Differential Constraints Compatible with Linearized Equations

Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints. One of the standard ways for determining particular solutions to partial differential equations is to reduce them to ordinary differential equations which are easier to solve. The classical work of Lie about group-invariant solutions generalizes well-known methods for finding similarity solutions and other basic reduction methods [1]. Bluman and Cole [2] proposed a generalization of Lie’s method for finding group-invariant solutions, which they named the “nonclassical” method. In this approach, one replaces the condition for the invariance of the given system of differential equations by the weaker condition for the invariance of the combined system consisting of the original differential equations along with the equations requiring the group invariance of the solutions. P.J. Olver and P. Rosenau proposed a generalization of the nonclassical method [3, 4]. They showed that many known reduction methods, including the classical and nonclassical methods, partial invariance, and separation of variables can be placed into a general framework. In their formulation, the original system of partial differential equations can be enlarged by appending additional differential constraints (side conditions), such that the resulting overdetermined system of partial differential equations satisfy compatibility conditions. This work discusses differential constraints compatible with the linearized equations of partial differential equations instead of the partial differential equations themselves. The relation between differential constraints and recursion operators are examined. For the type of equations in the form qt = P (x, t, q, qx, qxx) and qt = P (q, qx, qxx, qxxx), recursion operators are obtained by integrating the compatible differential constraints. A new type Copyright c ©1998 by A. Satır Differential Constraints Compatible with Linearized Equations 365 of integrable equations, which are generalizations of the integrable equations of Fokas and Svinolupov, are given. Results are also compared with Fokas’ generalized symmetry [5] and Mikhailov-Shabat-Sokolov’s formal symmetry approaches [6, 7]. We can describe the differential constraint method for evolutionary equations [9] qt = P (x, t, q, qx, qxx, . . .) (1) in the following way. First we linearize the given differential equation. In other words, we replace q (and its derivatives) in (1) by q+ ǫΨ and differentiate both sides of the resulting expression with respect to ǫ and take the limit ǫ → 0, i.e., Ψt = DP (Ψ) (2) where DP is the Fréchet derivative [1]. The equation above can also be written as Ψt = N