Abstract
An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence of a recursion operator therefore guarantees the complete integrability of the DDE. The algorithm is based in part on the concept of dilation invariance and uses our earlier algorithms for the symbolic computation of conservation laws and generalized symmetries. The algorithm has been applied to a number of well-known DDEs, including the Kacvan Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion operators are shown. The algorithm has been implemented in Mathematica, a leading computer algebra system. The package DDERecursionOperator.m is briefly discussed.
Highlights
A number of interesting problems can be modeled with nonlinear differentialdifference equations (DDEs) [1]-[3], including particle vibrations in lattices, currents in electrical networks, and pulses in biological chains
Nonlinear differential-difference equations (DDEs) play a role in queuing problems and discretizations in solid state and quantum physics, and arise in the numerical solution of nonlinear partial differential equations (PDEs)
The study of complete integrability of nonlinear DDEs largely parallels that of nonlinear partial differential equations (PDEs) [4]-[7]
Summary
A number of interesting problems can be modeled with nonlinear differentialdifference equations (DDEs) [1]-[3], including particle vibrations in lattices, currents in electrical networks, and pulses in biological chains. As in the continuous case, the existence of large numbers of generalized (higher-order) symmetries and conserved densities is a good indicator for complete integrability. Albeit useful, such predictors do not provide proof of complete integrability. The existence of a recursion operator, which allows one to generate an infinite set of such symmetries stepby-step, confirms complete integrability. We use the dilation invariance of DDEs in the construction of densities, higher-order symmetries, and recursion operators. Our Mathematica package InvariantsSymmetries.m [14] computes densities and generalized symmetries, and aids in automated testing of complete integrability of semi-discrete lattices.
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