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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 of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.

Chapter 1. Contact Geometry, Measurement, and Thermodynamics

Abstract
This paper has a long story and goes back to the middle of 80s but its recent version is based on the series of lectures I gave during the Summer school Wisla 18.
Valentin V. Lychagin

Chapter 2. Lectures on Geometry of Monge–Ampère Equations with Maple

Abstract
The main goal of these lectures is to give a brief introduction to application of contact geometry to Monge–Ampère equations.
Alexei Kushner, Valentin V. Lychagin, Jan Slovák

Chapter 3. Geometry of Monge–Ampère Structures

Abstract
These lectures were designed for the Summer school Wisła -18 ‘Nonlinear PDEs, their geometry, and applications’ of Bałtycki Instytut Matematyki, in Wisła, Poland, 20–30th August, 2018
Volodya Rubtsov

Chapter 4. Introduction to Symbolic Computations in Differential Geometry with Maple

Abstract
We discuss here computations in differential geometry with Maple. Our primary tool for that will be the package Differential Geometry (DG for short), which contains a lot of facilities to perform computations with vector fields, differential forms, tensors, Lie algebras, etc.
Sergey N. Tychkov

Chapter 5. On the Geometry Arising in Some Meteorological Models in Two and Three Dimensions

Abstract
Using the formalism of Monge–Ampère operators, Roubtsov and Roulstone have shown in  that a complex geometry on phase space arises naturally in some two-dimensional Hamiltonian models of nearly geostrophic flows in hydrodynamics. The aim of this note is to show how a similar approach describes the geometry associated with a variety of semi-geostrophic and quasi-geostrophic models in two and three dimensions.
Bertrand Banos, Volodya Roubtsov, Ian Roulstone

Chapter 6. Gas Flow with Phase Transitions: Thermodynamics and the Navier–Stokes Equations

Abstract
In this paper we study one-dimensional viscous gas flows described by the Navier–Stokes equations. Thermodynamics of the gas obeys the van der Waals law. This implies that phase transitions can occur along the flow of such gas. The corresponding solutions are found as asymptotic expansions with respect to parameters a and b of the van der Waals equation. The zeroth and the first-order terms are obtained by means of symmetry methods and the corresponding space-time domains of phase transitions and different phases are shown.
Anton A. Gorinov, Valentin V. Lychagin, Mikhail D. Roop, Sergey N. Tychkov

Chapter 7. Differential Invariants in Thermodynamics

Abstract
Due to the first and second law of thermodynamics, the state of a thermodynamic system is described by a Legendrian manifold. This Legendrian manifold is locally determined by the information gain function. We describe the algebra of rational differential invariants of the information gain function under the action of two different Lie groups appearing naturally as a result of measuring random vectors, and we discuss our results in the context of ideal and van der Waals gases.
Eivind Schneider

Chapter 8. Monge–Ampère Grassmannians, Characteristic Classes and All That

Abstract
We study the topology of integral Monge–Ampère grassmannians and related characteristic classes.
Valentin V. Lychagin, Volodya Roubtsov

Chapter 9. Weak Inverse Problem of Calculus of Variations for Geodesic Mappings and Relation to Harmonic Maps

Abstract
In this paper, we study the relation between geodesic and harmonic mappings. Harmonic mappings are defined between Riemannian manifolds as critical points of the energy functional, on the other hand, geodesic mappings are defined in a more general setting (manifolds with affine connections). Using the well-established formalism of calculus of variations on fibred manifolds, we solve the weak inverse problem for the equation of geodesic mappings and get a variational equation, which is a consequence of the geodesic mappings equation. For the connection on the target manifold, we get the expected result that it is a metric connection. However, we find that the connection on the source manifold need not be metric. The interesting result is that the metric, which induces the connection on the target manifold can change between fibres and these changes are related to the connection on the source manifold. These results hint onto a possibility for a more general structure on the fibred manifold, than usually assumed.
Stanislav Hronek

Chapter 10. Integrability of Geodesics of Totally Geodesic Metrics

Abstract
Analysis of the geodesics in the space of the signature (1, 3) that splits in two-dimensional distributions resulting from the Weyl tensor eigenspaces—hyperbolic and elliptic ones—described in [V.V. Lychagin, V. Yumaguzhin, Differential invariants and exact solutions of the Einstein equations, Anal. Math. Phys. 1664-235X 1–9 (2016)] is presented. The cases when geodesic equations are integrable are identified. A similar analysis is performed for the model coupled to electromagnetism described in [V.V. Lychagin, V. Yumaguzhi, Differential invariants and exact solutions of the Einstein–Maxwell equation, Anal. Math. Phys. 1, 19–29, (2017)].
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