A Variational Approach of Creeping Solitons with Hartman-Grobman Theorem in Complex Ginzburg-Landau Equation

Nur Izzati Khairudin, Farah Aini Abdullah, Yahya Abu Hassan, Agus Suryanto


The behavior of quintic nonlinear dispersion coefficient of creeping soliton in a spatial domain with hyperbolicity analysis of Hartman-Grobman Theorem by using variational approach is studied. Complex Ginzburg-Landau equation (CGLE) is used in the analysis as we relate the creeping soliton with Hartman- Grobman Theorem. We evaluated our work based on perturbed Jacobian matrix from system of three supercritical ordinary differential Euler-Lagrange equations, in which the eigenvalues of the stability matrix touch the imaginary axis. As a consequence in unfolding the bifurcation of creeping solitons, the equilibrium structure ultimately chaotic at the variation of the coefficient µ away from the critical value, µc . This leads to hyperbolicity loss of Hartman-Grobman Theorem in the dissipative system driven out the oscillatory instability of µ exceeded the criticality parameter corresponding to the Hopf bifurcations as the system is highly complex. This overall approach restrict to numerical investigation of the space time hyperbolic variation of CGLE.

Keywords— Dissipative solitons, complex Ginzburg-Landau equation.

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