Exact front, soliton and hole solutions for a modified complex Ginzburg–Landau equation from the harmonic balance method

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg–Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg–Landau equation in terms of Fourier series.