# The symbolic computation of integrability structures for partial differential equations

**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