CONSERVATION LAWS, FIRST INTEGRALS AND CONSERVATIVE DENSITIES OF THE GENERALIZED NONLINEAR GERDJIKOV–IVANOV EQUATION

The generalized Gerdjikov–Ivanov equation is considered. In recent years, this equation has been intensively studied, since this equation is used to describe pulse propagation in optical fiber. Unlike the classical Gerdjikov–Ivanov equation, the equation under study does not pass the Painlevé test and the Cauchy problem for this equation cannot be solved by the inverse scattering method. This version of the Gerdjikov –Ivanov equation has only a limited number of conservation laws. Using multipliers and direct calculations, conservation laws for the equation under consideration are constructed in this paper and two conservation laws are found without restrictions on the parameters of the equation. One more additional conservation law is found under an additional restriction on the parameters of the equation. In this paper, first integrals for ordinary differential equations are also obtained by reducing the conservation laws to the variables of a traveling wave in the generalized Gerdjikov–Ivanov equation. Analytical solutions of the equation under consideration are found. Exact solutions of the generalized Gerdjikov–Ivanov equation are presented in the form of optical solitons, as well as through the Jacobi elliptic functions. Using auxiliary integrals, conserved quantities for an optical soliton are calculated. Conservative densities correspond to physical quantities: power, momentum, and energy. The obtained conserved quantities are of practical use in numerical and neural network modeling of pulse propagation processes in optical fiber.

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