Joseph Krasil’shchik & Alexander Verbovetsky 
The Symbolic Computation of Integrability Structures for Partial Differential Equations 

This is the first book devoted to the task of computing integrability structures by computer. The symbolic computation of integrability operator is a computationally hard problem and the book covers a huge number of situations through tutorials. The mathematical part of the book is a new approach to integrability structures that allows to treat all of them in a unified way. The software is an official package of Reduce. Reduce is free software, so everybody can download it and make experiments using the programs available at our website.

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Introduction.- Computational problems in the geometry of PDEs.- Old and new Reduce software for integrability of PDEs.- Internal coordinates and total derivatives.- Conservation laws and nonlocal variables.- Cosymmetries.- Symmetries.- The tangent covering.- Recursion operators for symmetries.- Variational symplectic structures.- Cotangent covering.- Variational Poisson structures.- Recursion operators for cosymmetries.- The Plebanski equation.- Discussion.

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Joseph Krasil’shchik is a principal researcher at the Institute of Control Sciences of Russian Academy of Sciences and a full professor at the Independent University of Moscow. 
Alexander Verbovetsky is a lecturer at the Independent University of Moscow. 
Raffaele Vitolo is an associate professor in mathematical physics at the Department of Mathematics and Physics ‘E. De Giorgi’ of the Università del Salento.