Exact Solutions of a Nonlinear Differential Equation with Third and Fifth Degree Nonlinearities for Description of Optical Pulses
https://doi.org/10.1134/S2304487X20010046
Abstract
A fourth-order nonlinear partial differential equation with the third- and fifth-degree nonlinearities has been considered. This equation can be used to describe pulses in optical fibers. The Cauchy problem for this equation cannot be solved by the inverse scattering transform method; for this reason, the equation is considered using the traveling wave variables. The substitution of a certain type solution gives a system of ordinary differential equations (ODEs) for the imaginary and real parts of the equation. The Painlevé test is applied to the resulting system of ODEs. According to the test, the considered ODE system does not have the Painlevé property because the expansion of the general solution into the Laurent series contains complex Fuchs indices. When using the Painlevé test, a condition for the velocity of a traveling wave at which the system is simplified to one ordinary fourth-order differential equation is obtained. The first integral is found for this equation. The method of the simplest equations is used to construct the exact solution of the considered ODE. The found solution has two arbitrary constants and is expressed in terms of the elliptic Weierstrass function. A special case where the solution has the form of a solitary wave is considered. Periodic and solitary wave solutions at different parameter values are illustrated.
Supporting agencies: The work was carried out with financial support Russian Foundation for Basic Research (RFBR), No. 18-29-10039
About the Authors
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
D. V. Safonova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Kudryashov N.A., Safonova D.V. Exact Solutions of a Nonlinear Differential Equation with Third and Fifth Degree Nonlinearities for Description of Optical Pulses. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(1):25-31. (In Russ.) https://doi.org/10.1134/S2304487X20010046
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