Narrow Results

A hierarchy of the KN equations with self-consistent sources is derived with the Lie algebra sl(4). As an application example, the integrable couplings of the KN soliton hierarchy with self-consistent sources are constructed by using of Kronecker product and loop algebra .

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak–Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation.

Based on a $2\times 2$ matrix $\bar{\partial }$ problem, we have obtained the Lax pair by constructing a spectral transformation matrix. The Hirota equation with self-consistent sources is derived by considering the nonanalytic part of the dispersion relation. Furthermore, explicit solutions including N-soliton solutions are computed by $\bar{\partial }$-dressing method, and N-soliton solutions of the Hirota equation with self-consistent sources are obtained by using the properties of Cauchy matrix. The dynamic behavior of soliton solutions is analyzed.

A non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) is first derived by using the source generation procedure. Moreover, its bilinear Bäcklund transformation is obtained. Furthermore, a new nonlinear coupled system of the non-isospectral KPESCS and its exact solutions are established. The new coupled system cannot only be reduced to the non-isospectral KPESCS but can also be reduced to the coupled non-isospectral KP equation.

In this paper, Grammian solutions to a (2+1)-dimensional modified Date–Jimbo–Kashiwara–Miwa (mDJKM) equation are presented by using Hirota bilinear method and perturbation expansion. Starting from the Grammian solutions, an integrable coupled mDJKM equation is then obtained and the corresponding Grammian solutions are first derived by utilizing the source generation procedure. Besides, we also construct and solve a coupled DJKM equation via source generation procedure. It is interesting that the coupled mDJKM system constitute a bilinear Bäcklund transformation for the coupled DJKM system. This means that the commutativity of source generation procedure and Bäcklund transformation is valid for the (2+1)-dimensional DJKM equation.

It is shown that the AKNS hierarchy with self-consistent sources can transform to KN hierarchy with self-consistent sources through a transformation operator and gauge transformation. Besides, there exists transformation in their conservation laws and Hamiltonian structures.