A new set and new relations of multiple soliton solutions of (2 + 1)-dimensional Sawada–Kotera equation

Abstract

A new transformation v=4(lnf)xx that can formulate a quintic linear equation and a pair of Hirota’s bilinear equations for the (2 + 1)-dimensional Sawada–Kotera (2DSK) Eq. (1) or u=4(lnf)x for 2DSK Eq. (2) is reported firstly, which enables one to obtain a new set of multiple soliton solutions of the 2DSK equation. They are not special cases of the known multiple solitons. The results presented in this paper show that the 2DSK equation not only possesses two sets of multiple soliton solutions, but also has a relation between them that the square of fn in n-soliton solution u=4(lnfn)x can also be obtained from the singularity limit of f2n in 2n-soliton solution u=2(lnf2n)x by using dual combination rules and a singular limit method. This property is unusual and the 2DSK equation is the first and only one found so far by us. Also, it establishes a connection of two equations because 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 under the case u=2(lnf)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 set of multiple soliton solutions is studied.