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Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

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Abstract

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a 0+a 1tan ξ+a 2tan 2 ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.

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  1. Department of Mathematics, Northeast Petroleum University, Daqing, 163318, China

    Cheng-Shi Liu

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  1. Cheng-Shi Liu
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Correspondence to Cheng-Shi Liu.

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Liu, CS. Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics. Found Phys 41, 793–804 (2011). https://doi.org/10.1007/s10701-010-9521-4

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  • DOI: https://doi.org/10.1007/s10701-010-9521-4

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