New Exact Solitary Wave Solutions of the Perturbed Cubic-Quartic Complex Ginzburg–Landau Equation with Different Nonlinear Refractive Index Structures

In this work, some new exact solitary wave solutions of the perturbed cubic-quartic complex Ginzburg–Landau equation (CQ-CGLE) are obtained. Using a traveling wave ansatz, the nonlinear CQ-CGLE is transformed into two nonlinear ordinary differential equations. These two ordinary differential equations are proved to be equivalent under a general certain form of the nonlinear refractive index structures \(F(|q|^{2} )\) $F(|q{|}^2)$ . And the solutions of these equations are obtained through a direct reduction to a first-order ordinary differential equation with well-known solutions. The obtained solutions are in the form of hyperbolic, Weierstrass, and Jacobi elliptic functions which degenerate to periodic and solitary wave solutions.