On integrability of the higher dimensional time fractional KdV-type equation

Abstract

In this article, we investigated a nonlinear higher dimensional time fractional Korteweg–de Vries-type (KdV) equation. This considered equation is usually used to describe shallow water waves phenomena in physics. Here we found firstly the symmetry of the higher dimensional time fractional KdV-type equation in the sense of the Riemann–Liouville (RL) fractional derivative with the aid of the fractional Lie symmetry method. Then, the one-parameter group of Lie point symmetry transformation and some special solutions of this considered equation, were obtained. Next, the optimal system of one-dimensional Lie subalgebra of this considered equation, was constructed. Subsequently, on the basis of the multiple-parameter Erdélyi–Kober fractional differential operator (FDO) and Erdélyi–Kober fractional integral operator (FIO), the original equation can be reduced into the lower dimensional fractional differential equation (FDE). Furthermore, it can be translated further into more lower dimensional FDE with new symmetry from the reduced equation. Finally, conservation laws of this discussed equation are also found through a new conservation theorem. These results are favorable support for us to understand this dynamic model in a deeper level.