Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube |
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| Authors: | T Bridgman W Hereman G R W Quispel P H van der Kamp |
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| Institution: | 1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, 80401-1887, USA
2. Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia
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| Abstract: | A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg–de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica. |
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