A new two-component integrable system with peakon solutions

Abstract

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.

References in corpus (4)

• Inverse Scattering Transform for the Camassa-Holm equation
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• Integrable system with peakon, weak kink, and kink-peakon interactional solutions
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Cited by in corpus (10)

• Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa-Holm equation
• A synthetical two-component model with peakon solutions
• A four-component Camassa-Holm type hierarchy
• On a Lagrangian reduction and a deformation of completely integrable systems
• On the Cauchy problem for a new integrable two-component system with peakon and weak kink solutions
• Persistence Properties and Unique Continuation for a Dispersionless Two-Component Camassa-Holm System with Peakon and Weak Kink Solutions
• Integrable U(1)-invariant peakon equations from the NLS hierarchy
• A four component cubic peakon (4CH) equations
• Local well-posedness and blow-up for a family of $U(1)$-invariant peakon equations
• A three-component Camassa-Holm system with cubic nonlinearity and peakons