Frist international symposium on computing in science & engineering

A number of interesting problems can be modeled with nonlinear differential-difference equations (DDEs) [1][3], including particle vibrations in lattices, currents in electrical networks, and pulses in biological chains. Nonlinear DDEs also play a role in queuing problems and discretizations in solid state and quantum physics, and arise in the numerical solution of nonlinear PDEs. The study of complete integrability of nonlinear DDEs largely parallels that of nonlinear partial differential equations (PDEs) [4]-[7]. Indeed, as in the continuous case, the existence of large numbers of generalized (higherorder) symmetries and conserved densities is a good indicator for complete integrability. Albeit useful, such predictors do not provide proof of complete integrability. Based on the first few densities and symmetries, quite often one can explicitly construct a recursion operator which maps higher-order symmetries of the equation into new higher-order symmetries. The existence of a recursion operator, which allows one to generate an infinite set of such symmetries step-by-step, then confirms complete integrability. There is a vast body of work on the complete integrability of DDEs. Consult, e.g., [5, 8] for additional references. In this article we describe an algorithm to symbolically compute recursion operators for DDEs. This algorithm builds on our related work for PDEs and DDEs [9]-[11] and work by Oevel et al [12] and Zhang et al [13]. In contrast to the general symmetry approach in [5], our algorithms rely on specific assumptions. For example, we use the dilation invariance of DDEs in the construction of densities, higher-order symmetries, and recursion operators. At the cost of generality, our algorithms can be implemented in major computer algebra systems. Our Mathematica package InvariantsSymmetries.m [14] computes densities and generalized symmetries, and therefore aids in automated testing of complete integrability of semi-discrete lattices. Our new Mathematica package DDERecursionOperator.m [15] automates the required computations for a recursion operator. The paper is organized as follows. In Section II, we present key definitions, necessary tools, and prototypical examples, namely the Kac-van Moerbeke (KvM) [16] and Toda [17, 18] lattices. An algorithm for the computation of recursion operators is outlined in Section III. Usage of our package is demonstrated on an example in Section IV. The paper concludes with two additional examples in Section V, namely the Ablowitz-Ladik (AL) [19] and RelativisticToda (RT) [20] lattices.