2021, 42(9): 948-957.
doi: 10.21656/1000-0887.420011
Abstract:
A class of KdV-Burgers equations with large Reynolds numbers and weak dispersions were discussed, which were mathematically expressed as a class of singularly perturbed KdV-Burgers equations. The interaction between the nonlinear term and the dispersion term in the KdV-Burgers equation forms a stable forward-propagation soliton. Through mathematical analysis, the propagation path and speed of the soliton were described. By means of the singularly perturbed expansion method, the asymptotic solution to the problem was constructed. First, the degenerate solution was obtained with the Riemann-Earnshaw method, and the simple wave was obtained. There is a velocity difference between any point of the simple wave shape and the initial point, which makes the wave form continuously distorted in the process of propagation, and finally forms the shock wave surface, namely discontinuity. There is a time-varying jump in the velocity of particles between both sides of the discontinuity. Second, a modified traveling wave transformation was built through substitution of variables at the discontinuity of the degenerate solution, to obtain soliton solutions of the expansion of internal solutions and prove the existence and uniqueness of the internal and external solutions. Finally, the residual term was estimated with the existence of the uniformly bounded inverse operator, and the uniform effectiveness of the asymptotic solution was obtained. The results show that, the perturbations of KdV-Burgers equations with large Reynolds numbers and weak dispersions concentrate on the neighbourhoods of the discontinuities of the degenerate solutions. The soliton links the particles across the 2 sides, and its propagation path is not a linear form of time and space, but leads along the discontinuity of the degenerate solution, forming a stable waveform.
