New Solutions of the Schwarz-Korteweg-de Vries Equation in 2+1 Dimensions with the Gauge Transformation

In this paper, we find the relationship between the solution of (1+1)-dimensional Korteweg-de Vries (KdV) equation and the solution of (2+1)-dimensional integrable Schwarz-Korteweg-de Vries(SKdV) equation with Möbius transformations, Miura transformation and other transformations. Furthermore, we can obtain the new solution of the SKdV equation in 2+1 dimensions by using the solution of KdV equation which is given. In fact, the corresponding solutions of the (2+1)-dimensional SKdV equation include three arbitrary smooth functions f1(z), f2(z) and f3(z). Consequently, the solutions exhibit a rich variety of qualitative behaviour. By making appropriate choices for the arbitrary functions, an exhibit large families of solitary waves, coherent structures and different types of bound states can be obtained.