A new integrable two-component system with cubic nonlinearity

Abstract

In this paper, a new integrable two-component system, m t = [m(u x v x − uv + uv x − u x v)] x , n t = [n(u x v x − uv + uv x − u x v)] x , where m = u − u xx and n = v − v xx , is proposed. Our system is a generalized version of the integrable system m t = [m(u 2 x − u 2)] x , which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.

A new two-component integrable system with peakon solutions

“…In [39], the authors introduced an integrable two-component extension of the dispersionless version of cubic nonlinear equation ( We remark that the dispersionless version of our two-component system (1.5) with b = 0 is not equivalent to system (6.1). System (1.5) in our paper is able to be reduced to the CH equation, but system (6.1) is not, which apparently implies that these two equations are not equivalent.…”

“…It is very interesting for us to study multi-component integrable generalizations of peakon equations. For example, in [4,[33][34][35][36][37], the authors proposed two-component generalizations of the CH equation (1.1) with b = 0, and in [38,39], the authors presented two-component extensions of the cubic nonlinear equation (1.3) and equation (1.2) with b = 0.…”

A reciprocal transformation for the Geng-Xue equation

“…A two-component extension of above equation (2) is presented in Ref. 12 and is closely related with the Wadati-Konno-Ichikawa hierarchy. 9 Recently, applying the symmetry approach to classifying nonlocal partial differential equations, 13 Vladimir Novikov discovered a CH type equation with cubic nonlinearity, which reads as…”

Two-component generalizations of the Camassa–Holm equation

“…Recently, various different systems of this kind have been proposed [26,33], which deserve further study.…”

“…For example, if we take a 1 = 1, = = a a 0 2 8 and c 6 = −1 in (47), we get the two-component CH system (26) in [31] as follows:…”