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Computational Problems and Dedicated Software

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The Symbolic Computation of Integrability Structures for Partial Differential Equations

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

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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.

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Notes

  1. 1.

    The directory names follow the Linux convention; this means that they have slashes / instead of backslashes ∖ like in Windows.

  2. 2.

    Not necessarily trivial, actually, but we consider the simplest case here.

  3. 3.

    Everywhere below we, as a rule, use a shorter notation D i for \(D_{x^i}\).

  4. 4.

    Strictly speaking, not all these conditions are essential for all constructions and computations below, but we prefer not to go into unnecessary technical details.

  5. 5.

    With respect to the projection π ∞ .

  6. 6.

    Here and below, the notation \({\widehat {\cdot }}\) means that the corresponding term is omitted.

  7. 7.

    Traditionally, Poisson structures on \(\mathbb {E}\) are called Hamiltonian operators in the theory of integrable systems.

  8. 8.

    In principle, the package allows to deal with odd independent variables too (see, e.g., [54, 55, 59], but we do not discuss these problems here.).

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Authors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences RAS, Independent University of Moscow, Moscow, Russia

    Joseph Krasil’shchik

  2. Independent University of Moscow, Moscow, Russia

    Alexander Verbovetsky

  3. Department of Mathematics and Physics ‘E. De Giorgi’, University of Salento, Lecce, Italy

    Raffaele Vitolo

Authors
  1. Joseph Krasil’shchik
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  2. Alexander Verbovetsky
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  3. Raffaele Vitolo
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Krasil’shchik, J., Verbovetsky, A., Vitolo, R. (2017). Computational Problems and Dedicated Software. In: The Symbolic Computation of Integrability Structures for Partial Differential Equations. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-71655-8_1

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