A new two-component integrable system with peakon solutions

作者：Xia Baoqiang; Qiao Zhijun^*

来源：Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2015, 471(2175): 20140750.

DOI：10.1098/rspa.2014.0750

• integrable system
• Lax pair
• peakon

A new two-component system with cubic nonlinearity and linear dispersion:
m(t) = bu(x) + 1/2 [m(uv - u(x)v(x))](x) - 1/2 m(uv(x) - u(x)v),
n(t) = bv(x) + 1/2 [n(uv - u(x)v(x))](x) + 1/2 n(uv(x) - u(x)v),
m= u - u(xx), n = v - v(xx),
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 multipeakon 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
m(t) = bu(x) + 1/2 [m(|u|(2) - |u(x)|(2))](x) - 1/2 m(uu(x)* - u(x)u*),
m= u - u(xx),
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.

• 出版日期2015-3-8