Asymptotic Perturbation Methods: For Nonlinear Differential Equations in Physics

Asymptotic Perturbation Methods Cohesive overview of powerful mathematical methods to solve differential equations in physics Asymptotic Perturbation Methods for Nonlinear Differential Equations in Physics addresses nonlinearity in various fields of physics from the vantage point of its mathematical description in the form of nonlinear partial differential equations and presents a unified view on nonlinear systems in physics by providing a common framework to obtain approximate solutions to the respective nonlinear partial differential equations based on the asymptotic perturbation method. Aside from its complete coverage of a complicated topic, a noteworthy feature of the book is the emphasis on applications. There are several examples included throughout the text, and, crucially, the scientific background is explained at an elementary level and closely integrated with the mathematical theory to enable seamless reader comprehension. To fully understand the concepts within this book, the prerequisites are multivariable calculus and introductory physics. Written by a highly qualified author with significant accomplishments in the field, Asymptotic Perturbation Methods for Nonlinear Differential Equations in Physics covers sample topics such as: Application of the various flavors of the asymptotic perturbation method, such as the Maccari method to the governing equations of nonlinear system Nonlinear oscillators, limit cycles, and their bifurcations, iterated nonlinear maps, continuous systems, and nonlinear partial differential equations (NPDEs) Nonlinear systems, such as the van der Pol oscillator, with advanced coverage of plasma physics, quantum mechanics, elementary particle physics, cosmology, and chaotic systems Infinite-period bifurcation in the nonlinear Schrodinger equation and fractal and chaotic solutions in NPDEs Asymptotic Perturbation Methods for Nonlinear Differential Equations in Physics is ideal for an introductory course at the senior or first year graduate level. It is also a highly valuable reference for any professional scientist who does not possess deep knowledge about nonlinear physics.

Table of Contents:
1 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR OSCILLATORS 1.1 Introduction 1.2 Nonlinear Dynamical Systems 1.3 The Approximate Solution 1.4 Comparison with the Results of the Numerical Integration 1.5 External Excitation in Resonance with the Oscillator 1.6 Conclusion 2 THE ASYMPTOTIC PERTURBATION METHOD FOR REMARKABLE NONLINEAR OSCILLATORS 2.1 Introduction 2.2 Periodic Solutions and their Stability 2.3 Global Analysis of the Model System 2.4 Infinite-Period Symmetric Homoclinic Bifurcation 2.5 A Few Considerations 2.6 A Peculiar Quasiperiodic Attractor 2.7 Building an Approximate Solution 2.8 Results from Numerical Simulation 2.9 Conclusion 3 THE ASYMPTOTIC PERTURBATION METHOD FOR VIBRATION CONTROL WITH TIME DELAY STATE FEEDBACK 3.1 Introduction 3.2 Time Delay State Feedback 3.3 The AP Method 3.4 Stability Analysis and Parametric Resonance Control 3.5 Suppression of the Two-Period Quasiperiodic Motion 3.6 Vibration Control for Other Nonlinear Systems 4 THE ASYMPTOTIC PERTURBATION METHOD FOR VIBRATION CONTROL WITH NONLOCAL DYNAMICS 4.1 Introduction 4.2 Vibration Control for the van der Pol Equation 4.3 Stability Analysis and Parametric Resonance Control 4.4 Suppression of the Two-Period Quasiperiodic Motion 4.5 Conclusion 5 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR CONTINUOUS SYSTEMS 5.1 Introduction 5.2 The Approximate Solution for the Primary Resonance of the nth Mode 5.3 The Approximate Solution for the Subharmonic Resonance of Order One-Half of the nth Mode 5.4 Conclusion 6 THE ASYMPTOTIC PERTURBATION METHOD FOR DISPERSIVE NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 6.1 Introduction 6.2 Model Nonlinear PDEs Obtained from the Kadomtsev-Petviashvili Equation 6.3 The Lax Pair for the Model Nonlinear PDE 6.4 A Few Considerations 6.5 A Generalized Hirota Equation in 2+1 dimensions 6.6 Model Nonlinear PDEs Obtained from the KP equation 6.7 The Lax Pair for the Hirota-Maccari Equation 6.8 Conclusion 7 THE ASYMPTOTIC PERTURBATION METHOD FOR PHYSICS PROBLEMS 7.1 Introduction 7.2 Derivation of the Model System 7.3 Integrability of the Model System of Equations 7.4 Exact Solutions for the C-Integrable Model Equation 7.5 Conclusion 8 THE ASYMPTOTIC PERTURBATION METHOD FOR ELEMENTARY PARTICLE PHYSICS 8.1 Introduction 8.2 Derivation of the Model System 8.3 Integrability of the Model System of Equations 8.4 Exact solutions for the C-Integrable Model Equation 8.5 A Few Considerations 8.6 Hidden Symmetry Models 8.7 Derivation of the Model System 8.8 Coherent Solutions 8.9 Chaotic and Fractal Solutions 8.10 Conclusion 9 THE ASYMPTOTIC PERTURBATION METHOD FOR ROGUE WAVES IN NONLINEAR SYSTEMS 9.1 Introduction 9.2 The Mathematical Framework 9.3 The Maccari System 9.4 Rogue Waves Physical Explanation according to Maccari System and Blowing Solutions 9.5 Conclusion 10 THE ASYMPTOTIC PERTURBATION METHOD FOR FRACTAL AND CHAOTIC SOLUTIONS IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 10.1 Introduction 10.2 A new Integrable System from the Dispersive Long Wave Equation 10.3 Nonlinear Coherent Solutions 10.4 Chaotic and Fractal Solutions 10.5 Conclusion 11 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR QUANTUM MECHANICS 11.1 Introduction 11.2 The NLS equation for a1>0 11.3 The NLS equation for a1<0 11.4 A Possible Extension 11.5 The Nonrelativistic Case 11.6 The Relativistic Case 11.7 Conclusion 12 COSMOLOGY 12.1 Introduction 12.2 A New Field Equation 12.3 Exact solution in the Robertson-Walker Metrics 12.4 Entropy Production 12.5 Conclusion 13 CONFINEMENT AND ASYMPTOTIC FREEDOM IN A PURELY GEOMETRIC FRAMEWORK 13.1 Introduction 13.2 The Uncertainty Principle 13.3 Confinement and Asymptotic Freedom for the Strong Interaction 13.4 The Motion of a Light Ray into a Hadron 13.5 Conclusion 14 THE ASYMPTOTIC PERTURBATION METHOD FOR A REVERSE INFINITE-PERIOD BIFURCATION IN THE NONLINEAR SCHRÖDINGER EQUATION 14.1 Introduction 14.2 Building an Approximate Solution 14.3 A Reverse Infinite-Period Bifurcation 14.4 Conclusion Conclusion Bibliography