NONLOCAL SYMMETRIES AND NONLOCAL RECURSION OPERATORS

An expose about covering method on differential equations was given. The general formulae to determine nonlocal symmetries were derived which are analogous to the prolongation formulae of generalized symmetries. In addition., a new definition of nonlocal recursion operators was proposed, which gave a satisfactory explaination in covering theory for the integro-differential recursion operators.

Nonlocal symmetries and negative hierarchies related to bilinear Bcklund transformation

In this paper, nonlocal symmetries defined by bilinear Baacklund transformation for bilinear potential Kd V（p Kd V）equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian form of Kd V（SKd V）equation, the nonlocal symmetry is localized and the Levi transformation is presented. Besides, based on three different types of nonlocal symmetries for potential Kd V equation, three sets of negative p Kd V hierarchies along with their bilinear forms are constructed. An impressive result is that the coefficients of the third type of（bilinear） negative p Kd V hierarchy（N 〉 0） are variable, which are obtained via introducing an arbitrary parameter by considering the translation invariance of the p Kd V equation.

Nonlocal symmetries,consistent Riccati expansion integrability,and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system

For the （2＋1）-dimensional Broer–Kaup–Kupershmidt（BKK） system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion（CRE） is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

Nonlocal Symmetries to Systems of Nonlinear Diffusion Equations

In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Thosesystems have physical applications in soil science, mathematical biology, and invariant curve flows in R^3. Lie point symmetries of the potential system, which cannot be projected to vector fields of the given dependent and independent variables, yield potential symmetries. The class of the system that admits potential symmetries is expanded.

THE PROLONGATION STRUCTURES AND NONLOCAL SYMMETRIES FOR MODIFIED BOUSSINESQ SYSTEM

The Wahlquist-Estabrook （WE） prolongation structures of modified Boussi-nesq （MB） system are studied from the coverings point of view. The realizations and classifications of one-dimensional coverings of this system are obtained completely. More-over the sufficient and necessary conditions for a vector field to be a nonlocal symmetry of this system are also demonstrated in the WE prolongation structures.

Nonlocal symmetries,soliton-cnoidal wave solution and soliton molecules to a(2+1)-dimensional modified KdV system

A(2+1)-dimensional modified KdV(2DmKdV)system is considered from several perspectives.Firstly,residue symmetry,a type of nonlocal symmetry,and the Bäcklund transformation are obtained via the truncated Painlevéexpansion method.Subsequently,the residue symmetry is localized to a Lie point symmetry of a prolonged system,from which the finite transformation group is derived.Secondly,the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability.Simultaneously,explicit soliton-cnoidal wave solutions are provided.Finally,abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

CTE Solvability, Nonlocal Symmetries and Exact Solutions of Dispersive Water Wave System

A consistent tanh expansion （CTE） method is developed for the dispersion water wave （DWW） system. For the CTE solvable DlVVC system, there are two branches related to tanh expansion, the main branch is consistent while the auxiliary branch is not consistent. From the consistent branch, we can obtain infinitely many exact significant solutions including the soliton-resonant solutions and soliton-periodic wave interactions. From the inconsistent branch, only one special solution can be found. The CTE related nonlocal symmetries are also proposed. The nonlocai symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one.

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.

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.

Nonlocal Symmetries and Exact Solutions for PIB Equation

In this paper, the symmetry group of the is studied by means of the classical symmetry method （2＋l）-dimensionM Painlevd integrable Burgers （PIB） equations 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 Conservation Laws of Nonlocal Camassa-Holm Type Equations

The two-component μ-Camassa-Holm equation, the μ-Camassa-Holm equation and the modified μ- Camassa-Holm equation are three nonlocal integrable models. In this paper, it is shown that the two-component μ-Camassa-Holm equation and the μ-Camassa-Holm equation with a linear dispersion admit a kind of nonlocal symmetries, and the modified μ-Camassa-Holm equation does not have such kind of nonlocal symmetry. An number of conservation laws to the modified μ-Camassa-Holm equation and μ-Camassa-Holm equation are obtained.

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera(SK)equation is constructed with the known Lax pair.By introducing suitable and simple auxiliary variables,the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further.On the other hand,the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular,by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.

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.

Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the B?cklund transformation(BT)for the modified KdV-sine-Gordon(mKdV-SG)equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations.Then through the parameter expansion procedure,an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived.Finally,several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.

Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation

In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction 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.

New solutions from nonlocal symmetry of the generalized fifth order KdV equation

The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.

Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties

In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by threedimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries(mKdV) equation, generalized coupled mKdV integrable system and nonisospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit N-fold Darboux transformation for integrable system(2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method.The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables.The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly.Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.