Pulsating, Creeping, and Erupting Solitons in Dissipative Systems

Abstract

We present three novel pulsating solutions of the cubic-quintic complex Ginzburg-Landau equation. They describe some complicated pulsating behavior of solitons in dissipative systems. We study their main features and the regions of parameter space where they exist. PACS numbers: 42.65.Tg, 05.45.Yv, 05.70.Ln, 47.20.Ky A soliton is a self-localized solution of a nonlinear partial differential equation describing the evolution of a nonlinear dynamical system with an infinite number of degrees of freedom. Solitons are usually attributed to integrable systems. In this instance, solitons remain unchanged during interactions, apart from a phase shift. They can be viewed as "modes" of the system, and, along with radiation modes, they can be used to solve initial value problems, using a nonlinear superposition of modes [1]. The two main features of solitons in integrable systems which are of interest to us are, first, that they are one-(or few-)parameter families and, second, that the superposition of solitons with zero velocity produces a pulsating solution [2], which is sometimes called a "breather."Reductions to integrable systems are extreme simplifications of the complex systems existing in nature. They can be considered as a subclass of the more general Hamiltonian systems [3]. Indeed, such a simplification allows us to analyze the systems quantitatively and to completely understand the behavior of the solitons. Solitons in Hamiltonian (but nonintegrable) systems can also be regarded as nonlinear modes, but in the sense that they allow us to describe the behavior of systems with an infinite number of degrees of freedom in terms of a few variables, thus allowing us to effectively reduce the number of degrees of freedom. Solitons in these systems collide inelastically and interact with radiation waves, thus showing that they are qualitatively different from those in integrable systems. However, as in the integrable case, the solitons are still a one-(or few-)parameter family of solutions. Another interesting property of Hamiltonian systems is that there are no pulsating solutions. If the system is near integrable, then the two-soliton solutions of the nonlinear Schrödinger equation (NLSE) which are initially excited will gradually split into two solitons or transform into a single soliton solution, depending on the type of the perturbation [4]. If the system is far from integrable, pulsations may exist if a single soliton solution is excited with a perturbation; however, they die out, so that the pulse gradually converges to a stationary soliton.Dissipative systems are more complicated than Hamiltonian ones in the sense that, in addition to nonlinearity and dispersion, they include energy exchange with external sources. The generic equation which describes dissipative systems above the point of bifurcation is the complex Ginzburg-Landau equation (CGLE) [5,6]. A review of experiments described by the CGLE is given in [7]. In optics, it describes laser systems [8][9][10][11], soliton transmission lines [12], nonline...

Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation

“…These were found in numerical simulations [8,9] and their existence has been experimentally confirmed in a passively mode-locked solid state laser [10]. These solitons possess the interesting property of exploding at a certain point, breaking down into multiple pieces, and subsequently recovering their original shape.…”

“…In addition to localized solutions with fixed shape there are pulsating solitons [9], whose profile changes periodically, with the propagation distance z. Another interesting discovery is the "exploding soliton" [8]. This localized solution belongs to the class of chaotic solutions.…”

“…The essential features of explosions, observed both theoretically [8,9] and experimentally [10], are: (1) Explosions occur intermittently. In the continuous model, they happen more or less regularly, but the period changes dramatically with a change of parameters.…”

“…Already in the first work on exploding (erupting) solitons [8], it was found that these localized objects exist over a wide range of the system parameters (see Fig. 5 of [8]).…”

“…5 of [8]). A quick comparison with the results of [11] shows that this range of parameters is, if not larger, then at least comparable with the range where stable stationary solitons exist.…”