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Uspekhi Mat. Nauk, 2020, Volume 75, Issue 3(453), Pages 55–106 (Mi umn9947)  

This article is cited in 2 scientific papers (total in 2 papers)

Quadratic conservation laws for equations of mathematical physics

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the focus of interest in this paper: the existence of other quadratic integrals, the Hamiltonian property of a linear system, and the complete integrability of such a system. For non-degenerate linear systems in a finite-dimensional space essentially exhaustive answers to all these questions are known. Results of a general nature are applied to linear evolution equations of mathematical physics: the wave equation, the Liouville equation, and the Maxwell and Schrödinger equations.
Bibliography: 60 titles.

Keywords: linear systems, Hilbert space, Hamiltonian system, quadratic invariants, Poisson bracket, equations of mathematical physics.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).


DOI: https://doi.org/10.4213/rm9947

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English version:
Russian Mathematical Surveys, 2020, 75:3, 445–494

Bibliographic databases:

UDC: 517.91+517.95+530.145
MSC: Primary 34G10, 37K05, 37K10; Secondary 35G05, 35G35, 35L65
Received: 05.03.2020

Citation: V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Uspekhi Mat. Nauk, 75:3(453) (2020), 55–106; Russian Math. Surveys, 75:3 (2020), 445–494

Citation in format AMSBIB
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\pages 445--494
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  • http://mi.mathnet.ru/eng/umn/v75/i3/p55

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Kozlov, “Quantization of linear systems of differential equations with a quadratic invariant in a Hilbert space”, Russian Math. Surveys, 76:2 (2021), 357–359  mathnet  crossref  crossref  isi  elib
    2. V. M. Buchstaber, A. V. Mikhailov, “Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves”, Russian Math. Surveys, 76:4 (2021), 587–652  mathnet  crossref  crossref  isi
  • Успехи математических наук Russian Mathematical Surveys
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