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.     

N-Soliton Solutions of （2＋1）-Dimensional Non-isospectral AKNS System

The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.     

Darboux Transformations and N-soliton Solutions of Two （2＋1）-Dimensional Nonlinear Equations

Two Darboux transformations of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawaka （ CDGKS） equation and （2＋1）-dimensional modified Korteweg-de Vries （mKdV） equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux trans- formations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the （2＋1）-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the （2＋1）-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.     

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.     

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.     

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.     

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.     

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.     

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.     

Riccati-type Baicklund transformations of nonisospectral and generalized variable-coefficient KdV equations

We extend the method of constructing Bgcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg-de Vries （KdV） equations as examples, their Backlund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Es- pecially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.     

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.     

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.     

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.     

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.     

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.     

Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with Symbolic Computation

In this paper, the (2+1)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the (2+1)-dimensional generalization of shallow water wave equation possesses the Painlevé property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation (DT) is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).