Abstract
Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.
Highlights
The nonclassical symmetry method, proposed by G
Bluman et al in [1], is one of generalization of the classical Lie symmetry method for obtaining exact solutions of nonlinear partial differential equations (PDEs) which can not be obtained through its classical symmetries [2,3]
Determining classical and nonclassical symmetries of a PDE is equivalent to exactly solving the so-called the systems of determining equations arising from the invariance of the PDE under such symmetry transformations
Summary
The nonclassical symmetry method, proposed by G. The problem of solving the system of determining equations is turned to determining the set of zero points of the corresponding polynomial system This is suitably dealt with within the frameworks of the Gröebner basis method and Wu’s method. The connections provide supplementary conditions for nonclassical symmetries so that the corresponding nonlinear determining system is easier to solve This leads to a more efficient method to determine the nonclassical symmetry of a PDE without calculating such characteristic set of the determining system and having these ‘observations’.
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