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Nonlinear waves are essential phenomena in scientific and engineering disciplines. The features of nonlinear waves are usually described by solutions to nonlinear partial differential equations (NLPDEs). This book was prepared to familiarize students with nonlinear waves and methods of solving NLPDEs, which will enable them to expand their studies into related areas. The selection of topics and the focus given to each provide essential materials for a lecturer teaching a nonlinear wave course. Chapter 1 introduces "mode" types in nonlinear systems as well as Bäcklund transform, an indispensable technique to solve generic NLPDEs for stationary solutions. Chapters 2 and 3 are devoted to the derivation and solution characterization of three generic nonlinear equations: nonlinear Schrödinger equation, Korteweg–de Vries (KdV) equation, and Burgers equation. Chapter 4 is devoted to the inverse scattering transform (IST), addressing the initial value problems of a group of NLPDEs. In Chapter 5, derivations and proofs of the IST formulas are presented. Steps for applying IST to solve NLPDEs for solitary solutions are illustrated in Chapter 6.
Preface
About the Author
List of Figures
Nonlinear Waves
Introduction
“Mode” Types in Nonlinear Systems (Riemann Invariants)
Analytical Solutions of Nonlinear Wave Equations via Bäcklund Transform (Stationary Solutions)
Korteweg–de Vries (KdV) equation
Burgers equation
The sine-Gordon equation
The Liouville equation
Cubic nonlinear Schrödinger equation
Problems
Formulation of Nonlinear Wave Equations in Plasma
Equations for Self-Consistent Description of Nonlinear Waves in Plasma
Nonlinear Schrödinger Equation
For electromagnetic wave
For electron plasma (Langmuir) wave
Korteweg–de Vries (KdV) Equation for Ion Acoustic Wave
Burgers Equation for Dissipated Ion Acoustic Wave
Upper Hybrid Soliton Generated in Ionospheric HF Heating Experiments
Plasma density perturbed by the parametrically excited upper hybrid waves
Nonlinear envelope equation of the upper hybrid waves
Problems
Characteristics of Nonlinear Waves
Nonlinear Schrödinger Equation (NLSE)
Characteristic features of solutions
Conservation laws
Scaling symmetry
Galilean invariance
Virial theorem (variance identity)
Eigen solutions
Periodic solutions
Solitary solution
Collapse of nonlinear waves
Korteweg–de Vries (KdV) Equation
Conservation laws
Potential and modified Korteweg–de Vries (pKdV and mKdV) equations
Propagating modes
Periodic solution
Soliton trapped in self-induced potential well
Solitary solution with Miura transform
Burgers Equation
Analytical solution via the Cole–Hopf transformation
Propagating modes
Problems
Inverse Scattering Transform (IST)
Scattering Problem
Gelfand–Levitan–Marchenko (GLM) Linear Integral Equation
Nonlinear Synthesis
Auxiliary equations
Lax pair and Lax equation
Operator form
Matrix form
Matrix AKNS pair and AKNS equation
Time Evolution of Scattering Data
Solving GLM Equation
Evolution of an Impulse and a Rectangular Pulse in the KdV System (1.1)
Evolution of an impulse
Evolution of a rectangular pulse
Problems
Basis of Inverse Scattering Transform
Derivation of the GLM Integral Equation
Derivation of the Residues of φ(x, λ)eiλy
Proof of the Jost Solutions Satisfying the Linear Schrödinger Eigen Equation
Solitary Waves
Illustration of IST via Solving the KdV Equation
Steps of Applying IST
Modeling Gaussian Pulses as Reflectionless Potentials of the Linear Schrödinger Equation (4.6)
Pulse Behavior in the Transition Region
Application of Inverse Scattering Transform (IST): Exemplified with the mKdV and sine-Gordon Equations
mKdV equation
sine-Gordon equation
Problems
Answers to Problems
Bibliography
Index