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- Title
Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations.
- Authors
Ionescu, Carmen; Constantinescu, Radu
- Abstract
This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case.
- Subjects
NONLINEAR differential equations; PARTIAL differential equations; NONLINEAR equations; ELLIPTIC equations; EQUATIONS; NONLINEAR optical spectroscopy
- Publication
Symmetry (20738994), 2024, Vol 16, Issue 3, p335
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym16030335