Multi-shock and soliton solutions of the Burgers equation employing Darboux transformation with the help of the Lax pair

Abstract

The purpose of this article is to derive a class of multi-soliton solutions of the Burgers equation employing Darboux transformation with the help of the Lax pair. By means of an effective transformation, the Burgers equation is reduced to a suitable form, and accordingly, the Lax pair of the said equation is derived utilising the Ablowitz–Kaup–Newell–Segur (AKNS) approach. Thus, the integrability of the Burgers equation is confirmed. To find an effective solution for the Burgers equation, for the first time, we apply the Darboux transformation through the Lax pair and explore new types of one-soliton solutions and two-soliton solutions of the Burgers equation. These solutions provide some new features of the Burgers equation. To the best of our knowledge, this is the first study in which a parabolic type of structure is found in the Burgers system. Moreover, taking these solutions as the seed solution, higher-order multi-soliton solution can also be generated. Finally, some important three-dimensional plots of the wave solutions are presented to visualise the dynamics of the model.