Advanced Methods of Mathematical Physics

In an introductory style with many examples, Advanced Methods of Mathematical Physics presents some of the concepts, methods, and tools that form the core of mathematical physics. The material covers two main broad categories of topics: 1) abstract topics, such as groups, topology, integral equations, and stochasticity, and 2) the methods of nonlinear dynamics.

Table of Contents:
GENERAL INTRODUCTION THEORY OF FINITE GROUPS A Brief Review of Set Theory Abstract Groups Homomorphisms Group Representations Introduction to Continuous Groups Applications to Physical Problems RUDIMENTS OF TOPOLOGY AND DIFFERENTIAL GEOMETRY Preliminaries Metric Spaces Topological Spaces Compactness Connectedness Homotopy Essentials of Differential Geometry INTEGRAL EQUATIONS, STURM-LIOUVILLE THEORY AND GREEN'S FUNCTIONS Terminology and Definitions Solution of Integral Equations Sturm-Liouville Theory The Green's Functions Applications to Physics Problems STOCHASTIC PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS Random Variable and Distribution Function of Random Variables Numerical Characteristics of Random Variables: Moments of the Distribution Function Stochastic Processes: Markov Process Stochastic Differential Equations: An Introduction Applications to Physical Problems METHODS OF NONLINEAR DYNAMICS I: PHASE PORTRAITS A Brief Survey of Nonlinear Operators and Differential Equations Solution of Nonlinear Differential Equations: Existence and Uniqueness Theorems Critical Point Analysis of Differential Equations Nonlinear Systems in the Plane METHODS OF NONLINEAR DYNAMICS II: STABILITY AND BIFURCATION Stability of Critical Points and Liapunov Functions Limit Cycle Index of a Critical Point and Bendixson Criterion for Periodic Solutions Bifurcation and Structural Stability Applications to Physical Problems SOME NONLINEAR DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS Van der Pol Equation Solitary-Wave Solutions of Nonlinear Differential Equations Solitary-Wave Solution of Nonlinear Schrodinger Equation Application to Physical Problems SOME NONLINEAR INTEGRAL EQUATIONS AND THEIR SOLUTIONS Inverse Scattering Transform Method Backlund Transformation and the Solution of KdV Equation The Lax Pair Method Hirota's Method of Bilinear Derivatives Painleve' Property and Painleve' Transcendents Kadomstev-Petviashvili (KP) Equation and Its Solutions Solution of Some Nonlinear Integral Equations EXACT SOLUTION OF SOME NONLINEAR DIFFERENTIAL EQUATIONS Riccati Equation Exact Solution of Some Other NLODEs Nonlinear diffusion Equations Exact Solution of Some Other NLPDEs Applications to Physical Problems SYMMETRIES OF DIFFERENTIAL EQUATIONS Symmetry Groups of Differential Equations: An Introduction Extended Transformations or Prolongations Extended Infinitesimal Transformations Invariance of an Ordinary Differential Equation Invariance of a Partial Differential Equation Symmetry Groups and Conservation Laws Noether's Theorem and Lie-Backlund Symmetries NORMAL MODES IN NONLINEAR DYNAMICAL SYSTEMS Normal Modes of Linear Systems: A Brief Survey Normal Modes of Nonlinear Systems: A Simple Generalization Types of Mode Interactions: A Group Theoretic Approach Bushes of Modes for a Dynamical System Dynamical Bush Equations: Normal Forms Theory APPENDIX: SOME NUMERICAL ASPECTS OF NONLINEAR DYNAMICAL SYSEMS VIS-À-VIS CHAOS