Exact solutions of the complex Ginzburg-Landau equation with law of four powers of nonlinearity

The complex Ginzburg-Landau equation having polynomial law of nonlinearity with four powers is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform. However the partial differential equation admits the translation groups in two independent variables and we look for the general solution using the traveling wave reduction. The first integral of nonlinear ordinary differential equation corresponding to the complex Ginzburg-Landau equation with the four powers of nonlinearity is found. This first integral is reduced to the nonlinear ordinary differential equation of the first order with the general solution expressed via the elliptic function. We demonstrate that the direct method allows us to obtain exact solutions without constraints on the parameters of the mathematical model. Partial cases of bright and dark optical solitons of the equation are given.

Automated summary

This paper considers the complex Ginzburg-Landau equation with a polynomial law of nonlinearity with four powers. The Cauchy problem for this equation cannot be solved using the inverse scattering transform. However, the equation admits the translation groups in two independent variables and the general solution is sought using the traveling wave reduction method. A first integral of the nonlinear ordinary differential equation corresponding to the complex Ginzburg-Landau equation is found and reduced to a first-order nonlinear ordinary differential equation with the general solution expressed in terms of elliptic functions. The direct method allows for exact solutions without constraints on the parameters of the mathematiocal model. Partial cases of bright and dark optical solitons of the equation are given.