Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

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.

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.

Dust acoustic waves in collisional uniform dense magnetoplasma

In order to study the characteristics of dust acoustic waves in a uniform dense dusty magnetoplasma system, a nonlinear dynamical equation is deduced using the quantum hydrodynamic model to account for dust–neutral collisions. The linear dispersion relation indicates that the scale lengths of the system are revised by the quantum parameter, and that the wave motion decays gradually leading the system to a stable state eventually. The variations of the dispersion frequency with the dust concentration, collision frequency, and magnetic field strength are discussed. For the coherent nonlinear dust acoustic waves, new analytic solutions are obtained, and it is found that big shock waves and wide explosive waves may be easily produced in the background of high dusty density, strong magnetic field, and weak collision. The relevance of the obtained results is referred to dense dusty astrophysical circumstances.

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.

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

The residual symmetry of the generalized Kaup-Kupershmidt （KKK） equation is obtained from the truncated Painleve 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. Frthermore, the gKK equation is proved to integrable in the sense of owning consistent Riecati expansion and some new Bgcklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.