Asymptotic Perturbation Methods: For Nonlinear Differential Equations in Physics

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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