Institute of Mathematics and Computer Science · Chișinău
Valery Dryumadoctor of physical and mathematical sciences
For fifty years he has done one thing: translating nonlinear equations into the language of geometry.
…that I may know what holds the world together at its innermost coreGoethe, Faust
He has been looking for that answer for sixty years, and he writes it down as the curvature of space.
Valery Dryuma constructs Riemannian spaces whose curvature is arranged so that it reproduces exactly the behaviour of the solutions of a nonlinear equation. A problem from fluid physics or soliton theory becomes a problem about geometry — and the reverse.
His first paper appeared in 1973, when he was twenty-seven. The most recent one in 2025. Between them lie more than a hundred publications and a single unbroken line of work.
My name is Zoya Belinskaya. Valery Dryuma is my father. I built this site: he does not talk about his own work.
In 1973 he published a solution of the two-dimensional Korteweg — de Vries equation in JETP Letters. He was twenty-seven. Mathematicians still cite that paper — 171 times over fifty years. He turned eighty in July.
He lives in Chișinău and goes to the Academy of Sciences every day, without days off. It has been that way all his life: work took the place of everything else. His most recent paper appeared in 2025.
He worked at the Mathematisches Institut of the University of Leipzig and at the physics department in Lecce. For years his papers appeared with the support of MURST in Italy, then RFBR, then DAAD. All of that has ended.
He is eighty now, in Chișinău, and nothing of that network remains. The manuscripts sit unpublished. Letters to institutes and journals come back as automated replies. Submission has changed beyond recognition in thirty years, and there is no money for conference fees or travel.
I am not a mathematician and I will not judge his results. What I can do is gather everything he has done in one place — the papers, the years, the citations — so that it can be found and read. This site is that place.
After that it takes a reader. If you work on the geometry of nonlinear equations, Riemann extensions or the Navier — Stokes equations, there is something here to open: 56 papers, 46 of them downloadable in full and free of charge. The most recent one appeared this year.
Read one. If it turns out to be close to your own work, write to me — a question, an objection, a disagreement. He will answer himself: a conversation about his subject is the one thing he no longer has.
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 references — and remains his best known work.
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.
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.
Integration of the cylindrical Kadomtsev — Petviashvili equation by the inverse scattering method.
Geometry of second-order equations and Finsler metrics. The central theme begins here: translating equations into the language of geometry.
Riemannian and Einstein — Weyl geometry in the theory of second-order ordinary differential equations. A chapter in a Kluwer volume.
Toward a theory of spaces of constant curvature.
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.
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. Three papers in one year.
Multidimensional Riemannian metrics for integrating the Navier — Stokes equations. Eighty years old. The work continues.
The equations of a fluid, written as a condition on the curvature of a space.
Whether the solutions of the Navier — Stokes equations always remain smooth has been an open question since the nineteenth century, and it is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.
Dryuma’s approach is geometric. He constructs Riemannian spaces arranged so that they turn out to be Ricci-flat exactly when the velocity and pressure functions satisfy the Navier — Stokes equations. The behaviour of the fluid becomes a property of curvature.
Metrics have been constructed in six, eight, twelve and fourteen dimensions — for Navier — Stokes, for the Euler equations, and for the Kadomtsev — Petviashvili equation.
In the 2024 paper the fourteen-dimensional space is written out explicitly and shown to be Ricci-flat on solutions of the system. It splits into a flat six-dimensional part and two dual quadruples of coordinates — Eulerian and Lagrangian: the two descriptions of a fluid turn out to be two halves of one geometry. The same paper applies the method to the rotation of a rigid body, up to the Kovalevskaya top.
The author’s own words belong here: what has been built, where it leads, and what is still missing. The space is left open on purpose.
| 2001 | Is my ODE a Painleve equation in disguise?препринт arXiv | |
| 2006 | On the dressing method for Dunajski anti-self-duality equationпрепринт arXiv | |
| 2008 | Towards the theory of Benney equationsпрепринт arXiv | |
| 2014 | Six-dimensional spaces defined by the equations of KN and KdVпрепринт arXiv |
| 1983 | Об интегрировании цилиндрического уравнения Кадомцева–Петвиашвили методом обратной задачи теории рассеянияMath-Net.Ru | |
| 1994 | Geometrical properties of the multidimensional nonlinear differential equations and the finsler metrics of phase spaces of dynamical systemsMath-Net.Ru | |
| 1997 | Investigation of dynamical systems using tools of the theory of invariants and projective geometryпрепринт arXiv | |
| 1997 | On equation of geodesic deviation and its solutionsBulletin of Moldavian Academy of Sciences, ser. math. N3, (1996) 31-48 | |
| 1998 | Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical Systemsпрепринт arXiv | |
| 1998 | On the Law of Transformation of Affine Connection and its Integration. Part 1. Generalization of the Lame equationsBuletinul Academiei de Stiinte a Republicii Moldova Matematica, v.1(26), 1998, p.55-68 | |
| 2001 | Приложения римановой геометрии и геометрии Эйнштейна–Вейля в теории обыкновенных дифференциальных уравнений второго порядкаMath-Net.Ru | |
| 2002 | On Equation for Initial Values in Theory of the Second Order Ordinary Differential Equationsпрепринт arXiv | |
| 2005 | On geometrical properties of the spaces defined by the Pfaff equationsBuletinul Academiei de Stiintse a Republicii Moldova, matematica, No. 1(47), 2005 | |
| 2005 | On geometrical properties of the spaces defined by the Pfaff equationsMath-Net.Ru | |
| 2005 | On the theory of spaces of constant curvatureпрепринт arXiv | |
| 2006 | К теории пространств постоянной кривизныMath-Net.Ru | |
| 2007 | On dual equation in theory of the second order ODE'sпрепринт arXiv | |
| 2007 | On geometry of gonometric family of cyclesпрепринт arXiv | |
| 2007 | On solutions of Rashevskii equationпрепринт arXiv | |
| 2007 | On the Equations of Nonstationary Transonic Gas Flowsпрепринт arXiv | |
| 2008 | On Geometry of the Rössler system of equationsпрепринт arXiv | |
| 2008 | On nonlinear equations connected with six-dimensional Plebanski spaceпрепринт arXiv | |
| 2008 | Riemann geometry in theory of the first order systems of equationsInternational Conference "Differential Equations and Topology", Abstracts, Moscow, June 17-22, p.35-36 | |
| 2010 | On nonlinear equations associated with developable, ruled and minimal surfacesпрепринт arXiv | |
| 2014 | On vector field generated by the Hopf map $S^3$ on $S^2$препринт arXiv | |
| 2014 | The Monge equation and topology of solutions of the second order ODEsConference of Mathematical Society of the Republic of Moldova, Chișinău | — |
| 2019 | On limit cycles of polynomial systems of the first-order ODEsBul. Acad. Ştiinţe Repub. Mold. Mat., 2(90), 113–126 | — |
| 2022 | Invariants of E. Cartan and their applications to the theory of differential equationsConference on Applied and Industrial Mathematics, CAIM 2022, Chișinău | — |
| 2001 | On the Riemannian and Einstein-Weyl Geometry in Theory of the Second Order Ordinary Differential EquationsBul.Acad.Sti.Rep.Moldova (Fiz.Teh.) 3 (1999) 95-102 | |
| 2003 | On Riemann extension of the Schwarzschild metricMath-Net.Ru | |
| 2003 | The Riemann Extensions in Theory of Ordinary Differential Equations and their ApplicationsTheoretical and Mathematical Physics, v.128 (1), (2001), 15-26 | |
| 2003 | The Riemann extensions in theory of differential equations and their applicationsMath-Net.Ru | |
| 2004 | On the Riemann Extension of the Schwarzschild MetricBul.Acad.Sti.Rep.Moldova (Fiz.Teh.) 3 (2003) 92-103 | |
| 2005 | On the Riemann Extension of the Gödel Space-Time metricпрепринт arXiv | |
| 2005 | On the Riemann extension of rotating Mikowsky space-time metricпрепринт arXiv | |
| 2005 | On the Riemann extension of the Gödel space-time metricMath-Net.Ru | |
| 2005 | Riemann extensions in theory of the first order systems of differential equationsпрепринт arXiv | |
| 2006 | 10-Dim Einstein spaces made up on basis of 6-Dim Ricci-flat spaces and 4-Dim Einstein spacesпрепринт arXiv | |
| 2006 | Around a theory of a Walker spacesпрепринт arXiv | |
| 2006 | Four-dimensional Einstein spaces on Six-dimensional Ricci-flat base spaceпрепринт arXiv | |
| 2006 | On solutions of a Heavenly equations and their generalizationsпрепринт arXiv | |
| 2007 | Dunajski generalization of the second heavenly equation: dressing method and the hierarchyJournal of Physics A, 40 (2007) 14383-14393 | |
| 2008 | Eight-dimensional Ricci-flat space related with the KP equationпрепринт arXiv | |
| 2009 | Multidimensional the Ricci-flat spaces defined by nonlinear equationsпрепринт arXiv | |
| 2011 | On the equations defining the Ricci-flows of manifoldsпрепринт arXiv | |
| 2017 | Duality and a Riemann metrics in theory of a second order ODEsConference of Mathematical Society of the Republic of Moldova, Chișinău | — |
| 2023 | Riemann and projective spaces in theory of the first and second order ODEsMathematics and Information Technologies: Research and Education, Chișinău | — |
| 2024 | The 14D Ricci-flat metric in the theory of fluid flows and the equations of rotation of a topNonlinear Analysis and Extremal Problems, Irkutsk | — |
| 2024 | The application of 14D Ricci-flat metrics to the equations of rotation of a topBul. Acad. Ştiinţe Repub. Mold. Mat., 3(106), 137–141 · DOI 10.56415/basm.y2024.i3.p137 |
This site is dedicated to my father and to his unfinished search.