Continuous limit, various exact solutions, kink soliton resonant phenomena and dynamical behaviors for a discrete Burgers equation

Abstract

The Burgers equation can describe some phenomena such as turbulence, shock wave and traffic flow dynamics problems. Under consideration in this paper is a discrete Burgers equation, which can be viewed as the discrete counterpart of Burgers equation. Firstly, we correspond this discrete equation to the famous continuous Burgers equation under the continuous limit. Secondly, based on its 2 × 2 matrix linear spectral problem, we construct the discrete generalized (m,N−m)-fold Darboux transformation (DT) to derive various exact solutions covering kink soliton, rational solutions, semi-rational solutions as well as their mixed superposition solutions. Furthermore, we discuss kink multi-soliton resonant phenomena through the graphics analysis and use asymptotic analysis technique to investigate the limit states of various exact solutions. Finally, the dynamical evolutions of kink soliton solutions are studied through numerical simulations. The results obtained in this paper may be useful for exploring physical phenomena governed by Burgers equation.