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In the paper, Ablowitz—Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as nn+1, where n is related to the Ablowitz—Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrodinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.

A direct method of constructing the Hamiltonian structure of the soliton hierarchy with self-consistent sources is proposed through computing the functional derivative under some constraints. The Hamiltonian functional is related with the conservation densities of the corresponding hierarchy. Three examples and their two reductions are given.

N-lump solutions of the Kadomtsev—Petviashvili I equation in non-uniform media are derived through the inverse scattering transform. The obtained solutions describe lump waves with time-dependent amplitudes and velocities. Dynamics of 1-lump wave and interactions of two lump wave are illustrated.

N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources and the hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scattering transform.

By means of Hirota method, N-soliton solutions of the modified KdV equation under the Bargmann constraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair of the modified KdV equation.

In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Schrodinger equation can be a summation of squares of wave functions from bilinear point of view.

The conservation laws of the Levi equation are presented. Two types of symmetry of the Levi equation hierarchy are deduced. Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.

In this paper, a non-isospectral differential-difference Kadomtsev–Petviashvilli equation (n-DΔKPE) is presented. Then, the Casoratian solutions of the n-DΔKPE are obtained by generalizing Casoratian conditions of the non-isospectral DΔKPE, single-soliton solution is also derived by using Hiorta's method.

The bilinear equation of the general nonlinear Schrodinger equation with derivative (GDNLSE) and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative (DNLSE) and its multisoliton solutions are given by reduction.

Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.

The new method for constructing the Wronskian entries is applied to the Boussinesq equation. The novel Wronskian solutions to it are obtained, including solitons, rational solutions, Matveev solutions, and complexitons.

A new method for constructing the Wronskian entries is proposed and applied to the differential-difference Kadomtsev–Petviashvilli (DΔKP) equation. The generalized Wronskian solutions to it are obtained, including rational solutions and Matveev solutions.

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.

A type of new loop algebra M is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.

A new simple loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of generalized Tu scheme, the multi-component SC hierarchy is obtained. Furthermore, an expanding loop algebra of the loop algebra is presented. Based on , the multi-component integrable coupling system of the multi-component SC hierarchy of soliton equations is worked out. How to design isospectral problem of mulitcomponent hierarchy of soliton equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.

By using the extended Hirota's method, the N-soliton-like solution of the Ito equation is obtained. Furthermore, we also investigate the soliton-like solution interaction and find singularity.

Bilinear form of the nonisospectral AKNS equation is given. The N-soliton solutions are obtained through Hirota's method.

A new simple loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component C-KdV hierarchy is obtained. Further,an expanding loop algebra of the loop algebra is presented. Based on , the multi-component integrable coupling system of the multi-component C-KdV hierarchy is worked out. The method can be used to other nonlinear evolution equations hierarchy.

In this letter, a new loop algebra is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.

Some novel solutions of the KdV equation are obtained through the modified bilinear Backlund transformation.

Some novel solutions are constructed for the sine-Gordon equation from the modified bilinear derivative Backlund transformation. The approach used here is general and can be applied to other soliton equations.

Comment on a recent paper on Commun. Theor. Phys. 38 (2002) pp.523-528. PACS numberss: 03.40, 12.90