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    Scaling invariant Lax pairs of nonlinear evolution equations
    (Taylor & Francis Ltd, 2012) Hickman, Mark; Hereman, Willy; Larue, Jennifer; Goktas, Unal
    A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called a Lax pair. Two equivalent representations are presented. The first uses a pair of differential operators which leads to a higher order linear system for the auxiliary function. The second uses a pair of matrices which leads to a first-order linear system. In this article, we present a method, which is easily implemented in MAPLE or MATHEMATICA, to compute an operator Lax pair for a set of PDEs. In the operator representation, the determining equations for the Lax pair split into a set of kinematic constraints which are independent of the original equation and a set of dynamical equations which depend on it. The kinematic constraints can be solved generically. We assume that the operators have a scaling symmetry. The dynamical equations are then reduced to a set of nonlinear algebraic equations. This approach is illustrated with well-known examples from soliton theory. In particular, it is applied to a three parameter class of fifth-order Korteweg-de Vries (KdV)-like evolution equations which includes the Lax fifth-order KdV, Sawada-Kotera and Kaup-Kuperschmidt equations. A second Lax pair was found for the Sawada-Kotera equation.
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    Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential Difference Equations
    (Springer-Verlag Berlin, 2012) Goktas, Unal; Hereman, Willy
    Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. The codes INVARIANTSSYMMETRIES.M and DDERECURSIONOPERATOR.M can aid researchers interested in properties of nonlinear DDEs.