Wave phenomena are ubiquitous in nature. Their mathematical modeling, simulation and analysis lead to fascinating and challenging problems in both analysis and numerical mathematics. These challenges and their impact on significant applications have inspired major results and methods about wave-type equations in both fields of mathematics. The Conference on Mathematics of Wave Phenomena 2018 held in Karlsruhe, Germany, was devoted to these topics and attracted internationally renowned experts from a broad range of fields. These conference proceedings present new ideas, results, and techniques from this exciting research area.

This conference series intends to illuminate the relationship between different types of waves. This second conference focused primarily on classical wave modeling of acoustic waves in solids and fluids, electromagnetic waves, as well as elastic wave modeling, and both direct and inverse problems are addressed. Topics included are: (1) Classical linear wave propagation modeling, analysis and computation: general, electromagnetic applications, acoustics of fluids, acoustics of solids; (2) classical nonlinear wave propagation modeling, analysis, and computation; (3) inverse scattering modeling: gneral and electromagnetic imaging, wood imaging, seismic imaging; (4) quantum and statistical mechanics; (5) signal processing and analysis.

This book is based on an undergraduate course taught at the IAS/Park City Mathematics Institute (Utah) on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques. The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small vibrations of a taut string, and solutions are constructed via d'Alembert's formula and Fourier series.The last part of the book discusses waves arising from conservation laws. After deriving and discussing the scalar conservation law, its solution is described using the method of characteristics, leading to the formation of shock and rarefaction waves. Applications of these concepts are then given for models of traffic flow. The intent of this book is to create a text suitable for independent study by undergraduate students in mathematics, engineering, and science. The content of the book is meant to be self-contained, requiring no special reference material. Access to computer software such as MathematicaR, MATLABR, or MapleR is recommended, but not necessary. Scripts for MATLAB applications will be available via the Web. Exercises are given within the text to allow further practice with selected topics.

Mathematical modeling in the form of differential equations is a branch of applied mathematics that includes topics from physics, engineering, environmental and computer science. The mathematical model is an approximate description of real processes. Mathematical modeling can be thought of as a three step process: 1) Physical situation; 2) Mathematical formulation; 3) Solution by purely operations of the mathematical problem; 4) Physical interpretation of the mathematical solution. Over the centuries, Step 2 took on a life of its own. Mathematics was studied on its own, devoid of any contact with a physical problem; this is known as pure mathematics. Applied mathematics and mathematical modeling deals with all three steps. Improvements of approximations or their extensions to more general situations may increase the complexity of mathematical models significantly. Before the 18th century, applied mathematics and its methods received the close attention of the best mathematicians who were driven by a desire to develop approximate descriptions of natural phenomena. The goal of asymptotic and perturbation methods is to find useful, approximate solutions to difficult problems that arise from the desire to understand a physical process. Exact solutions are usually either impossible to obtain or too complicated to be useful. Approximate, useful solutions are often tested by comparison with experiments or observations rather than by rigorous mathematical methods. Hence, the authors will not be concerned with rigorous proofs in this book. The derivation of approximate solutions can be done in two different ways. First, one can find an approximate set of equations that can be solved, or, one can find an approximate solution of a set of equations. Usually one must do both. Models of natural science show that the possibilities of applying differential equations for solving problems in the disciplines of the natural scientific cycle are quite wide. This book represents a unique blend of the traditional analytical and numerical methods enriched by the authors developments and applications to ocean and atmospheric sciences. The overall viewpoint taken is a theoretical, unified approach to the study of both the atmosphere and the oceans. One of the key features in this book is the combination of approximate forms of the basic mathematical equations of mathematical modeling with careful and precise analysis. The approximations are required to make any progress possible, while precision is needed to make the progress meaningful. This combination is often the most elusive for student to appreciate. This book aims to highlight this issue by means of accurate derivation of mathematical models with precise analysis and MATLAB applications. This book is meant for undergraduate and graduate students interested in applied mathematics, differential equations and mathematical modeling of real world problems. This book might also be interested in experts working in the field of physics concerning the ocean and atmosphere.

Nonlinear Waves in Integrable and Nonintegrable Systems presents cutting-edge developments in the theory and experiments of nonlinear waves. Its comprehensive coverage of analytical and numerical methods for nonintegrable systems is the first of its kind. This book is intended for researchers and graduate students working in applied mathematics and various physical subjects where nonlinear wave phenomena arise (such as nonlinear optics, Bose-Einstein condensates, and fluid dynamics).

Mathematical Modelling of Waves in Multi-Scale Structured Media presents novel analytical and numerical models of waves in structured elastic media, with emphasis on the asymptotic analysis of phenomena such as dynamic anisotropy, localisation, filtering and polarisation as well as on the modelling of photonic, phononic, and platonic crystals.

The author uses mathematical techniques to give an in-depth look at models for mechanical vibrations, population dynamics, and traffic flow.