Two new Wronskian conditions for the (3 + 1)-dimensional Jimbo-Miwa equation

Keywords: (3 + 1)-dimensional Jimbo–Miwa equation Hirota bilinear form Wronskian determinant solution a b s t r a c t In this paper, we obtain two kinds of sufficient conditions consisting of systems of linear partial differential equations, which guarantee that the corresponding Wroskian determinant solves the (3 + 1)-dimensional Jimbo–Miwa equation in the Hirota bilinear form. Our results suggest that more general conditions could be derived by further study. Soliton equations are one special kind of nonlinear partial differential equations, which are characterized by the existence of solitary wave solutions. It is challenging to obtain explicit solutions to most nonlinear partial differential equations. One well known soliton equation is the Korteweg-de Vries (KdV) equation, which can go back to the observation by Russell. After the celebrated inverse scattering method was constructed in the 1960s, the area of soliton equations and integrable systems grew very rapidly, and have deeply influenced many branches of mathematics and physics. Compared with the analytical approach, the Hirota bilinear method is more straightforward and easily handled to get the explicit soliton solutions. Furthermore , the beauty of algebra hidden in the soliton equations is found by M. Sato, who got his s function and Grassman-nian solutions from the Hirota bilinear form. The (3 + 1)-dimensional Jimbo–Miwa equation u xxxy þ 3u xx u y þ 3u x u xy þ 2u yt À 3u xz ¼ 0 ð1:1Þ was firstly investigated by Jimbo–Miwa and its soliton solutions were obtained in [11]. It is the second member in the entire KP hierarchy and it was studied in a series of papers [12–16]. Ma [15] proposed a direct approach to solve Eq. (1.1). Wazwaz [16] employed the Hirota's bilinear method to obtain multiple-soliton solutions. We recall Hirota's bilinear operators [17] defined by