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For the first time, exact analytical solutions are available for some of the most interesting types of problems arising in physics and applied mathematics. Hydrodynamics of Unstable Media outlines the theory of "quasi-gaseous" unstable media, and demonstrates how the theory can be used to obtain analytical solutions. Until now, the solutions to many of the examples considered in this new book have been numerical or partial solutions. By developing a new theory of hydrodynamics for unstable media, the authors provide a means for scientists and mathematicians to solve problems exactly and analytically. Solutions to some 50 problems are provided! Hydrodynamics of Unstable Media features interdisciplinary discussions by internationally recognized authors.
Table of Contents
Quasi-Chaplygin Equations and Their Solutions Introduction The Linear Approximation Symmetric Self-Similar Solutions Ellipse-like and Ellipsoid-like Solutions Three Forms of the Quasi-Chaplygin Equations General Theory in the One-Dimensional Case System Reduction to the Laplace Equation The Evolutionary Selection Principle of Spontaneous Solutions The Four Simplest Perturbations Maxima and Minima of Density Profiles Parametric Representation of the Coordinate Length-Periodic Perturbations Hill-Shaped, Well-Shaped, and Wave-Shaped Solitary Perturbations Elementary Algebraic Solutions Concerning the Flow Integral and Particle Distribution Functions A Solution to the General Cauchy Problem The Media of the Chaplygin Gas Type The Chaplygin Gas The Bunemann Plasma Instability Tearing Instability of Plasma Current Sheet Parametric Plasma Instability in an External Field Heating-Radiational Instability of Plasma in an Electric Field Media of the "Drops on the Ceiling" Type Overturned Shallow Water Instability of a Cold Surface Stream in the Ocean Self-Focusing of Light in a Nonlinear Medium Modulation Instability of the Langmuir Waves in a Plasma Self-Contraction of Wave Packets and the Lighthill Criterion Instability of Waves in Deep Water and Instability of a Gravitating Gas Slab Separation of an Electron Beam into Bunches Separation of an Electron Beam into Slabs and Filaments A Model Mechanism for the Birth of Galaxies Media with Negative Azimuthal Numbers Decay of a Liquid Jet into Drops Plasma Pinch Gas-Dynamic Particle Acceleration in a Pinch Electrodynamic Particle Acceleration in a Pinch and Neutrons Relativistic Plasma Pinch as a Source of Cosmic Rays Long-Wave Perturbations of Solitons and Solitrons The Korteweg-de Vries (KdV) Solitons The Knoidal KdV Waves Two-Dimensional Solitons of the Kadomtsev-Petviashvili Equation Solitons of Benjamin-Ono and of Sine-Gordon Equations Instability of Solitrons of the Generalized KdV Equation Geometric Optics Approximation for Solitons Solitons of the Nonlinear Schrödinger Equation (NLSE) and Solitrons of the Generalized NLSE Energy Principle of Stability in the Theory of Generalized NLSE Solitrons Nonlinear Description of Hydrodynamic Discontinuity Instabilities The Tangential Velocity Discontinuity Instability Problems on TVDI with Bounding Walls The Corrugation Instability of Shock Waves Flame Front Instability An Equation for the Front of Solidification of a Fluid Stationary Patterns on the Front of Solidification A Mechanism for Development of Column-like Separateness and "Heaps of Logs" in Rocks Supplementary Topics in the Theory of Unstable Media The Chaplygin Problem and Its Solution Method Inclusion of the Dispersion Corrections to the Chaplygin Media A List of Equations with the Hilbert Operator Tables of Formulas and Functions Short List of Results Bibliography Index
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CHOICE – Outstanding Academic Title – Award Winner
CHOICE – 2018 Outstanding Academic Title – Award Winner
Shingo Research and Professional Publication Award Winner
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