increasingly important, especially in application to highly nonlinear Schrodinger equations with their functions of nonlinearity drastically different from the simplest known cubic nonlinearity. It has been recently demonstrated by Kaplan that for a certain class of nonlinearities, bistable (and, in general, multistable) solitary waves can exist which carry the same energy but have distinctly different profiles and speeds of propagation. The issue of the stability of these new solutions is of prime importance for their physical feasibility.In this Rapid Communication we show that the stability against small perturbations alone (as well as instability against large perturbations alone) does not provide a comprehensive description of stability of solitary-wave solutions of highly nonlinear Schrodinger equations. In order to explore this issue, we introduce here the notion of "robust" solitons as distinct from "weak" solitons in the sense that the latter are stable against (sufficiently) small perturbations, whereas the former are stable against any possible perturbation, including large perturbations; in particular, perturbations in the form of collisions with other solitary waves. (Solitons of a cubic Schrodinger equa-tion are robust in this sense. ) By studying a wide variety of nonlinear models, many of which exhibit bistability for certain ranges of the parameters, we have fou...

Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation

“…This type of bistability was introduced in Refs. [10,11] by Gatz and Hermann to distinguish it from the earlier definition [12][13][14] which implies the existence of different solutions possessing the same value of one invariant of motion (e.g., the power) for different values of the internal parameter, typically the nonlinear propagation constant β. The analysis carried out by Herrmann, however, is limited to stationary solitons, while a full family of moving dark solitons can exist.…”

Criteria for the experimental observation of multidimensional optical solitons in saturable media

“…Evolution of the amplitude E of the electromagnetic wave in a lossless Kerr-like medium with anomalous GVD obeys the well-known scaled equation [24,26,27,28] …”

“…As said above, two different forms of the saturation of the Kerr nonlinearity were previously considered in detail theoretically, with ∆n(I) in rational form [24,26,27,28],…”

“…It was shown that both rational [24,25,26,27,28] and cubic-quintic (CQ) [29,30,31] variants of the saturation readily support stable twodimensional (2D) and three-dimensional (3D) solitons. A difference between them is that the former cannot stabilize "spinning" solitons with an intrinsic vorticity, but the CQ nonlinearity makes it possible, in the 2D [32,33,34] and even 3D [35] cases.…”

Bistable dark solitons of a cubic-quintic Helmholtz equation

“…This intrinsic phenomenon [12] is potentially useful for switching applications exploiting spatial solitons in planar waveguides [13], as opposed to cavity solitons [6][7][8]14]. For a wide class of nonlinearity, there is often a parameter regime where one finds the coexistence of degenerate bright solitons -that is, beam solutions with different propagation constants but the same power [15,16]. While the ubiquitous Kerr nonlinearity is excluded from this category, materials with more involved refractive nonlinearities (e.g., cubic-quintic and saturable) offer greater flexibility for potential device designs.…”