Exact Solution of Fourth Order Differential Equations for Description of Optical Pulses
https://doi.org/10.1134/S2304487X20050120
Abstract
The nonlinear fourth-order partial differential equation is considered with power and nonlocal nonlinearities. This equation is used to describe the propagation of pulses in an optical fiber. Since the Cauchy problem for this equation cannot be solved by the inverse scattering transform method, the reduction of this partial differential equation to an ordinary differential equation (ODE) is considered. To construct the reduction, the traveling wave variables are used. Using a traveling wave solution, a system of ODEs is obtained composed of the imaginary and real parts of the equation. The Painlevé test is performed to check the integrability of this reduction. It is established that the constructed system of ODEs does not have the Painlevé property. Using the Fuchs indices obtained during the second step of the Painlevé test, the values of traveling wave velocity is found at which the model is simplified. In this case, only one four-order equation remains in the system of ODEs, and the simplest equation method is used to find exact solutions of it. As a result, solutions expressed in terms of the Jacobi elliptic function and the exponential function are constructed. The found exact solutions have two arbitrary constants and have the form of periodic and solitary waves.
Supporting agencies: The work was carried out with financial support The Russian Foundation for Basic Research (project No. 18-29-10039) and the Ministry of Science and Higher Education of the Russian Federation (draft state assignment No. 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 Fourth Order Differential Equations for Description of Optical Pulses. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(5):412-419. (In Russ.) https://doi.org/10.1134/S2304487X20050120
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