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This paper studied the planar, solitary, and spiral waves of the coupled Burgerscomplex 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. [ABSTRACT FROM AUTHOR]

In this paper, we numerically investigate the dynamics of the solitary waves in the presence of damping and external stochastic forces for the improved Boussinesq equation. The hydrodynamical and Stokes damping will be considered respectively. Since the analytic solutions for the considered problems are not available, Monte Carlo method will be used in random space and finite volume element method will be used in physical space. Numerical results demonstrate that solitary wave profile is not strongly affected by the Gaussian white noise in our study. The average of the velocity of the solitary waves is very close to that for the deterministic case. [ABSTRACT FROM AUTHOR]