Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations

@article{Gktas2011SymbolicCO,
  title={Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations},
  author={{\"U}nal G{\"o}ktas and Willy A. Hereman},
  journal={ArXiv},
  year={2011},
  volume={abs/1104.4582}
}
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… 

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References

SHOWING 1-10 OF 44 REFERENCES

Symbolic computation of polynomial conserved densities, generalized symmetries, and recursion operators for nonlinear differential-difference equations

Algorithms for the symbolic computation of polynomial conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are

Algorithmic computation of symmetries, invariants and recursion operators for systems of nonlinear evolution and differential-difference equations

New straightforward algorithms for the symbolic computation of higher-order symmetries, conservation laws and recursion operators of nonlinear evolution equations and lattice equations are presented.

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

TLDR
A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented and the code is tested on several well-known partial differential equations from soliton theory.

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

TLDR
A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented and the resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability ofPolynomial PDE's that can be written as nonlinear evolution equations.

Symbolic algorithms for the Painleve test, special solutions, and recursion operators for nonlinear PDEs.

TLDR
The algorithms and implementations of three MATHEMATICA packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs are discussed.

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

TLDR
The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations and with the Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equation.

Computation of densities and fluxes of nonlinear differential‐difference equations

  • M. HickmanW. Hereman
  • Mathematics, Computer Science
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2003
TLDR
Direct methods to find conserved densities and fluxes of differential‐difference equations for the Kac‐van Moerbeke (KvM) and modified Volterra lattices and a Miura map which connects both lattices is explicitly constructed based on homotopic deformation are presented.