Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference equations

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 algo-rithm 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 Kac-van Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion opera-tors are shown. The algorithm has been implemented in Mathematica, a leading com-puter algebra system. The package DDERecursionOperator.m is briefly discussed.

[1]  B. Fuchssteiner,et al.  Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure , 1991 .

[2]  M. Toda.  Vibration of a Chain with Nonlinear Interaction , 1967 .

[3]  Michael D. Colagrosso,et al.  Algorithmic integrability tests for nonlinear differential and lattice equations , 1998, solv-int/9803005.

[4]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[5]  W. Strampp,et al.  On the correspondence between symmetries and conservation laws of evolution equations , 1982 .

[6]  A. Fordy.  APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS (Graduate Texts in Mathematics) , 1987 .

[7]  C. Moorehead.  All rights reserved , 1997 .

[8]  W. Marsden.  I and J , 2012 .

[9]  Mark J. Ablowitz,et al.  Nonlinear differential−difference equations , 1975 .

[10]  Willy Hereman,et al.  Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations , 1997, J. Symb. Comput..

[11]  R. Sahadevan,et al.  Belov–Chaltikian and Blaszak–Marciniak lattice equations: Recursion operators and factorization , 2003 .

[12]  Yuri B. Suris,et al.  INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES , 1997, solv-int/9709005.

[13]  Y. Suris.  The Problem of Integrable Discretization: Hamiltonian Approach , 2003 .

[14]  M. Hickman,et al.  Computation of densities and fluxes of nonlinear differential‐difference equations , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[16]  Willy Hereman,et al.  Symbolic computation of polynomial conserved densities, generalized symmetries, and recursion operators for nonlinear differential-difference equations , 2005 .

[17]  Willy Hereman,et al.  Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations , 1999, Adv. Comput. Math..

[18]  M. Wester.  Computer algebra systems: a practical guide , 1999 .

[19]  G. Teschl.  Jacobi Operators and Completely Integrable Nonlinear Lattices , 1999 .

[20]  W. Hereman,et al.  Integrability Tests for Nonlinear Evolution Equations , 1999, solv-int/9904022.

[21]  Mark Kac,et al.  On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices , 1975 .

[22]  Alexey Borisovich Shabat,et al.  Symmetry Approach to the Integrability Problem , 2000 .

[23]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[24]  H. Flaschka.  The Toda lattice. II. Existence of integrals , 1974 .

[25]  W. Hereman,et al.  Algorithmic Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations , 1998, solv-int/9802004.

[26]  Morikazu Toda,et al.  Theory Of Nonlinear Lattices , 1981 .

[27]  B. Fuchssteiner,et al.  Mastersymmetries and Multi-Hamiltonian Formulations for Some Integrable Lattice Systems , 1989 .

[28]  Willy Hereman,et al.  Computation of conservation laws for nonlinear lattices , 1998, solv-int/9801023.