Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$ order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in the Abel chain always ends at the $(N-1)^{th}$ order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived. The Boussinesq equation arises in several physical applications, the first one was propagation of long waves in shallow water [3]. There have been several generalizations of the Boussinesq equation such as the modified Boussinesq equation, or the dispersive water wave. Another generalized Boussinesq equation is utt − uxx + (f(u) + uxx)xx = 0, (1) which has the classical Boussinesq equation as an special case for f(u) = u2 +u. Recently 2 conditions for the finite-time blow-up of solutions of (1) have been investigated by Liu [8]. In this work we classify the Lie symmetries of (1) and we study the class of functions f(u) for which this equation is invariant under a Lie group of point transformations. Most of the required theory and description of the method can be found in [2, 10, 11]. Motivated by the fact that symmetry reductions for many PDE’s are known that are not obtained using the classical Lie group method, there have been several generalizations of the classical Lie group method for symmetry reductions. Clarkson and Kruskal [4] introduced an algorithmic method for finding symmetry reductions, which is known as the direct method. Bluman and Cole [1] developed the nonclassical method to study the symmetry reductions of the heat equation. The basic idea of the method is to require that both the PDE (1) and the surface condition Copyright c○1998 by M.L. Gandarias an