The (2+1) -dimensional nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane is analyzed by using the classical Lie group approach.Using the group theory,some types of general exact Rossby wave solutions can be obtained whence a special Rossby wave solution is known. Especially,it is found that the only effect of the time-dependent background basic wind on the Rossby waves is the accumulate motion in the zonal direction. Some types of exact explicit similarity Rossby wave solutions with both nonconstant linear and nonlinear shears are also given.

By means of a simple new approach, a general Kadomtsev-Petviashvili (KP) family with an arbitrary function of group invariants of arbitrary order is proposed. It is proved that the general KP family possesses a common infinite dimensional Kac-Moody-Virasoro Lie point symmetry algebra. The known fourth order one can be re-obtained as a special example. The finite transformation group is presented in a clearer form. The Kac-Moody-Virasoro group invariant solutions and the Kac-Moody group invariant solutions of the KP family are determined by the Boussinesq and KdV families, respectively.

Using the generalized conditional symmetry approach, a complete list of canonical forms for the Korteweg de-Vries type equations with which possessing derivative-dependent functional separable solutions(DDFSSs)is obtained. The exact DDFSSs of the resulting equations are explicitly exhibited.

By means of infinite-dimensional Kac-Moody-Vicasoro symmetry group transformation. Rich localized (3+1). Dimensional excitations such as dromions, ring-shape and bubble-like excitations are obtained for a matrix system which is produced by extending the Lax pair of the celebrated self-dual Yang-Mills field. Adundant (3+1). dimensional localized excitations can also be found in other types of nonlinear systems.