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

Nikolay A. Kudryashov (), Sofia F. Lavrova and Daniil R. Nifontov
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Nikolay A. Kudryashov: Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia
Sofia F. Lavrova: Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia
Daniil R. Nifontov: Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia

Mathematics, 2023, vol. 11, issue 23, 1-20

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.

Keywords: Gerdjikov–Ivanov equation; phase portraits; conservation laws; periodic and solitary wave; optical soliton; partial differential equations; first integral; exact solutions; solitary wave (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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