Spatio-Temporal Pulsating Dissipative Solitons through Collective Variable Methods

Abstract

A semi-analytical approach for the pulsating solutions of the 3D complex Cubic-quintic Ginzburg-Landau Equation (CGLE) is presented in this article. A collective variable approach is used to obtain a system of variational equations which give the evolution of the light pulses parameters as a function of the propagation distance. The collective coordinate approach is incomparably faster than the direct numerical simulation of the propagation equation. This allows us to obtain, efficiently, a global mapping of the 3D pulsating soliton. In addition it allows describing the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics.

Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation

“…In reference to our work [5] [9] [18], we use the Collective variable theory [21] to identify the different types of solutions. The main idea of this approach is to associate collective variables with the pulse parameters of interest for which equations of motion may be derived.…”

“…Using the bare approximation to the 2D CSHE (see all the details in [5] [9] [18] [21]) we get the six collective variables that evolve according to the following set of six coupled ordinary differential equations:…”

“…Using the collective variable approach, we have expanded the regions of coexistence of 3D dissipative stationary and pulsating solitons in the complex Ginz-Journal of Applied Mathematics and Physics burg-Landau equation with the cubic-quintic nonlinearity [18]. Recently, with the same approach, we have demonstrated the stationary dissipative solutions of the 2D complex Swift-Hohenberg equation.…”

Effects of Dissipative Terms on Dissipative Soliton Resonance Curve

“…This equation admits stationary [14] pulsating [15] and many other types of soliton solutions [16]. Thus, transitions between them occur in the form of sequences of bifurcations [2] [15]. Finding and classification of the different types of CGLE's solutions are not easy.…”

“…Higher-order dissipative terms are responsible for the nonlinear transmission characteristics of the cavity, which allows, for example, passive mode locking. This equation admits stationary [14] pulsating [15] and many other types of soliton solutions [16]. Thus, transitions between them occur in the form of sequences of bifurcations [2] [15].…”