ANALYTICAL SOLUTIONS OF THE GENERALIZED TRIKI-BISWAS EQUATION

Abstract

The mathematical model is considered for describing the propagation of pulses in a nonlinear optical medium, which is described by the generalized Triki-Biswas equation. The Cauchy problem for the nonlinear partial differential equation under study is not solved by the method of inverse scattering transformation, therefore, a transition is made to the traveling wave variable. The resulting ordinary differential equation is considered as a system of two equations for the real and imaginary parts of the original equation. After a series of transformations related to finding the first integrals of the equations under consideration, the system of equations is transformed to a nonlinear ordinary differential equation of the first order, the solution of which cannot be expressed in a general form using elementary functions. The method of transformation of the dependent and independent variables is applied, with the help of which the solution of the considered differential equation is written using the Jacobi elliptic functions in an implicit form. We study the question of the existence of degenerate solutions depending on the values of the parameters of the original differential equation. A degenerate case is presented when the solution has the form of a solitary wave and is written in an implicit form. The solutions found in the form of periodic and solitary waves are illustrated for various values of the parameters of the model under study.