The symbolic computation of integrability structures for partial differential equations

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The Springer's page of the book

Authors: Joseph Krasil'shchik, Alexander Verbovetsky, and Raffaele Vitolo.

Published in the Springer series Texts and Monographs in Symbolic Computations, 2018. The Springer's page of the book.

In this book the authors present a unified mathematical approach for the symbolic computation of integrability structures of partial differential equations. By "integrability structures" we mean one of the following: Hamiltonian operators, symplectic operators, recursion operators for symmetries, recursion operators for cosymmetries.

The computations are carried out within the computer algebra system Reduce by the packages CDE and CDIFF.

Many example programs are discussed in the book, you can find here a zip file of all examples. The examples concern the computation of:

  • linearization of a PDE and its adjoint
  • higher (or generalized) symmetries
  • conservation laws and cosymmetries
  • hamiltonian operators
  • Schouten bracket of hamiltonian operators (only local operators at the moment)
  • symplectic operators (the symplectic property can only be checked for local operators at the moment)
  • recursion operators for symmetries
  • Nijenhuis tensor of recursion operators for symmetries (only local operators at the moment)
  • recursion operatos for cosymmetries.

The above computations are performed for well-known examples of non-linear partial differential equations in two independent variables, like

  • Burgers
  • Korteweg-de Vries
  • dispersionless Boussinesq
  • Camassa-Holm
  • Gibbons-Tsarev
  • Witten-Dijkgraaf-Verlinde-Verlinde

and in more than two independent variables, like

  • Khokhlov-Zabolotskaya
  • Kadomtsev-Petviashvili
  • Pavlov
  • Universal Hyerarchy
  • rdDym
  • Plebanski
  • General Heavenly
  • Husain