Isolated periodic wave trains in a generalized Burgers–Huxley equation

• Published in: Electronic Journal of Qualitative Theory of Differential Equations
• Main Authors: Qinlong Wang, Yu'e Xiong, Wentao Huang, Valery Romanovski
• Format: Article
• Language: English
• Published: University of Szeged 2022-01-01
• Subjects: generalized burgers–huxley equation; isolated periodic wave solution; positive definite lyapunov function; degenerate singular point
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=9524

We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer $n$ in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.