Exact Solution of Sixth Order Nonlinear Differential Equations for Description of Optical Pulses
https://doi.org/10.1134/S2304487X20060085
Abstract
A sixth order partial differential equation with nonlocal and power nonlinearities has been discussed. The equation is used to describe optical pulses. Since the Cauchy problem for this partial differential equation cannot be solved by the method of the inverse scattering transform, the traveling wave reduction of this equation has been considered. The use of the traveling wave variables separates the imaginary and real parts of the equation; as a result, a system of ordinary differential equations (ODEs) has been obtained. The Painlevé test is used to check the integrability of this system. The system of ODEs does not pass the Painlevé test, since complex Fuchs indices exist. Conditions for some parameters of the equation, under which the system of ODEs becomes consistent, have been obtained. Taking into account these conditions, one ODE of the sixth order has been evaluated. The method of the simplest equations has been used to construct solutions. The constructed solutions have been expressed in terms of the exponential function and the Jacobi elliptic function and have the form of solitary and periodic waves. The found wave solutions of the equation have been illustrated for some values of the parameters.
Supporting agencies: The work was carried out with financial support Russian Foundation for Basic Research (project No. 18-29-10039) and the Ministry of Science and Higher Education of the Russian Federation (draft state task 0723-2020-0036)
About the Authors
D. V. Safonova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Safonova D.V., Kudryashov N.A. Exact Solution of Sixth Order Nonlinear Differential Equations for Description of Optical Pulses. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(6):521-528. (In Russ.) https://doi.org/10.1134/S2304487X20060085
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