Bifurcation and Stability Analysis of Pulsating Solitons

Abstract

Pulsating soliton solutions bifurcation analysis of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE) is presented. The approach is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Thanks to the collective variable approach, we investigated the influence of the nonlinear gain and the saturation of the Kerr nonlinearity on the pulsations of the solitons. Research has shown that the transformation between pulsating soliton and fronts can be realized through a series of period-doubling bifurcations. The complete bifurcation diagrams of the total energy have been obtained for a definite range of the nonlinear gain and the saturation of the Kerr nonlinearity values. The detailed analysis reveals that when the saturation of the Kerr nonlinearity increases one-period pulsating solution bifurcates to double-period pulsations. While the increase of the nonlinear gain leads the double-period pulsations to return into one-period pulsation before transforming into a stationary pulsating solitons.