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Symbolic computation of conservation laws, generalized symmetries, and recursion operators for nonlinear differential-difference equations. (English) Zbl 1245.34001

Luo, Albert C.J. (ed.) et al., Dynamical systems and methods. Based on the 3rd conference on nonlinear science and complexity (NSC), Ankara, Turkey, July 27–31, 2010. Berlin: Springer (ISBN 978-1-4614-0453-8/hbk; 978-1-4614-0454-5/ebook). 153-168 (2012).
For systems of nonlinear differential-difference equations (DDEs) the authors suggest algorithms implemented by a new Mathematica program for the symbolic computation of polynomial conservation laws, generalized symmetries and recursion operators. The computer codes can be used to test the complete integrability for nonlinear DDE systems provided they are polynomial or can be presented in such a form by a suitable transformation. However, this software does not always allow to conclude that the considered DDE is not completely integrable, explained by the fact that polynomial conserved quantities and generalized symmetries could not be found. Although the suggested algorithms successfully works for various Volterra-Toda lattices and Ablowitz-Ladik lattices, the recursion operator algorithm fails for the Belov-Chaltikian and Blaszak-Marciniak lattices.
For the entire collection see [Zbl 1244.37004].

MSC:

34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
34A33 Ordinary lattice differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
68W30 Symbolic computation and algebraic computation
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