Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

In nonlinear physics, it is very difficult to study interactions among different types of nonlinear waves. In this paper,the nonlocal symmetry related to the truncated Painleve′ expansion of the（2＋1）-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system. Then the corresponding group invariant solutions are found, from which interaction solutions among different types of nonlinear waves can be found.Furthermore, the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation（BT） is obtained. From this BT, novel interactive solutions among different nonlinear excitations are found.

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 from residual symmetry of the Boussinesq equation

The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained： a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.

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.

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.

A Note on Similarity Solutions of Navier-Stokes Equations

This note looks at the two similarity solutions of the Navier-Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.

New interaction solutions of the Kadomtsev–Petviashvili equation

The residual symmetry relating to the truncated Painlev6 expansion of the Kadomtsev-Petviashvili （KP） equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.

Exact Solutions of(2+1)-Dimensional HNLS Equation

In this paper, we use the classical Lie group symmetry method to get the Lie point symmetries of the （2＋1）-dimensional hyperbolic nonlinear Schr6dinger （HNLS） equation and reduce the （2＋1）-dimensional HNLS equation to some （1 ＋ 1 ）-dimensional partial differential systems. Finally, many exact travelling solutions of the （2＋1）-dimensional HNLS equation are obtained by the classical Lie symmetry reduced method.

Multiple soliton solutions and symmetry analysis of a nonlocal coupled KP system

A nonlocal coupled Kadomtsev–Petviashivili(ncKP) system with shifted parity(P_(s)^(x)) and delayed time reversal(T_(d)) symmetries is generated from the local coupled Kadomtsev–Petviashivili(cKP) system. By introducing new dependent variables which have determined parities under the action of P_(s)^(x)T_(d)^(d), the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the c KP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.

Quasiperiodic Solutions for Nonlinear Differential Equations of Second Order with Symmetry

In this paper,we study the existence of quasi-periodic solutions and the bound- edness of solutions for a wide class nonlinear differential equations of second order.Using the KAM theorem of reversible systems and the theory of transformations,we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.

Nonlocal Symmetry,CTE Solvability and Interaction Solutions of Whitham–Broer–Kaup Equations

Whitham–Broer–Kaup(WBK) equations in the shallow water small-amplitude regime is hereby under investigation. Nonlocal symmetry and Bcklund transformation are presented via the truncated Painlevé expansion.This residual symmetry is localised to Lie point symmetry by the properly enlarged system. The finite symmetry transformation of the prolonged system is computed. Based on the CTE method, WBK equations are linearized and new analytic interaction solutions between solitary waves and cnoidal waves are given with the aid of solutions for the linear equation.

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 B¨acklund 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.

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.

Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.