Creeping solitons and Hartman-Grobman theorem

In this paper, hyperbolicity analysis of the 5th order nonlinear gain of creeping soliton in the cubic-quintic complex Ginzburg-Landau equation (CGLE) is studied. To analyze the hyperbolicity of creeping soliton in dissipative system, we relate it to the Hartman-Grobman Theorem. We analyzed our problem based on perturbed variational eigenvalues approach in the reduced supercritical ordinary differential equations (ODEs) in the Euler-Lagrange system, in which the real eigenvalues of the ODEs are less than zero. It is found that the problem of unfolding the bifurcation of creeping solitons gives the critical value of 5th order nonlinear gain μc, which is the hyperbolicity loss as the external parameter μ is varied about the critical value. We restricted ourselves to the numerical space-time hyperbolic variation of creeping solitons and point out common features of our system with the Hartman-Grobman hyperbolic theorem when μ is varied.