Calculation of Homoclinic/Heteroclinic Orbits Arisen from the Complex Ginzburg-Landau Equation

The complex Ginzburg-Landau equation (CGLE) is a canonical model for dissipative, weakly nonlinear systems. Soliton and hole solutions are related to homoclinic/heteroclinic orbits of a three-dimensional dynamical system. The harmonic balance methods which are effective in computing periodic solutions for arbitrary parametric values usually fail in finding approximate solutions of such orbits.Recently, we developed a perturbation-incremental (PI) method to find approximate analytical solutions of homoclinic/heteroclinic orbits for planar systems. In this proposal, we will extend the PI method to obtain approximate solutions of such orbits in the three-dimensional dynamical system which is related to the CGLE. Since the PI method is able to find both stable and unstable solutions, it will give a global picture of the dynamical behaviour.An outcome of the proposal will be an efficient but simple method for finding homoclinic/heteroclinic orbits of ordinary differential equations, and soliton and hole solutions of the CGLE.