Asymptotic Perturbation Methods

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.

The book 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.

1 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR OSCILLATORS1.1 Introduction1.2 Nonlinear Dynamical Systems1.3 The Approximate Solution1.4 Comparison with the Results of the Numerical Integration1.5 External Excitation in Resonance with the Oscillator1.6 Conclusion2 THE ASYMPTOTIC PERTURBATION METHOD FOR REMARKABLE NONLINEAR OSCILLATORS2.1 Introduction2.2 Periodic Solutions and their Stability2.3 Global Analysis of the Model System2.4 Infinite-Period Symmetric Homoclinic Bifurcation2.5 A Few Considerations2.6 A Peculiar Quasiperiodic Attractor2.7 Building an Approximate Solution2.8 Results from Numerical Simulation2.9 Conclusion3 THE ASYMPTOTIC PERTURBATION METHOD FOR VIBRATION CONTROL WITH TIME DELAY STATE FEEDBACK3.1 Introduction3.2 Time Delay State Feedback3.3 The AP Method3.4 Stability Analysis and Parametric Resonance Control3.5 Suppression of the Two-Period Quasiperiodic Motion3.6 Vibration Control for Other Nonlinear Systems4 THE ASYMPTOTIC PERTURBATION METHOD FOR VIBRATION CONTROL WITH NONLOCAL DYNAMICS4.1 Introduction4.2 Vibration Control for the van der Pol Equation4.3 Stability Analysis and Parametric Resonance Control4.4 Suppression of the Two-Period Quasiperiodic Motion4.5 Conclusion5 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR CONTINUOUS SYSTEMS5.1 Introduction5.2 The Approximate Solution for the Primary Resonance of the nth Mode5.3 The Approximate Solution for the Subharmonic Resonance of Order One-Half of the nth Mode5.4 Conclusion6 THE ASYMPTOTIC PERTURBATION METHOD FOR DISPERSIVE NONLINEAR PARTIAL DIFFERENTIALEQUATIONS6.1 Introduction6.2 Model Nonlinear PDEs Obtained from the Kadomtsev-Petviashvili Equation6.3 The Lax Pair for the Model Nonlinear PDE6.4 A Few Considerations6.5 A Generalized Hirota Equation in 2+1 dimensions6.6 Model Nonlinear PDEs Obtained from the KP equation6.7 The Lax Pair for the Hirota-Maccari Equation6.8 Conclusion7 THE ASYMPTOTIC PERTURBATION METHOD FOR PHYSICS PROBLEMS7.1 Introduction7.2 Derivation of the Model System7.3 Integrability of the Model System of Equations7.4 Exact Solutions for the C-Integrable Model Equation7.5 Conclusion8 THE ASYMPTOTIC PERTURBATION METHOD FOR ELEMENTARY PARTICLE PHYSICS8.1 Introduction8.2 Derivation of the Model System8.3 Integrability of the Model System of Equations8.4 Exact solutions for the C-Integrable Model Equation8.5 A Few Considerations8.6 Hidden Symmetry Models8.7 Derivation of the Model System8.8 Coherent Solutions8.9 Chaotic and Fractal Solutions8.10 Conclusion9 THE ASYMPTOTIC PERTURBATION METHOD FOR ROGUE WAVES IN NONLINEAR SYSTEMS9.1 Introduction9.2 The Mathematical Framework9.3 The Maccari System9.4 Rogue Waves Physical Explanation according to Maccari System and Blowing Solutions9.5 Conclusion10 THE ASYMPTOTIC PERTURBATION METHOD FOR FRACTAL AND CHAOTIC SOLUTIONS IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS10.1 Introduction10.2 A new Integrable System from the Dispersive Long Wave Equation10.3 Nonlinear Coherent Solutions10.4 Chaotic and Fractal Solutions10.5 Conclusion11 THE ASYMPTOTIC PERTURBATION METHOD FOR NONLINEAR QUANTUM MECHANICS11.1 Introduction11.2 The NLS equation for a1>011.3 The NLS equation for a1<011.4 A Possible Extension11.5 The Nonrelativistic Case11.6 The Relativistic Case11.7 Conclusion12 COSMOLOGY12.1 Introduction12.2 A New Field Equation12.3 Exact solution in the Robertson-Walker Metrics12.4 Entropy Production12.5 Conclusion13 CONFINEMENT AND ASYMPTOTIC FREEDOM IN A PURELY GEOMETRIC FRAMEWORK13.1 Introduction13.2 The Uncertainty Principle13.3 Confinement and Asymptotic Freedom for the Strong Interaction13.4 The Motion of a Light Ray into a Hadron13.5 Conclusion14 THE ASYMPTOTIC PERTURBATION METHOD FOR A REVERSE INFINITE-PERIOD BIFURCATION IN THE NONLINEAR SCHRÖDINGER EQUATION14.1 Introduction14.2 Building an Approximate Solution14.3 A Reverse Infinite-Period Bifurcation14.4 ConclusionConclusionBibliography