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Asymptotic Perturbation Methods: For Nonlinear Differential Equations in Physics

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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.

Author(s): Attilio Maccari
Publisher: Wiley-VCH
Year: 2023

Language: English
Pages: 254
City: Weinheim

Cover
Title Page
Copyright
Contents
About the Author
Foreword
Introduction
Chapter 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
Chapter 2 The Asymptotic Perturbation Method for Remarkable Nonlinear Systems
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
Chapter 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 Perturbation Method
3.4 Stability Analysis and Parametric Resonance Control
3.4.1 The Frequency–Response Curve Is
3.5 Suppression of the Two‐period Quasiperiodic Motion
3.6 Vibration Control for Other Nonlinear Systems
Chapter 4 The Asymptotic Perturbation Method for Vibration Control by 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
Chapter 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
Chapter 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 Remarks
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
Chapter 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
7.4.1 Nonlinear Wave
7.4.2 Solitons
7.4.3 Dromions
7.4.4 Lumps
7.4.5 Ring Solitons
7.4.6 Instantons
7.4.7 Moving Breather‐Like Structures
7.5 Conclusion
Chapter 8 The Asymptotic Perturbation Model 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
8.4.1 Nonlinear Wave
8.4.2 Solitons
8.4.3 Dromions
8.4.4 Lumps
8.4.5 Ring Solitons
8.4.6 Instantons
8.4.7 Moving Breather‐like Structures
8.5 A Few Considerations
8.6 Hidden Symmetry Models
8.7 Derivation of the Model System
8.8 Coherent Solutions
8.8.1 Nonlinear Wave
8.8.2 Solitons
8.8.3 Dromions
8.8.4 Lumps
8.8.5 Ring Solitons
8.8.6 Instantons
8.8.7 Moving Breather‐like Structures
8.9 Chaotic and Fractal Solutions
8.9.1 Chaotic–Chaotic and Chaotic–Periodic Patterns
8.9.2 Chaotic Line Soliton Solutions
8.9.3 Chaotic Dromion and Lump Patterns
8.9.4 Nonlocal Fractal Solutions
8.9.5 Fractal Dromion and Lump Solutions
8.9.6 Stochastic Fractal Dromion and Lump Excitations
8.10 Conclusion
Chapter 9 The Asymptotic Perturbation Method for Rogue Waves
9.1 Introduction
9.2 The Mathematical Framework
9.3 The Maccari System
9.4 Rogue Wave Physical Explanation According to the Maccari System and Blowing Solutions
9.5 Conclusion
Chapter 10 The Asymptotic Perturbation Method for Fractal and Chaotic Solutions
10.1 Introduction
10.2 A New Integrable System from the Dispersive Long‐wave Equation
10.3 Nonlinear Coherent Solutions
10.3.1 Nonlinear Wave
10.3.2 Solitons
10.3.3 Dromions
10.3.4 Lumps
10.3.5 Ring Solitons
10.3.6 Instantons
10.3.7 Moving Breather‐Like Structures
10.4 Chaotic and Fractal Solutions
10.4.1 Chaotic–Chaotic and Chaotic–Periodic Patterns
10.4.2 Chaotic Line Soliton Solutions
10.4.3 Chaotic Dromion and Lump Patterns
10.4.4 Nonlocal Fractal Solutions
10.4.5 Fractal Dromion and Lump Solutions
10.4.6 Stochastic Fractal Excitations
10.4.7 Stochastic Fractal Dromion and Lump Excitations
10.5 Conclusion
Chapter 11 The Asymptotic Perturbation Method for Nonlinear Relativistic and Quantum Physics
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
Chapter 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
Chapter 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
Chapter 14 The Asymptotic Perturbation Method for a Reverse Infinite‐Period Bifurcation in the Nonlinear Schrodinger Equation
14.1 Introduction
14.2 Building an Approximate Solution
14.3 A Reverse Infinite‐Period Bifurcation
14.4 Conclusion
Conclusion
References
Index
EULA