# Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions

Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions 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 $$\dot{x} =y$$ x ˙ = y , $$\dot{y} =-c/d y +f(x)(f'(x)+r)$$ y ˙ = - c / d y + f ( x ) ( f ′ ( x ) + r ) where f is a polynomial with $$\deg f \ge 2$$ deg f ≥ 2 , $$c,d>0$$ c , d > 0 and r are real constants. All results are of interest by themselves. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Qualitative Theory of Dynamical Systems Springer Journals

# Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions

, Volume 17 (2) – Jun 27, 2017
11 pages

/lp/springer_journal/algebraic-traveling-wave-solutions-darboux-polynomials-and-polynomial-HrFaLGZbZR
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Dynamical Systems and Ergodic Theory; Difference and Functional Equations
ISSN
1575-5460
eISSN
1662-3592
DOI
10.1007/s12346-017-0245-0
Publisher site
See Article on Publisher Site

### 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 $$\dot{x} =y$$ x ˙ = y , $$\dot{y} =-c/d y +f(x)(f'(x)+r)$$ y ˙ = - c / d y + f ( x ) ( f ′ ( x ) + r ) where f is a polynomial with $$\deg f \ge 2$$ deg f ≥ 2 , $$c,d>0$$ c , d > 0 and r are real constants. All results are of interest by themselves.

### Journal

Qualitative Theory of Dynamical SystemsSpringer Journals

Published: Jun 27, 2017

### References

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