Nonlinear Differential Equations of the Third and Fourth orders with Exact Solutions Expressed in terms of the Weierstrass Elliptic Function
https://doi.org/10.1134/S2304487X19050031
Abstract
The classification of ordinary differential equations with exact solutions is a classical mathematical problem. In this work, the classification problem is considered for ordinary differential equations with solutions expressed in terms of the Weierstrass elliptic function. The algorithm of search for such equations is as follows. First, the order of the singularity of the solution is chosen. Then, the order of the sought nonlinear differential equation is set. Next, Newton polygons are used to write the general form of the nonlinear differential equation taking into account the singularity of the solution and the given order for the nonlinear differential equation. After that, limitations for the parameters are found so that the general form of the nonlinear differential equation has an exact solution expressed in terms of the Weierstrass elliptic function. Theorems used to look for parameter limitations are presented. The nonlinear autonomous ordinary differential equations of the third and fourth orders are constructed using the described algorithm. Moreover, nonlinear autonomous differential equations and their solutions expressed in terms of the Weierstrass elliptic function are presented.
Supporting agencies: The study was carried out at the expense of the RFBR grant (project No. 18-29-10039)
About the Authors
S. F. Lavrova
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
A. A. Kutukov
National Research Nuclear University MEPhI(Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Lavrova S.F., Kudryashov N.A., Kutukov A.A. Nonlinear Differential Equations of the Third and Fourth orders with Exact Solutions Expressed in terms of the Weierstrass Elliptic Function. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(5):428-436. (In Russ.) https://doi.org/10.1134/S2304487X19050031
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