A new two-component system with cubic nonlinearity and linear dispersion: \begin{ eqnarray*} \left\{ \begin{ array}{ l} m_t=bu_{ x}+\frac{ 1}{ 2}[m(uv-u_xv_x)]_x-\frac{ 1}{ 2}m(uv_x-u_xv), \\ n_t=bv_{ x}+\frac{ 1}{ 2}[ n(uv-u_xv_x)]_x+\frac{ 1}{ 2} n(uv_x-u_xv), \\m=u-u_{ xx},~~ n=v-v_{ xx}, \end{ array}\right. \end{ eqnarray*} 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 nontrivial 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 \begin{ eqnarray*} m_t=bu_{ x}+\frac{ 1}{ 2}[m(|u|^2-|u_x|^2)]_x-\frac{ 1}{ 2}m(uu^\ast_x-u_xu^\ast), \quad m=u-u_{ xx}, \end{ eqnarray*} is obtained by imposing the complex conjugate reduction $v=u^\ast$ to the two-component system. The complex valued $N$-peakon solution and kink wave solution to this complex equation are also derived.
Categories: nlin.SI