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2024 | Book

Equations of Fluid Dynamics and the Shallow Water Approximation Read chapter Read first chapter

Analytical Properties of Nonlinear Partial Differential Equations

with Applications to Shallow Water Models

Authors: Alexei Cheviakov, Peng Zhao

Publisher: Springer International Publishing

Book Series : CMS/CAIMS Books in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Nonlinear partial differential equations (PDE) are at the core of mathematical modeling. In the past decades and recent years, multiple analytical methods to study various aspects of the mathematical structure of nonlinear PDEs have been developed. Those aspects include C- and S-integrability, Lagrangian and Hamiltonian formulations, equivalence transformations, local and nonlocal symmetries, conservation laws, and more. Modern computational approaches and symbolic software can be employed to systematically derive and use such properties, and where possible, construct exact and approximate solutions of nonlinear equations. This book contains a consistent overview of multiple properties of nonlinear PDEs, their relations, computation algorithms, and a uniformly presented set of examples of application of these methods to specific PDEs. Examples include both well known nonlinear PDEs and less famous systems that arise in the context of shallow water waves and far beyond. The book will beof interest to researchers and graduate students in applied mathematics, physics, and engineering, and can be used as a basis for research, study, reference, and applications.

Table of Contents

Frontmatter
Chapter 1. Equations of Fluid Dynamics and the Shallow Water Approximation
Abstract
In this section, we briefly summarize some common notation used in this book:
Alexei Cheviakov, Peng Zhao
Chapter 2. Integrability and Other Analytical Properties of Nonlinear PDE Systems
Abstract
Integrability of a nonlinear partial differential equation or a system of such equations, in its basic sense, refers to the possibility of construction of a solution for every initial/boundary value problem within a certain class of problems. However, no common definition of integrability has been recognized to date. In particular, a model that possesses advanced analytical properties may be called integrable (or conjectured to be integrable) in a certain specific sense.
Alexei Cheviakov, Peng Zhao
Chapter 3. Shallow Water Models and Their Analytical Properties

In this chapter, we review several well known PDE models that arise in the context of shallow water modeling and in multiple other contexts.

Alexei Cheviakov, Peng Zhao
Backmatter
Metadata
Title
Analytical Properties of Nonlinear Partial Differential Equations
Authors
Alexei Cheviakov
Peng Zhao
Copyright Year
2024
Electronic ISBN
978-3-031-53074-6
Print ISBN
978-3-031-53073-9
DOI
https://doi.org/10.1007/978-3-031-53074-6

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