Darboux Transformation and Explicit Solutions for Discretized Modified Korteweg-de Vries Lattice Equation

The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.

Darboux Transformation and Explicit Solutions for Drinfel'd-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.

Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation

The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.

Sign-invariant and Explicit Solutions of Nonlinear Reaction-Diffusion Systems

Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.

Symmetry Reductions and Explicit Solutions for a Generalized Zakharov-Kuznetsov Equation

Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.

Explicit Solutions for a New (2+1)-Dimensional Coupled mKdV Equation

An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two （1＋1）-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.

DARBOUX TRANSFORMATION OF A NONLINEAR EVOLUTION EQUATION AND ITS EXPLICIT SOLUTIONS

In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-difference equation is given. In order to solve the differential-difference equation, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equation are given.

Explicit solutions of Boussinesq-Burgers equation

Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.

New Explicit Solutions of (1+1)-Dimensional Variable-Coefficient Broer-Kaup System

By using the compatibility method, many explicit solutions of the (1+1)-dimensional variable-coefficientBroer-Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponentialfunction, and Airy function.Some figures of the solutions are given by the symbolic computation system Maple.

Reduction and New Explicit Solutions of （2＋1）-Dimensional Breaking Soliton Equation

Abstract By applying the Lie group method, the （2＋l）-dimensional soliton equation is reduced to some （1＋1）-dimensional nonlinear equations. Based upon some new explicit solutions of the （2＋1）-dimensional breaking soliton equation are obtained.

Explicit Solutions for a （2＋1）-Dimensional Toda Lattice with Two Discrete Variables

An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems is found. As an application, explicit solutions of the （2＋1）-dimensional Toda equation with two discrete variables are obtained.

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

Explicit solutions to some nonlinear physical models by a two-step ansatz

Explicit solutions are derived for some nonlinear physical model equations by using a delicate way of two-step ansatz method.

Bcklund Transformations and Explicit Solutions of (2+1)-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.

Darboux Transformations and New Explicit Solutions for a Blaszak-Marciniak Three-Field Lattice Equation

In this paper, the two different Darboux transformations for a Blaszak-Marciniak （BM） three-field lattice equation are constructed. As the applications of the obtained Darboux transformations, new explicit solutions for the BM lattice are given. We also discuss some properties for these new explicR solutions. Our analysis shows that the explicit solutions possess new characters.

Group Analysis, Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular, an exact solution ＆ provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.

Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev~ equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.

Explicit solutions for a class of nonlinear BSDEs and their nodal sets

In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.

New explicit and exact solutions of the Benney-Kawahara-Lin equation

In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.