Bistable Optical Solitons

We showed recently [1] that a single soliton solution of the highly-nonlinear Schrödinger equation becomes multi-stable for a certain class of nonlinearities. This implies that more than one amplitude profile and propagation constant of a single soliton may exist for the same total power carried by the soliton. The multistable solitons exist only if the nonlinear component of the susceptibility as a function of intensity is either changing its sign or its derivative has a sufficiently sharp peak. This can result in such effects as bistable (or multistable, in general) self-trapping of light in nonlinear media as well as bistable propagation of soliton pulses in nonlinear optical fibers. Both of these processes may be described by the same nonlinear Schrödinger equation with a nonlinear term in the form Ef(|E|2) where f(|E|2) is an arbitrary function of the field intensity |E|2 with f(0) = 0. The so called cubic nonlinear Schrödinger equation with f(|E|2) proportional |E|2, corresponds to Kerr-nonlinearity in optical propagation and does not give rise to bistable solitons). By evaluating the propagation constant of a solitary solution δ as a function of its total power P for each given f(|E|)2), we found a class of nonlinear functions f(I) for which the dependence δ(P) becomes multivalued. We conjectured earlier [1] that stable (unstable) solitary solutions are those with dδ/dP > 0 (dδ/dP < 0). The first computer results by Enns and Rangnekar [2] supported this conjecture for a certain non-Kerr nonlinearity. In our further collaborative research [3], we discovered that depending on the nonlinear function f(I), the stable solitary solutions (dδ/dP > 0) fall into two classes: “weak” solitons (stable only against sufficiently small perturbation), and “robust” solitons (stable against any perturbation, in particular, against collisions with other solitons). It was shown [3] that while the condition dδ/dP > 0 is sufficient for the “weak” stability, some extra conditions imposed on the function f(I) warrant existence of “robust” solitons (one of them e.g. is f(I)/I2 = o(1) as I → ∞; stability against collapsing behavior). It was shown most recently by Enns [4] that at least for some of the nonlinear functions of (|E|2) giving rise to bistable solitons, the nonlinear Schrödinger equation passes the so called Painleve test and is therefore completely integrable. This confirms that the “robust” solitary solutions are indeed solitons. The propagation of single pulses in nonlinear fibers with an appropriate nonlinearity may provide the first known opportunity to obtain a temporal (dynamic) bistability as contrasted to the known kinds of optical bistability which are so far based on the equilibrium regimes.