A new two-mode coupled Burgers equation: conditions for multiple kink solution and singular kink solution to exist

In this paper, we establish new two-mode coupled Burgers? equations which are introduced to the first time. We derive the multiple kink and singular solutions for new two-mode coupled Burgers? equation. We find the necessary conditions in which these solutions exist. We investigate the necessary conditions to have kink and singular kink solutions for the new two-mode coupled Burgers? equations. We write an explicit form for the single-soliton solution of the two-mode coupled Burgers? equations. We find the necessary conditions in which the multiple kink and multiple singular kink solutions exist. The two-front solutions of the two-mode coupled Burgers? equations are presented. The two-kink solutions and the two-singular-kink solutions are obtained. The three-front solutions of the two-mode coupled Burgers? equations are presented. The three-kink solution and the three-singular-kink solution are obtained. In general, the N-front solutions and the singular N-kink solution of the two-mode coupled Burgers? equations are given. To show the reliability of the proposed solution, we present some of our numerical results.