Conservation Laws and Symmetries of the Levi Equation

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.     

On Camassa-Holm Equation with Self-Consistent Sources and Its Solutions

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.     

Integrability and Gauge Fixing for Bosonic String on AdS3 × S^3

The integrability and the TST transformation of the bosonic string on AdS3× S^3 are studied. The Lax connection for gauge fixed theory is constructed and proved to be fiat.     

Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

The new soliton solutions for the variable-coefficient Boussinesq system, whose applications are seen in fluid dynamics, are studied in this paper with symbolic computation. First, the Painleve analysis is used to investigate its integrability properties. For the identified case we give, the Lax pair of the system is found, and then the Darboux transformation is constructed. At last, some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics.     

N-Fold Darboux Transformation and Bidirectional Solitons for Whitham-Broer-Kaup Model in Shallow Water

Under investigation in this paper is the Whitham Broer-Kaup （WBK） system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer-Kaup system whose Lax pair can be derived by the Ablowitz-Kaup Newell-Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions （including the head-on and overtaking collisions） for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.     

Various Methods for Constructing Auto-Baicklund Transformations for a Generalized Variable-Coefficient Korteweg-de Vries Model from Plasmas and Fluid Dynamics

In this paper, under the Painleve-integrable condition, the auto-Biicklund transformations in different forms for a variable-coefficient Korteweg-de Vries model with physical interests are obtained through various methods including the Hirota method, truncated Painleve expansion method, extendedvariable-coefficient balancing-act method, and Lax pair. Additionally, the compatibility for the truncated Painleve expansion method and extended variable-coetfficient balancing-act method is testified.     

Non-autonomous discrete Boussinesq equation:Solutions and consistency

A non-autonomous 3-component discrete Boussinesq equation is discussed. Its spacing parameters pn and qm are related to independent variables n and m, respectively. We derive bilinear form and solutions in Casoratian form. The plain wave factor is defined through the cubic roots of unity. The plain wave factor also leads to extended non-autonomous discrete Boussinesq equation which contains a parameter δ. Tree-dimendional consistency and Lax pair of the obtained equation are discussed.     

Pseudo-differential Operators and Generalized Lax Equations in Symbolic Computation

In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.     

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.     

N-Soliton Solutions of Modified KdV Equation under Bargmann Constraint

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.     

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.     

Rosochatius Deformed Soliton Hierarchy with Self-Consistent Sources

Integrable Rosochatius deformations of finite-dimensional integrable systems are generalized to the soliton hierarchy with self-consistent sources. The integrable Rosochatius deformations of the Kaup-Newell hierarchy with self-consistent sources, of the TD hierarchy with self-consistent sources, and of the Jaulent-Miodek hierarchy with self- consistent sources, together with their Lax representations are presented.     

A New （2＋1）-Dimensional Integrable Equation

A new nonlinear partial differential equation （PDE） in 2＋1 dimensions is obtained from the mKP equation by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal resealing. In order to demonstrate integrability property of the new equation, the corresponding Lax pair is obtained by applying the reduction technique to the Lax pair of the mKP equation.     

Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from Fluid Mechanics,Ocean Dynamics,and Plasma Mechanics

This paper is to investigate a variable-coefficient modified Kortweg-de Vries （vc-mKdV） model, which describes some situations from fluid mechanics, ocean dynamics, and plasma mechanics. By the AblowRz-Kaup-NewellSegur procedure and symbolic computation, the Lax pair of the vc-MKdV model is derived. Then, based on the aforementioned Lax pair, the Darboux transformation is constructed and a new one-soliton-like solution is obtained as weft Features of the one-soliton-like solution are analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation.     

A New （2＋1）-Dimensional KdV Equation and Its Localized Structures

A new （2＋1）-dimensional KdV equation is constructed by using Lax pair generating technique. Exact solutions of the new equation are studied by means of the singular manifold method. Bgcklund transformation in terms of the singular manifold is obtained. And localized structures are also investigated.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

A General Approach to the Construction of Conservation Laws for Birkhoffian Systems in Event Space

For a Birkhoffing system in the event space, a general approach to the construction of conservation laws is presented. The conservation laws are constructed by finding corresponding integrating factors for the parametric equations of the system. First, the parametric equations of the Birkhoffian system in the event space are established, and the definition of integrating factors for the system is given; second the necessary conditions for the existence of conservation laws are studied in detail, and the relation between the conservation laws and the integrating factors of the system is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the results.     

Collisions of Two Soatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schroeinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect （and also diminished for the linear defocusing effect）. In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.     

Lie symmetry and Mei continuum conservation law of system＊

Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.