A System of Linear Algebraic Equations for Giving Dark Multi-Soliton Solutions of the mKdV Equation

A system of linear algebraic equations for giving dark multi-soliton solutions of the mKdV equation is formulated based upon the method of Darboux transformation. The dark 1- and 2-soliton sol u tions are given explicitly.     

A New Three-Dimensional Lie Algebra and a Modified AKNS Hierarchy of Soliton Equations

A new three-dimensional Lie algebra and its corresponding loop algebra are constructed, from which a modified AKNS soliton-equation hierarchy is obtained.     

Bilinear Forms and Soliton Solutions for the Reduced Maxwell-Bloch Equations with Variable Coefficients in Nonlinear Optics

Investigation in this paper is given to the reduced Maxwell-Bloch equations with variable coefficients,describing the propagation of the intense ultra-short optical pulses through an inhomogeneous two-level dielectric medium.We apply the Hirota method and symbolic computation to study such equations. With the help of the dependent variable transformations, we present the variable-coefficient-dependent bilinear forms. Then, we construct the one-, two-and N-soliton solutions in analytic forms for them.     

Complete soliton solutions of the ZS/AKNS equations of the inverse scattering method

Starting with the (n-1)-soliton solution of a non-linear evolution equation (NLEE) and the corresponding Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur (ZS/AKNS) wavefunctions, we obtain the n-soliton solution of the NLEE and the corresponding ZS/AKNS wavefunctions. This is then used to obtain complete soliton solutions of the NLEE and the ZS/AKNS equations.     

Zakharov-Shabat Equations for Dark Soliton Solutions to the NLS Equation

For dark soliton solutions of the NLS equation, an inverse scattering transform is redeveloped. Deductions are essentially simplified in terms of an auxiliary spectral parameter from the beginning. Equations of inverse scattering transform in the form of Zakharov-Shabat are found to be simpler than those in the form of Marchenko. An explicate expression for the dark N-soliton solution and its asymptotic behaviors in the limits as t →±∞ are simply derived.     

A Unified Explicit Construction of 2N-Soliton Solutions for Evolution Equations Determined by 2×2 AKNS System

An explicit N-fold Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving alinear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.     

Multiple Soliton Solutions of the High Order Broer-Kaup Equations

Using the extended homogeneous balance method, which is very concise and primary, we find the multiple soliton solutions of the high order Broer-Kaup equations. The method can be generalized to dealing with high-dimensional Broer-Kaup equations and other class of nonlinear equations.     

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and the method of solving a Riccati map by exploiting two known particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler’s lattice equation (or Q4).     

A Generalized Representation Formula for Systems of Tensor Wave Equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433, 2007) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic Equ. 4(3):401–433, 2007) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in (J. Amer. Math. Soc. 23(2):345–382, 2009) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.     

Algebraic Method for Constructing Exact Discrete Soliton Solutions of Toda and mKdV Lattices

In this paper, with the aid of symbolic computation, we present a new method for constructing soliton solutions to nonlinear differentiM-difference equations. And we successfully solve Toda and mKdV lattice.     

On the Kac-Moody ⊗ Virasoro Algebraic Structures of the Symmetries of Super AKNS Integrable Systems

AKNS systems, as one of the integrable systems, are supersymmetrized. After the calculation of the corresponding recursion operator, two sets of symmetries are obtained. Their Lie brackets constit u te Kac-Moody ⊗ Virasoro algebra.     

Generalized Regularly Discontinuous Solutions of the Einstein Equations

The physical consistency of the match of piecewise-C ⁰ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.     

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

In this paper, the short-wave model equations are investigated, which are associated with the Camassa- Holm （CH） and Degasperis Procesi （DP） shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformatioas back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems.     

Single Peak Soliton and Periodic Cusp Wave of the Generalized Schrodinger-Boussinesq Equations

In this paper,we study peakon,cuspon,smooth sohton and periodic cusp wave of the generalized Schrodinger-Boussinesq equations.Based on the method of dynamical systems,the generalized Schrodinger-Boussinesq equations are shown to have new the parametric representations of peakon,cuspon,smooth soliton and periodic cusp wave solutions.Under different parametric conditions,various sufficient conditions to guarantee the existence of the above solutions are given.     

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given.The research of Victor Katsnelson was supported by the Minerva Foundation.     

New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like,period-form solutions of nonlinear evolution equations (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Left. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Left. A 285 (2001) 355]. Abundant new families of the exact solutions of Bogoyavlenskii‘s generalized breaking soliton equation are obtained by using this method and symbolic computation system Maple.     

Structure of the Solutions of Generalized Kadomtsev–Petviashvili Equations

We consider generalizations of the earlier results, obtained for one-dimensional equations of the Kadomtsev–Petviashvili (KP) class, to two- and three-dimensional KP-class equations with an arbitrary nonlinearity index with allowance for the higher-order dispersion correction and terms describing dissipation and instability. The asymptotics of soliton and nonsoliton solutions are derived. Constructing phase portraits in the 8-dimensional space based on the results of a qualitative analysis of generalized Korteweg–de Vries (KdV) equations is discussed.     

New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like,period-form solutions of nonlinear evolution equations (NEEs). This method is more powerful than the extended tanhfunction method [Phys. Lett. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A 285 (2001) 355]. Abundant new families of the exact solutions of Bogoyavlenskii‘s generalized breaking soliton equation are obtained by using this method and symbolic computation system Maple.     

Single Peak Soliton and Periodic Cusp Wave of the Generalized Schrödinger-Boussinesq Equations

In this paper, we study peakon, cuspon, smooth soliton and periodic cusp wave of the generalized Schrödinger-Boussinesq equations. Based on the method of dynamical systems, the generalized Schrödinger-Boussinesq equations are shown to have new the parametric representations of peakon, cuspon, smooth soliton and periodic cusp wave solutions. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.