Wronskian, Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Abstract

Under investigation in this paper is a \((2 + 1)\)-dimensional extended shallow water wave equation. Bilinear form is obtained via the generalized dependent variable transformation. The Nth-order analytic solutions are, respectively, obtained via the Wronskian and Pfaffian techniques. Soliton solutions are constructed through the Nth-order solutions. Discussions on the propagation of the solitons indicate that the soliton solutions with \(\varphi (y)\) are more general than those without \(\varphi (y)\), and \(\varphi (y)\) could affect the features of the soliton solutions, where \(\varphi (y)\) is a real function related to the aforementioned transformation. One-periodic wave solutions are obtained via the Hirota–Riemann method. Relation between the one-periodic wave solutions and one-soliton solutions is studied, which indicates that the one-periodic wave solutions can approach to the one-soliton solutions under certain condition.