Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model

Abstract

This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge.

Soliton solutions of the improved perturbed nonlinear Schrödinger equation having parabolic law with non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion terms

“…The use of the NLSE in many application areas, especially modeling ultrashort pulse propagation in optical fibers, modeling the behavior in quantum gases, explaining the behavior of wave guides and optical solitons, modeling waves in plasma physics, understanding and improving the performance of fiber optic sensors and laser systems, is considered a powerful tool for modeling and understanding complex physical processes [1][2][3][4]. Investigating analytical solutions of the NLSE provides a deeper understanding of the formation, evolution, and interactions of optical solitons, as well as enabling theoretical predictions about the behavior of specific physical systems [5][6][7][8][9][10][11][12][13]. Therefore, different methods have been improved to produce analytical solutions of the NLSE like the new Kudryashov scheme [14][15][16], the improved generalized Kudryashov method [17], enhanced modified extended tanh application [18][19][20], modified simple equation technique [21,22] and the F-expansion technique [23,24].…”

“…If we substitute equation (14) and equation (11) into equation (5), we produce a polynomial. If all coefficients of W W cosh sinh i j ( ) ( ), i = 0, K7, j = 0, 1 to zero in the polynomial, then the algebraic system is derived.…”