Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Backlund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.     

Nonlocal Symmetry and Explicit Solution of the Alice-Bob Modified Korteweg-de Vries Equation

Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries(ABm Kd V) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-m Kd V equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, "the shifted parity and delayed time reversal to describe two-place events".     

Nonlocal Symmetry and Explicit Solution of the Alice-Bob Modified Korteweg-de Vries Equation

Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries (ABmKdV) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-mKdV equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, "the shifted parity and delayed time reversal to describe two-place events".     

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Bäcklund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via the Bäcklund transformation (BT) and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the G_N(t) for the original diagonal one.     

Nonlocal symmetries and similarity reductions for Korteweg-de Vries-negative-order Korteweg-de Vries equation

The nonlocal symmetries are derived for the Korteweg–de Vries–negative-order Korteweg–de Vries equation from the Painlevétruncation method.The nonlocal symmetries are localized to the classical Lie point symmetries for the enlarged system by introducing new dependent variables.The corresponding similarity reduction equations are obtained with different constant selections.Many explicit solutions for the integrable equation can be presented from the similarity reduction.     

CRE Solvability,Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.     

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.     

Infinitely Many Lax Pairs of the Korteweg-de Vries Equation

Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs.In this letter, we take the well-known KdV equation as a typical example. Using infinitely many symmetries, the infinitely many inhomogeneous linear Lax pairs of KdV equation can be obtained. And considering the Darboux transformations for the KdV equation leads to the infinitely many inhomogeneous nonlinear Lax pairs.     

On Construction and Solution of the Fifth-Order Korteweg-de Vries Equation

The Lax's criterion has an attempt at construction of the fifth-order Korteweg-de Vries equation. One-soliton solution of this equation is obtained via using the inverse scattering technique.     

Residual Symmetry of the Alice-Bob Modified Korteweg-de Vries Equation

Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie's first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th B?cklund transformation in the form of the determinant. Moreover, the P_sT_d(the shifted parity and delayed time reversal) symmetric exact solutions(including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation.     

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

In this paper, we put our focus on a variable-coe~cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.     

Nonlocal Symmetries and Exact Solutions for PIB Equation

In this paper, the symmetry group of the (2+1)-dimensional Painlev? integrable Burgers (PIB) equations is studied by means of the classical symmetry method. Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-dimensional group invariant solutions. Furthermore, by using the conservation laws of the reduced equations, we obtain nonlocal symmetries and exact solutions of the PIB equations.     

Nonlocal Symmetries and Exact Solutions for PIB Equation

In this paper,the symmetry group of the(2+1)-dimensional Painlevé integrable Burgers(PIB) equations is studied by means of the classical symmetry method.Ignoring the discussion of the infinite-dimensional subalgebra,we construct an optimal system of one-dimensional group invariant solutions.Furthermore,by using the conservation laws of the reduced equations,we obtain nonlocal symmetries and exact solutions of the PIB equations.     

Rational Solutions with Non-zero Asymptotics of the Modified Korteweg-de Vries Equation

Using the relation between the mKdV equation and the KdV-mKdV equation, we derive non-singular rational solutions for the mKdV equation. The solutions are given in terms of Wronskians. Dynamics of some solutions is investigated by means of asymptotic analysis. Wave trajectories of high order rational solutions are asymptotically governed by cubic curves.     

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Bäcklund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variablecoefficients can affect the conserved density, associated flux, andappearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.     

Nonlocal Symmetries and Interaction Solutions for Potential Kadomtsev-Petviashvili Equation

The nonlocal symmetry for the potential Kadomtsev-Petviashvili(pKP)equation is derived by the truncated Painleve analysis.The nonlocal symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variable.Thanks to localization process,the finite symmetry transformations related with the nonlocal symmetry are obtained by solving the prolonged systems.The inelastic interactions among the multiple-front waves of the pKP equation are generated from the finite symmetry transformations.Based on the consistent tanh expansion method,a nonauto-B(a|¨)cklund transformation(BT)theorem of the pKP equation is constructed.We can get many new types of interaction solutions because of the existence of an arbitrary function in the nonauto-BT theorem.Some special interaction solutions are investigated both in analytical and graphical ways.     

Localization of Nonlocal Symmetries and Symmetry Reductions of Burgers Equation

The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.     

CTE Solvability,Exact Solutions and Nonlocal Symmetries of the Sharma-Tasso-Olver Equation

In this letter, we prove that the STO equation is CTE solvable and obtain the exact solutions of solitons fission and fusion. We also provide the nonlocal symmetries of the STO equation related to CTE. The nonlocal symmetries are localized by prolonging the related enlarged system.