KdV6 equation can be described as the Kupershmidt deformation of the KdV equation (see 2008, Phys. Lett. A 372: 263). In this paper, starting from the bi-Hamiltonian structure of the discrete integrable system, we propose a generalized Kupershmidt de
formation to construct new discrete integrable systems. Toda hierarchy, Kac-van Moerbeke hierarchy and Ablowitz-Ladik hierarchy are considered. The Lax representations for these new deformed systems are presented. The generalized Kupershmidt deformation for the discrete integrable systems provides a new way to construct new discrete integrable systems.
With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $tau_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and $
{tau_B}$ and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its $t_A$-reduction and $tau_B$ reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.
Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with self-consistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon
solution, N-soliton, N-cuspon, N-positon and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.
The Degasperis-Procesi equation with self-consistent sources(DPESCS) is derived. The Lax representation and the conservation laws for DPESCS are constructed. The peakon solution of DPESCS is obtained.
