The stability of the solitary-wave solutions of the nonlinear cubic–quintic Schrödinger equation (NLCQSE) is examined numerically. The solutions are found not to be solitons, but quasi-soliton behaviour is found to persist over wide regions of parameter space. Outside these regions dispersive and explosive behaviour is observed in solitary-wave interactions.

No abstract

No 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...

No abstract