Nonlocal Symmetry and Interaction Solutions of a Generalized Kadomtsev-Petviashvili Equation

A generalized Kadomtsev-Petviashvili equation is studied by nonlocal symmetry method and consistent Riccati expansion (CRE) method in this paper. Applying the truncated Painlevé analysis to the generalized Kadomtsev-Petviashvili equation, some Bäcklund transformations (BTs) including auto-BT and non-auto-BT are obtained. The auto-BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion. Then the nonlocal symmetry is localized to the corresponding nonlocal group by introducing two new variables. Further, by applying the Lie point symmetry method to the prolonged system, a new type of finite symmetry transformation is derived. In addition, the generalized Kadomtsev-Petviashvili equation 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 be found by other traditional methods. Moreover, figures are given out to show the properties of the explicit analytic interaction solutions.     

CTE Solvability,Nonlocal Symmetry and Explicit Solutions of Modified Boussinesq System

A consistent tanh expansion (CTE) method is used to study the modified Boussinesq equation. It is proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by the Painlevé analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependent variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initial value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among solitons and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wave interaction behaviors are studied both in analytical and graphical ways.     

Invariant interaction solutions for a supersymmetric mKdV equation

The bosonic supersymmetric modified KdV (BSmKdV) system is obtained by the bosonization approach. The nonlocal symmetry for the BSmKdV equation is obtained by the truncated Painlevé method. By introducing multiple new fields, the finite symmetry transformation for the BSmKdV equation is derived by applying Lie’s first principle to the prolonged systems. The similarity reductions related to the nonlocal symmetry are studied. The interaction solutions among the solitons and other complicated waves, including Painlevé II waves and periodic cnoidal waves, are presented through the reduction theorems. The concrete soliton-cnoidal interaction solutions are illustrated in detail by using the mapping and deformation method.     

Nonlocal Symmetry Reductions,CTE Method and Exact Solutions for Higher-Order KdV Equation

The nonlocal symmetries for the higher-order KdV equation are obtained with the truncated Painlev′e method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing suitable prolonged systems.The finite symmetry transformations and similarity reductions for the prolonged systems are computed. Moreover, the consistent tanh expansion(CTE) method is applied to the higher-order KdV equation. These methods lead to some novel exact solutions of the higher-order KdV system.     

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.     

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.     

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

Nonlocal symmetries related to the Bäcklund 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.     

Bäcklund transformations,nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the $n$th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type $N$-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.     

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.     

Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation

In this paper, nonlocal symmetry of the (2+1) dimensional modified generalized long dispersive wave system and its applications are investigated. The nonlocal symmetry related to the eigenfunctions in Lax pairs is derived, and infinitely many nonlocal symmetries are obtained. By introducing three potentials, the prolongation is found to localize the given nonlocal symmetry. Various finite-and infinite-dimensional integrable models are constructed by using the nonlocal symmetry constraint method. Moreover, applying the general Lie symmetry approach to the enlarged system, the finite symmetry transformation and similarity reductions are computed to give novel exact interaction solutions. In particular, the explicit soliton-cnoidal wave solution is obtained for the modified generalized long dispersive wave system, and it can be reduced to the two-dark-soliton solution in one special case.     

A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions

From a two-vortex interaction model in atmospheric and oceanic systems, a nonlocal counterpart with shifted parity and delayed time reversal is derived by a simple AB reduction. To obtain some approximate analytic solutions of this nonlocal system, the multi-scale expansion method is applied to get an AB-Burgers system. Various exact solutions of the AB-Burgers equation, including elliptic periodic waves, kink waves and solitary waves, are obtained and shown graphically.To show the applications of these solutions in describing correlated events, a simple approximate solution for the two-vortex interaction model is given to show two correlated dipole blocking events at two different places. Furthermore, symmetry reduction solutions of the nonlocal AB-Burgers equation are also given by using the standard Lie symmetry method.     

Residual Symmetry Reduction and Consistent Riccati Expansion of the Generalized Kaup-Kupershmidt Equation

The residual symmetry of the generalized Kaup-Kupershmidt(gKK) equation is obtained from the truncated Painlevé expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Furthermore, the g KK equation is proved to integrable in the sense of owning consistent Riccati expansion and some new B¨acklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.     

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.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

Kaup-Kupershmidt方程的非局域对称及新的精确解

利用机械化算法得到了Kaup-Kupershmidt方程的非局域对称、约化,通过解约化方程得到了该方程的一些新的精确解.     

Symmetry Analysis and Conservation Laws to the (2+1)-Dimensional Coupled Nonlinear Extension of the Reaction-Diffusion Equation

In this paper, a detailed Lie symmetry analysis of the (2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method, which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.     

Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through the parameter expansion procedure, infinitely many nonlocal symmetries of the ShG equation can be generated accordingly. Making advantages of the consistent conditions introduced when solving the nonlocal symmetires, some new nonlocal conservation laws of the ShG equation related to the nonlocal symmetries are obtained straightforwardly. Finally, taking the nonlocal symmetries as symmetry constraint conditions imposing on the BT, some new finite and infinite dimensional nonlinear systems are constructed.     

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively.Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion(CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.     

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.     

Residual symmetries,consistent-Riccati-expansion integrability,and interaction solutions of a new (3+1)-dimensional generalized Kadomtsev—Petviashvili equation

With the aid of the Painlevé analysis, we obtain residual symmetries for a new (3+1)-dimensional generalized Kadomtsev—Petviashvili (gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the (3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.