Valery Dryumadoctor of physical and mathematical sciences

Born on 25 July.

Analytic solution of the two-dimensional Korteweg — de Vries equation. He was twenty-seven; the paper has been cited for fifty years, 171 times. Scholarpedia, in its article on the Kadomtsev — Petviashvili equation, records it: “In 1974 Valery Dryuma showed how the KP equation could be written in Lax form, providing a strong hint that the equation was integrable.” His name comes first, before Zakharov and Shabat. The editors of JETP Letters have placed the paper in their “Golden Fund” — a selected collection of the most significant papers in the journal’s history. In the note he wrote for that collection himself, he describes what he did: “The author established for the first time that the scope of the inverse scattering method can be substantially extended, and that it can be applied successfully to the exact integration of multidimensional nonlinear differential equations.”

On solutions of the cylindrical Korteweg — de Vries equation.

Group-theoretical interpretation of integrable nonlinear equations.

Doctor of Physical and Mathematical Sciences. Field: differential equations and dynamical systems.

Integration of the cylindrical — that is, axisymmetric — Kadomtsev — Petviashvili equation by the inverse scattering method. Printed on the first page: “Communicated by Academician S. P. Novikov, 10 May 1982.” Only a full member of the Academy could submit to Doklady, vouching for the paper with his own name; Sergei Novikov is the only Soviet mathematician to have won the Fields Medal and the founder of the algebro-geometric theory of integrable systems.

A lecture, “Finsler’s metrics of phase space of nonlinear dynamical systems and their applications”, at the International Conference on Differential Geometry and its Applications. On the organising committee: Makoto Matsumoto, head of the Japanese school of Finsler geometry. Their eight-year collaboration begins here.

Geometry of second-order equations and Finsler metrics. The central theme begins here: translating equations into the language of geometry.

Mathematisches Institut, Universität Leipzig — joint work with L. A. Bordag. The same year: Dipartimento di Fisica and INFN Lecce, Consortium EINSTEIN, work with B. G. Konopelchenko. From then until 2004 his papers appear with the support of MURST Italy, the European INTAS grants, the Cariplo Foundation in Milan, the Physical Department of Sapienza in Rome and the University of Lecce — whose support, in his own words in print, “allowed him to continue his scientific activity”.

“The theory of second-order differential equations based on Finsler geometry”, with Makoto Matsumoto, in the journal of Kyoto University. Its opening paragraph records how they met in Bucharest and that “the intimate contacts between the present authors have continued” ever since.

Riemannian and Einstein — Weyl geometry in the theory of second-order ordinary differential equations. A chapter in a Kluwer volume.

“Is my ODE a Painlevé equation in disguise?” — joint work with Jarmo Hietarinta. A point transformation changes an equation beyond recognition, so telling whether one is secretly a Painlevé equation is close to impossible. They build an invariant test — a quantity unchanged by any such transformation — and give it for Painlevé I to IV. The machinery is R. Liouville’s theory of invariants, the same he uses twenty-three years later to draw an elliptic curve out of a fluid flow. In the acknowledgements he thanks the Physics Department of the University of Turku for financial support and hospitality.

A lecture, “On the Riemann extension of the space-time with vanishing curvature invariants”, at the Fourth European Congress of Mathematics, 27 June – 2 July. The Congress meets once every four years and is second only to the International Congress in standing.

Toward a theory of spaces of constant curvature. Two preprints of the same year build four- and ten-dimensional Einstein spaces on a six-dimensional Ricci-flat base: the same geometric machinery, turned towards the equations of gravitation.

On spaces related to the Navier — Stokes equations. The first paper on the theme he is still working on.

The Ricci-flat spaces related to the Navier — Stokes equations.

His first talk at the scientific session of the Russian Academy of Sciences Council on Nonlinear Dynamics, held at the Shirshov Institute of Oceanology: on solving the equations of incompressible flow in Lagrangian variables. He has spoken there again in 2019, 2020, 2022, 2023 and 2025. The same year he lectured at the Steklov Mathematical Institute, at the conference for L. I. Sedov’s 110th anniversary; a year earlier, at the conference for D. V. Anosov’s 80th birthday.

On limit cycles of polynomial systems of the first order — this is Hilbert’s sixteenth problem, open since 1900. The same construction, turned on another famous question.

A talk on the geometric properties of the Navier — Stokes equations at the international conference on laser and plasma research at the National Research Nuclear University MEPhI. The fourteen-dimensional Ricci-flat metric is presented to a Russian audience for the first time: from conditions on the curvature tensor and the geodesic equations he derives the equation of a hypersurface that matters for the topology of the solutions.

A talk, “The Riemann spaces related to the Navier-Stokes equations”, at the conference in memory of Academician N. N. Yanenko at Suranaree University of Technology. Thailand runs these conferences together with the Institute of Theoretical and Applied Mechanics in Novosibirsk and the Keldysh Institute of Applied Mathematics. He speaks at them by Zoom in 2021, 2022, 2024 and 2026.

A satellite conference of the International Congress of Mathematicians 2022: the “Lobachevsky Readings” at Kazan Federal University, in the institute that carries Lobachevsky’s own name. His talk: a geometric approach to the Navier — Stokes system, using Cartan’s invariants to build a three-dimensional generalisation of the classical Taylor — Green vortex.

Geodesic lines of the fourteen-dimensional metric that is Ricci-flat on solutions of the Navier — Stokes equations: talks at the online conference AGMA-2023, at the international AMiTaNS conference in Albena, Bulgaria, and at Maple Conference 2023, where the organiser, Maplesoft, published his slides. In December, the RAS Council on Nonlinear Dynamics once more.

A theorem on a fourteen-dimensional metric, Ricci-flat on solutions of the Navier — Stokes equations, and its application to the rotation of a rigid body. Nine papers and talks in one year: Thailand, Irkutsk, Odesa, Chișinău, Bremen (Dynamics Days Europe), Yibin in China (NSC 2024), Yaroslavl (Integrable Systems and Nonlinear Dynamics, co-organised by the Steklov Institute) and Salt Lake City.

Multidimensional Riemannian metrics for integrating the Navier — Stokes equations. Talks in Yaroslavl and at the 34th session of the RAS Council on Nonlinear Dynamics, where he builds examples of solutions from theta functions. Eighty years old. The work continues.

A talk at Yanenko 105, Suranaree University of Technology: the fourteen-dimensional space whose Ricci tensor vanishes on solutions of the Navier — Stokes equations, its geodesics, and the path they open to linear equations of Schrödinger type.

The survey “Riemann spaces associated with the Navier-Stokes equations: a survey of the geometric approach” is accepted on 10 October by the journal Transport Phenomena, DOI 10.1515/tp-2026-0119. One text gathers the geometric approach he has developed over fifteen years: a fourteen-dimensional space, Ricci-flat precisely on solutions of the Navier — Stokes equations. Preprint: arXiv:2609.13314.