Abstract
In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.
Highlights
Introduction and MotivationSymmetry plays an important role in the construction of solutions for nonlinear (PDE), a continuous symmetry of a partial differential equation (PDE) system is a transformation that leaves invariant for the solution manifold of the system, i.e., it maps any solution of the system into a solution of the same system
Symmetry plays an important role in the construction of solutions for nonlinear (PDE), a continuous symmetry of a PDE system is a transformation that leaves invariant for the solution manifold of the system, i.e., it maps any solution of the system into a solution of the same system
Lou and Hu started from the recursion operator and its inverse to construct the nonlocal symmetry for PDEs [11]
Summary
We list some important definitions and theorems [7] which will be used later. A one-parameter Lie group of point transformations (x∗)i = f i(x, u; ),. ∂ku), i.e., it maps any family of solution surfaces of PDE system (5). In this case, the one-parameter Lie group of point transformations (4) is called a point symmetry of the PDE system (5). Be the infinitesimal generator of the Lie group of point transformations (4). For PDE system (5), if we derive the infinitesimal generators (9) of classical Lie point symmetries, the corresponding finite symmetry transformations are given as the solution of the initial value problem dxd ξ(x, t, u), x( = 0) = x, dtd τ(x, t, u), t( = 0) = t,.
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