Abstract
Solving nonclassical symmetry of partial differential equations (PDEs) is a challenging problem in applications of symmetry method. In this paper, an alternative method is proposed for computing the nonclassical symmetry of PDEs. The method consists of the following three steps: firstly, a relationship between the classical and nonclassical symmetries of PDEs is established; then based on the link, we give three principles to obtain additional equations (constraints) to extend the system of the determining equations of the nonclassical symmetry. The extended system is more easily solved than the original one; thirdly, we use Wu’s method to solve the extended system. Consequently, the nonclassical symmetries are determined. Due to the fact that some constraints may produce trivial results, we name the candidate constraints as “potential” ones. The method gives a new way to determine a nonclassical symmetry. Several illustrative examples are given to show the efficiency of the presented method.
Highlights
The classical Lie symmetry (CLS) method, proposed by Sophus Lie in 1870s, has been widely used to solve nonlinear partial differential equations (PDEs) in mathematics, physics, and mechanics [1, 2]
We establish a set of intrinsical link identities (26) between the determining equations of the CLS and nonclassical symmetry through the corresponding determining polynomial system
We observed that the infinitesimal functions of a nonclassical symmetry of PDEs are zero points of some partial terms of the differential polynomials involved in the identities
Summary
The classical Lie symmetry (CLS) method, proposed by Sophus Lie in 1870s, has been widely used to solve nonlinear PDEs in mathematics, physics, and mechanics [1, 2]. In paper [16], the authors discussed the problem of compatibility of a PDE of second order with several invariance surface conditions They revealed the relationships of some reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables with nonclassical and weak point symmetries. One obtains the corresponding symmetries of the PDEs. In this article, we explore a constraints method for solving the system of determining equations through finding a relationship between CLS and nonclassical symmetries of PDEs. Based on the link, we get three principles to obtain additional equations (constraints) to the system of determining equations. (R4) Assembling (R1)–(R3) and combining Wu’s method of differential form, we propose a method of solving the system of determining equations that lead to determining nonclassical symmetries of PDEs. The given method can be used in nonclassical symmetry computation and in symmetry classification problems
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