Nonlinear PDEs, Their Geometry, and Applications

1. Front Matter

   Pages i-xvii

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2. Lectures

   1. Front Matter

      Pages 1-1

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   2. Contact Geometry, Measurement, and Thermodynamics

      • Valentin V. Lychagin

      Pages 3-52

   3. Lectures on Geometry of Monge–Ampère Equations with Maple

      • Alexei Kushner, Valentin V. Lychagin, Jan Slovák

      Pages 53-94

   4. Geometry of Monge–Ampère Structures

      • Volodya Rubtsov

      Pages 95-156

   5. Introduction to Symbolic Computations in Differential Geometry with Maple

      • Sergey N. Tychkov

      Pages 157-182

3. Participants Contributions

   1. Front Matter

      Pages 183-183

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   2. On the Geometry Arising in Some Meteorological Models in Two and Three Dimensions

      • Bertrand Banos, Volodya Roubtsov, Ian Roulstone

      Pages 185-207

   3. Gas Flow with Phase Transitions: Thermodynamics and the Navier–Stokes Equations

      • Anton A. Gorinov, Valentin V. Lychagin, Mikhail D. Roop, Sergey N. Tychkov

      Pages 209-222

   4. Differential Invariants in Thermodynamics

      • Eivind Schneider

      Pages 223-232

   5. Monge–Ampère Grassmannians, Characteristic Classes and All That

      • Valentin V. Lychagin, Volodya Roubtsov

      Pages 233-257

   6. Weak Inverse Problem of Calculus of Variations for Geodesic Mappings and Relation to Harmonic Maps

      • Stanislav Hronek

      Pages 259-265

   7. Integrability of Geodesics of Totally Geodesic Metrics

      • Radosław A. Kycia, Maria Ułan

      Pages 267-275

4. Back Matter

   Pages 277-279

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This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations.


Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge ofdifferential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.

• Nonlinear Partial Differential Equations
• Probability theory
• Gaussian distribution
• Summer School Wisla
• Monge-Ampere Equations
• Contact geometry
• Tensor invariants
• Recursion operators
• Navier-Stokes equations
• Geometry of Nonlinear PDEs
• Baltic Institute of Mathematics
• Thermodynamics
• Meterology
• Maple mathematics
• partial differential equations

• The Faculty of Science, Masaryk University, Brno, Czech Republic

  Radosław A. Kycia

• Baltic Institute of Mathematics, Warszawa, Poland

  Maria Ułan

• Department of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, Norway

  Eivind Schneider

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