New similarity reductions and exact solutions of the Date–Jimbo–Kashiwara–Miwa equation

Abstract

We study the \((2+1)\)-dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation by the CK direct method. In the literature, no one has used the CK direct method to solve the DJKM equation. However, in the process of solving the DJKM equation by the CK direct method, it is almost impossible to solve for all \(\beta\) and \(z\), and we therefore use a certain method to find the concrete expressions of \(\beta\) and \(z\) more easily. Finally, some new one-dimensional similarity reductions and new exact solutions of the DJKM equation are obtained via a large amount of complex and tedious calculations; these solutions can contain some arbitrary functions of \(t\).