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.
In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equanons is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the （2＋1）-dimensional Burgers equation are presented.
We investigate the extended （2＋ 1）-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.
A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations.The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation,the coupled Korteweg-de Vries system,the Kadomtsev–Petviashvili equation,the Ablowitz–Kaup–Newell–Segur equation and the potential Korteweg-de Vries equation.The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.
We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.
By means of the classical symmetry method, a hyperbolic Monge-Ampere equa- tion is investigated. The symmetry group is studied and its corresponding group invariant solutions are constructed. Based on the associated vector of the obtained symmetry, the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampere equation, from which two interesting classes of solutions to the hyperbolic Monge-Ampere equation are obtained successfully.
We investigate the Lax equation that can be employed to describe motions of long waves in shallow water under gravity.A nonlocal symmetry of this equation is given and used to find exact solutions and derive lower integrable models from higher ones.It is interesting that this nonlocal symmetry links with its corresponding Riccati-type pseudopotential.By introducing suitable and simple auxiliary dependent variables,the nonlocal symmetry is localized and used to generate new solutions from trivial solutions.Meanwhile,this equation is reduced to an ordinary differential equation by means of this nonlocal symmetry and some local symmetries.
