Time-dependent defects in integrable soliton equations.

We study (1 + 1)-dimensional integrable soliton equations with time-dependent defects located at x = c(t), where c(t) is a function of class C 1. We define the defect condition as a Bäcklund transformation evaluated at x = c(t) in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.

A new two-component integrable system with peakon solutions.

A new two-component system with cubic nonlinearity and linear dispersion: [Formula: see text]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 [Formula: see text]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.