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Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions

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Abstract

In this paper we completely characterize the existence of algebraic traveling wave solutions for the celebrated Kolmogorov–Petrovskii–Piskunov/Zeldovich equation. To do it, we find necessary and sufficient conditions in order that a polynomial linear differential equation has a polynomial solution and we classify all the Darboux polynomials of the planar system \(\dot{x} =y\), \(\dot{y} =-c/d y +f(x)(f'(x)+r)\) where f is a polynomial with \(\deg f \ge 2\), \(c,d>0\) and r are real constants. All results are of interest by themselves.

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Acknowledgements

Partially supported by FCT/Portugal through the project UID/MAT/04459/2013.

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  1. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Claudia Valls

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  1. Claudia Valls
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Correspondence to Claudia Valls.

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Valls, C. Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions. Qual. Theory Dyn. Syst. 17, 429–439 (2018). https://doi.org/10.1007/s12346-017-0245-0

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  • DOI: https://doi.org/10.1007/s12346-017-0245-0

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