Xia Baoqiang, Qiao Zhijun
School of Mathematics and Statistics , Jiangsu Normal University , Xuzhou, Jiangsu 221116, People's Republic of China.
Department of Mathematics , University of Texas-Pan American , Edinburg TX 78541, USA.
Proc Math Phys Eng Sci. 2015 Mar 8;471(2175):20140750. doi: 10.1098/rspa.2014.0750.
A new two-component system with cubic nonlinearity and linear dispersion: [Formula: see text]where 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 =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 [Formula: see text]is obtained by imposing the complex conjugate reduction =* to the two-component system. The complex-valued -peakon solution and kink wave solution to this complex equation are also derived.
[公式：见原文]，其中 是任意实常数。该系统通过其拉克斯对、双哈密顿结构和无穷多个守恒律被证明是可积的。从几何角度看，该系统描述了一个非平凡的单参数伪球面族。在 =0 的情况下，推导了该双分量系统的尖峰孤子（peakon）和多尖峰孤子解。特别地，明确求解了双尖峰孤子动力系统并详细研究了它们的相互作用。此外，通过对双分量系统施加复共轭约化 =*，得到了一个具有线性色散的新型可积三次非线性方程 [公式：见原文]。还推导了该复方程的复值 -尖峰孤子解和扭结波解。
