A novel (2+1)-dimensional integrable KdV equation with peculiar solution structures*

The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extension, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation. A novel (2+1)-dimensional KdV extension, the cKP3-4 equation, is obtained by combining the third member (KP3, the usual KP equation) and the fourth member (KP4) of the KP hierarchy. The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair. The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable. Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations. For instance, the soliton molecules and the missing D-Alembert type solutions (the arbitrary travelling waves moving in one direction with a fixed model dependent velocity) including periodic kink molecules, periodic kink-antikink molecules, few-cycle solitons, and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.