A new two-component integrable system with peakon solutions

摘要

A new two-component system with cubic nonlinearity and linear dispersion: mt=bux+12[m(uv−uxvx)]x−12m(uvx−uxv),nt=bvx+12[n(uv−uxvx)]x+12n(uvx−uxv),m=u−uxx,n=v−vxx,where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure and infinitely many conservation laws. Geometrically, this system describes a non-trivial one-parameter family of pseudo-spherical surfaces. In the case b=0, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion mt=bux+12[m(|u|2−|ux|2)]x−12m(uux∗−uxu∗),m=u−uxx,is obtained by imposing the complex conjugate reduction v=u* to the two-component system. The complex-valued N-peakon solution and kink wave solution to this complex equation are also derived.