On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

Opuscula Mathematica

article

Ivan Tsyfra¹ 

[1] AGH University of Science and Technology, Faculty of Applied Mathematics

关键词: ordinary differential equation;    partial differential equation;    integrability;    symmetry;    quadrature;    Lie transformation group.;   

DOI  :  10.7494/OpMath.2021.41.5.685

学科分类：环境科学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

【 授权许可】

CC BY-NC