New hierarchy of multiple soliton solutions for the (2+1)-dimensional Sawada-Kotera equation

Abstract

A new transformation $u=4 ({\rm ln}f)_x$ that can formulate a quintic linear equation and a pair of Hirota's bilinear equations for the (2+1)-dimensional Sawada-Kotera (2DSK) equation is reported firstly, which enables one to obtain a new hierarchy of multiple soliton solutions of the 2DSK equation. It tells a crucial fact that a nonlinear partial differential equation could possess two hierarchies of multiple soliton solutions and the 2DSK equation is the first and only one found in this paper. The quintic linear equation is solved by a pair of Hirota's bilinear equations, of which one is the (2+1)-dimensional bilinear SK equation obtained by $u=2 ({\rm ln}f)_{x}$, and the other is the bilinear KdV equation. The (1+1)-dimensional SK equation does not possess this property. As another example, a (3+1)-dimensional nonlinear partial differential equation possessing a pair of Hirota's bilinear equations, however only bearing one hierarchy of multiple soliton solutions is studied.