Computation of stationary pulse solutions of the cubic-quintic complex Ginzburg-Landau equation by a perturbation-incremental method

Stationary pulse solutions of the cubic-quintic complex Ginzburg-Landau equation are related to heteroclinic orbits in a three-dimensional dynamical systems and they are usually obtained using numerical simulation. The harmonic balance method has severe limitation in computing homoclinic/heteroclinic orbits since the period of such orbits is infinite. In this paper, we present a perturbation-incremental method to find such stationary pulse solutions. With the introduction of a nonlinear transformation, perturbed analytical pulse solutions are obtained in terms of trigonometric functions. Such formulation makes it possible to apply the harmonic balance method to find accurate approximate solutions of the corresponding heteroclinic orbits with arbitrary parametric values. Zero-order analytical solutions from the perturbation step and approximate solutions from the incremental step are compared with that from the bifurcation package AUTO, and they are in good agreement.