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.
A new (gamma_n,sigma_k)-KP hierarchy with two new time series gamma_n and sigma_k, which consists of gamma_n-flow, sigma_k-flow and mixed gamma_n and sigma_k evolution equations of eigenfunctions, is proposed. Two reductions and constrained flows of
(gamma_n,sigma_k)-KP hierarchy are studied. The dressing method is generalized to the (gamma_n,sigma_k)-KP hierarchy and some solutions are presented.
When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in cite{kd} offers an useful way to construct new integr
able system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax Pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.
We first derive an integrable deformed hierarchy of short pulse equation and their Lax representation. Then we concentrated on the solution of integrable deformed short pulse equation (IDSPE). By proposing a generalized reciprocal transformation, we
find a new integrable deformed sine-Gordon equation (IDSGE) and its Lax representation. The multisoliton solutions, negaton solutions and positon solutions for the IDSGE and the N-loop soliton solutions, N-negaton and N-positon solutions for the IDSPE are presented. In the reduced case the new N-positon solutions and N-negaton solutions for short pulse equation are obtained.
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.
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturb
ation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.
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.
We show that the KdV6 equation recently studied in [1,2] is equivalent to the Rosochatius deformation of KdV equation with self-consistent sources (RD-KdVESCS) recently presented in [9]. The $t$-type bi-Hamiltonian formalism of KdV6 equation (RD-KdVE
SCS) is constructed by taking $x$ as evolution parameter. Some new solutions of KdV6 equation, such as soliton, positon and negaton solution, are presented.
