Advanced Methods of MATHEMATICAL PHYSICS R.S. Kaushal D. Parashar Alpha Science International Ltd.
Contents Preface Abbreviations, Notations and Symbols vii xi 1. General Introduction 1 2. Theory of Finite Groups 5 2.1 A Brief Review of Set Theory 6 Cartesian products 6 Mappings 7 Binary compositions 7 Composition tables 8 2.2 Abstract Groups 8 Preliminaries 8 Cyclic groups 10 Permutation group (S,,) 12 Groups of symmetry 14 Conjugate elements and classes 18 Subgroups 19 Center of a group 20 Cosets 20 Normal subgroups 22 Factor groups 23 2.3 Homomorphisms 24 Automorphisms 26 Inner and outer automorphisms 27 Group of automorphisms 27 Direct product of groups 28 Semidirect product of groups 29 2.4 Group Representations 30 Invariant subscpaces 32 Reducible representations 32 Irreducible representations 34 Unitary representations 36 Schur's lemmas 37 The orthogonality theorem 39 Characters of a representation 41 2.5 Introduction to Continous Groups 42 Lie groups 43 Lie groups of transformations 44 Infinite continous groups 45 Generators of a Lie group 45
xiv Contents- 2.6 Applications to Physical Problems 48 Permutation group (S,,) 48 Unitary group (SU(»)) 5/ Symmetry group of a square 55 2.7 Summary and Further Reading 59 Problems 60-3. Rudiments of Topology and Differential Geometry 63 3.1 Preliminaries 63 Denumerable and countable sets 64 Lower and upper bounds 66 Neighbourhoods, open and closed sets 66 Continuity 68 Limit points 69 Bolzano-Weierstrass theorem 70 Isolated, dense and perfect sets 71 3.2 Metric Spaces 72 Euclidean space 73 Hilbert space 74 Distance between sets 74 Open and closed spheres 75 Equivalence of metric spaces 76 3.3 Topological Spaces 76 Definition 76 Union and intersection of topologies 78 Limit points 79 Closure of a set 79 Interior, exterior and boundary of a set 80 Base for a topology 82 Housdorff spaces 82 Relative topologies 82 3.4 Compactness 83 Some definitions 83 Heine-Borel theorem 84 3.5 Connectedness 86 Separated sets 86 Connected sets 86 Connected spaces 87 3.6 Homotopy 88 Homotopic paths 88 Simply connected spaces 90 The fundamental group 91 3.7 Essentials of Differential Geometry 94 Some basic concepts and definitions 94 Differentiable manifolds 96 Diffeomorphism 98 Vector fields 98 Differential forms 101 3.8 Summary and Further Reading 103 Problems 104
Contents xv 4. Integral Equations, Sturm-Liouville Theory and Green's Functions 107 4.1 Terminology and Definitions 107 Fredholm integral equations 108 Volterra integral equations 109 Differentiation of a function under the integral sign 110 Relation between differential and integral equations 111 4.2 Solution of Integral Equations 113 The Liouville-Neumann series method 113 The Fredholm method 124 The Hilbert-Schmidt theory 134 A3 Sturm-Liouville Theory 142 Adjoint differential sytstem 143 The Sturm-Liouville problems: Eigenvalues and eigenfunctions 145 4.4 The Green's Functions 148 Determination of G(x, t) 149 Connection with inhomogeneous Sturm-Liouville equation 152 4.5 Applictions to Physics Problems" 153 The influence function 154 The Abel's integral equation 156 4.6 Summary and Further Reading 158 Problems 159 5. Stochastic Processes and Stochastic Differential Equations 162 5.1 Random Variable and Distribution Function of Random Variables 163 Some basic definitions and results 163 Multidimensional distribution functions 168 Functions of random variables 171 The Stieltjes integral 174 5.2 Numerical Characterstics of Random Variables: Moments of the distribution function 176 Mathematical expectation 176 Variance 178 Covariance and covariance matrix 182 Characteristic function of random variables 185 5.3 Stochastic Processes: Markov process 756 Classification of stochastic processes 187 Markov processes: Fokker-Planck equation 191 General theory of continuous (Markov) processes 193 5.4 Stochastic differential equations: an introduction 199 5.5 Applications to physical problems 203 5.6 Summary and further reading 205 Problems 208 6. Methods of Nonlinear Dynamics I: Phase Portraits 212 6.1 A Brief Survey of Nonlinear Operators and Differential Equations 213
xvi Contents- 6.2 Solution of nonlinear Differential Equations: Existence and Uniqueness Theorems 275 6.3 Critical Point Analysis of Differential Equations 223 Generalization to the case of n variables 223 Definitions: linear systems 224 Definitions (continued): A two dimensional linear system 225 Further remarks on linear systems 237 6.4 Nonlinear Systems in the Plane 238 Linearization at a critical point 239 Volterra-Lotka system 241 General remarks 244 Problems 245 7. Methods of Nonlinear Dynamics II: Stability and Bifurcation 249 7.1 Stability of Critical Points and Liapunov Functions 249 Stability for non-autonomous systems 249 Stability for autonomous systems 253 Liapunov functions 256 Stability and linear aproximation in two dimensions 261 7.2 Limit Cycle 264 7.3 Index of a Critical Point and Bendixson Criterion for Periodic Solutions 265 Index of a critical point 268 Bendixson's criterion for periodic solutions 270 7.4 Bifurcation and Structural Stability 273 Phenomenon of bifurcation 273 One-dimensional bifurcation 277 Hopf bifurcation 281 Structural stability 283 Chaos and stange attractor 284 7.5 Applications to Physical Problems 257 Conservative systems 287 Hamiltonian systems 294 7.6 Summary and Further Reading 297 Problems 297 8. Some Nonlinear Differential Equations and their Solutions 300 8.1 Van der Pol Equation 300 Lienard systems and Vab der pol equation 300 Dependence of the solution on the parameter e 303 Large parameter behaviour of the solution 304 8.2 Solitary-Wave Solutions of Nonlinear Differential Equations 310 A brief introduction 310 Korteweg-de Vries (KdV) equation and its solutions 313 Some remarks about KdV and modified KdV equations 325
8.3 Solitary-Wave Solution of Nonlinear Schrodinger Equation 326 8.4 Application to Physical Problems 329 Applications of Van der Pol equation 329 Applictions of KdV and NLS equations 333 8.5 Summary and Further Reading 334 Problems 334 - Contents xvii 9. Some Nonlinear Integral Equations and their Solutions 339 9.1. Inverse Scattering Transform Method 339 A brief introduction 339 Two typical examples of u(x) 342 Inverse scattering problem 346 Connection with the KdV equation 349 9.2 Backlund Transformation and the Solution of KdV Equation 352 9.3 The Lax Pair Method 357 9.4 Hirota's Method of Bilinear Derivatives 367 9.5 Painleve' Property and Painleve' Transcendents 364 9.6 Kadomstev-Petviashvili (KP) Equation and its Solutions 367 9.7 Solution of Some Nonlinear Integral Equations 368 Existence theorems for NL integral equations 368 Some representative nonlinear integral equations 370 9.8 Summary and Further Reading 376 Problems 377 10. Exact Solution of Some Nonlinear Differential Equations 378 10.1 Riccati Equation 378 Riccati equation and the linear differential equation of second order 379 Solution of the original Riccati equation 381 Further remarks on Riccati equation 383 10.2 Exact Solution of Some Other NLODEs 355 10.3 Nonlinear Diffusion Equations 357 Case when F(Q = C 388 Case when F(Q = exp (Q 391 10.4 Exact Solution of Some Other NLPDEs 392 10.5 Applications to Physical Problems 396 Classical mechanics 396 Quantum mechanics 396 Astrophysics 397 10.6 Concluding Discussion 399 Problems 400
xviii Contents- 11. Symmetries of Differential Equations 402 11.1 Symmetry Groups of Differential Equations: An Introduction 402 Symmetries of algebraic equations 402 Groups and differential equations 404 Some basic results and definitions 405 11.2 Extended Transformations or Prolongations 419 11.3 Extended Infinitesimal Transformations 425 Case of one dependent and one independent variables 426 Case of one dependent and p independent variables 426" 11.4 Invariance of an Ordinary Differential Equation 427 11.5 Invariance of a Partial Differential Equation 430 11.6 Symmetry Groups and Conservation Laws 434 11.7 Noether's Theorem and Lie-Backlund Symmetries 435 11.8 Summary and Further Reading 440 Problems 441 12. Normal Modes in Nonlinear Dynamical Systems 443 12.1 Normal Modes of Linear Systems: A Brief Survey 444 12.2 Normal Modes of Nonlinear Systems: A Simple Generalization 448 12.3 Types of Mode Interactions: A Group Theoretic Approach 454 An overview 454 Various modes and their interactions 455 12.4 Bushes of Modes for a Dynamical System 459 Interactions between modes of an N-particle nonlinear system 454 Bush of dynamical variables 461 12.5 Dynamical Bush Equations: Normal Forms Theory 463 An introduction to normal form theory 463 Indicator of a resonance 465 Indicator of resonance for Hamiltonian systems 466 12.6 Concluding Discussion 469 Appendix: Some Numerical Aspects of Nonlinear Dynamical Systems Vis-a-Vis Chaos 470 Al. Introduction 470 A2. Phase Flow and Maps 471 A3. Characterization of the Chaotic Motion 478 A4. Routes to Chaos 486 A5. Fractals and Hausdorff Dimension 493 A6. Application to Physical problems 495 A7. Summary and Further Reading 497 Bibliography 499 Index 503