Planar, solitary, and spiral waves of the Burgers-CGL equations for flames governed by a sequential reaction

Abstract

This paper studied the planar, solitary, and spiral waves of the coupled Burgers-complex Ginzburg-Landau (Burgers-CGL) equations, which were derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction having two flame fronts corresponding to two reaction zones with a finite separation distance between them. First, some exact solutions including the planar and solitary waves for the one-dimensional Burgers-CGL equations that are obtained by subtle transforms and the hyperbolic tangent function expansion method. Second, some spiral waves for the two-dimensional Burgers-CGL equations are investigated. The existence of the spiral waves is proved rigorously by Schauder’s fixed point theorem applied to a class of functions, and the approximate solutions are also obtained by the power series expansion method. Furthermore, some numerical simulations are carried out near 0<r<1, since the core of the spiral wave is a singular point in the view of mathematics, and the results verify the theoretical analysis.