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2017 | Book

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The Symbolic Computation of Integrability Structures for Partial Differential Equations

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

Publisher: Springer International Publishing

Book Series : Texts & Monographs in Symbolic Computation

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents
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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Computational Problems and Dedicated Software
Abstract
In this chapter, we give an overview of the basic computational problems that arise in the study of geometrical aspects related to nonlinear partial differential equations and in the study of their integrability in particular.
We also discuss the historical development and the latest features of the Reduce software that we will use to solve the above computational problems: CDIFF, developed around 1990 by our colleagues P.K.H. Gragert, P.H.M. Kersten, G.F. Post, and G.H.M. Roelofs of the University of Twente and the CDE package, developed by one of us (RV).
Finally, we review other publicly available software that is currently used in similar computational tasks.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 2. Internal Coordinates and Total Derivatives
Abstract
We describe here a general coordinate setting that allows one to deal with computational problems arising in geometry of PDEs listed in Sect. 1.​1 and in the theory of integrable systems, in particular. For the convenience of reading, we expose in the beginning and in more detail the needed theoretical material that was concisely presented in Sect. 1.​1. The same scheme of exposition is adopted in all the forthcoming chapters.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 3. Conservation Laws and Nonlocal Variables
Abstract
We discuss here the notion of conservation laws and briefly the theory of Abelian coverings over infinitely prolonged equations. Computation of conservation laws is also closely related to that of cosymmetries , and we shall continue this discussion in Chap. 4 below. In this chapter we give the solution to Problems 1.​7, 1.​13, and 1.​15 posed in Chap. 1.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 4. Cosymmetries
Abstract
The most common use of cosymmetries is related to construction of conservation laws, because the generating functions of conservation laws are cosymmetries, but they also play an important role in the theory of the tangent (Chap. 6) and the cotangent (Chap. 9) coverings. We give the solution to Problems 1.​8, 1.​9, 1.​10 and 1.​13 in this chapter.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 5. Symmetries
Abstract
Symmetries of \(\mathbb {E}\subset J^\infty (n,m)\) are vector fields that preserve solutions of \(\mathbb {E}\). Effectively, this means that they preserve the Cartan distribution on the equation at hand. We discuss symmetries and related notions in this chapter and describe solutions to Problems 1.​5, 1.​13, and 1.​14.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 6. The Tangent Covering
Abstract
The tangent covering is an equation naturally related to the initial equation \(\mathbb {E}\) and which covers the latter and plays the same role in the category of differential equations that the tangent bundle plays in the category of smooth manifolds. It is used to construct recursion operators for symmetries of \(\mathbb {E}\) and symplectic structures on \(\mathbb {E}\). In this chapter we give the solution to Problem 1.​18 and also prepare a basis to solution of Problems 1.​20 (Chap. 7) and 1.​22 (Chap. 8).
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 7. Recursion Operators for Symmetries
Abstract
A recursion operator for symmetries of an equation \(\mathcal {E}\) is a \(\mathcal {C}\)-differential operator \(\mathcal {R}\colon \varkappa =\mathcal {F}(\mathcal {E};m) \to \varkappa \) that takes symmetries of \(\mathcal {E}\) to themselves. We expose below a computational theory of such operators based on the tangent covering techniques. The simplest version of this theory relates to local operators, but in reality all recursion operators, except for the case of linear equations with constant coefficients, are nonlocal. Such operators, in general, act on shadows of symmetries only. Unfortunately, to the best of our knowledge, a self-contained theory for these operators (as well as for nonlocal operators of other types that are considered below) does not exist at the moment, but some reasonable ideas can be applied to particular classes of examples nevertheless. In this chapter, we give the solution to Problems 1.​20 and 1.​28.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 8. Variational Symplectic Structures
Abstract
A variational symplectic structure on an equation \(\mathcal {E}\) is a \(\mathcal {C}\)-differential operator \(\mathcal {S}\colon \varkappa = \mathcal {F}(\mathcal {E};m)\to \hat {P} = \mathcal {F}(\mathcal {E};r)\) that takes symmetries of \(\mathcal {E}\) to cosymmetries and enjoys additional integrability properties. We expose here the computational theory of local symplectic structures and consider some instructive examples of nonlocal ones. In this chapter we give the solution to Problems 1.​22, 1.​23, and 1.​28.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 9. Cotangent Covering
Abstract
The cotangent covering of an equation \(\mathcal {E}\) is a natural object to define and compute Hamiltonian operators and recursion operators for cosymmetries of \(\mathcal {E}\). It is a counterpart of the cotangent bundle in the world of differential equations. In this chapter we give the solution to Problem 1.​19.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 10. Variational Poisson Structures
Abstract
A variational Poisson structure on a differential equation \(\mathcal {E}\) is a \(\mathcal {C}\)-differential operator that takes cosymmetries of \(\mathcal {E}\) to its symmetries and possesses the necessary integrability properties. In the literature on integrable systems, Poisson structures are traditionally called Hamiltonian operators. We expose here the computational theory of local variational Poisson structures for normal equations. In this chapter the solutions of Problems 1.​24, 1.​25, 1.​26, and 1.​28 is presented.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Chapter 11. Recursion Operators for Cosymmetries
Abstract
These \(\mathcal {C}\)-differential operators are somewhat dual (see Remark 11.1 below) to recursion operators for symmetries considered in Sect. 7. They send cosymmetries of an equation \(\mathcal {E}\) to themselves. (Actually, recursion operators for cosymmetries take solutions of the equation \(\tilde {\ell }_{\mathcal {E}}^*(\psi )=0\) in some covering over \(\mathcal {E}\), i.e. shadows of cosymmetries, to objects of the same nature.) Though these operators are not so popular in applications as their symmetry counterparts, we expose briefly our approach to compute them. In this chapter we give the solution to Problems 1.​27 and 1.​28.
Joseph Krasil’shchik, Alexander Verbovetsky, Raffaele Vitolo
Backmatter
Metadata
Title
The Symbolic Computation of Integrability Structures for Partial Differential Equations
Authors
Prof. Dr. Joseph Krasil'shchik
Alexander Verbovetsky
Prof. Dr. Raffaele Vitolo
Copyright Year
2017
Electronic ISBN
978-3-319-71655-8
Print ISBN
978-3-319-71654-1
DOI
https://doi.org/10.1007/978-3-319-71655-8

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