The purpose of this book is to help scientists and engineers explore the nonlinear
dynamics of plasmas while developing their proficiency with the tools of nonlinear
science. Nonlinear science owes much of its origin to the early studies of plasma
physicists. For example, two key plasma physics problems from the nineteen six-
sixties led first to what is now the standard estimate for the criterion for the onset
of stochasticity in a Hamiltonian system and secondly to the inverse scattering
method of solving the initial value problem for integrable nonlinear wave equations.
The method for gauging the onset of a significant volume of stochasticity in the
phase space arose from the study of the structures of the magnetic fields in toroidal
plasma confinement vessels. The problem of determining the confinement time for
charged particles in the magnetic mirror trap led physicists to the standard map.
The standard map, as its name implies, is a universal description of Hamiltonian
chaos showing the beautifully intricate mixtures of integrable and chaotic particle
trajectories. For the closed two degree-of-freedom system the surface of section tech-
technique is used to develop an equivalent two-dimensional area preserving map from
which the integrable structures and chaotic sea in the phase space can be found.
These methods and numerous examples are developed in Chapters 1 to 4.
The concept of robust, localized intrinsically nonlinear solutions of a class of
nonlinear wave equations which appear in many fields of science and engineering
was developed by the celebrated team of Martin Kruskal and Norman Zabnsky
in the late nineteen sixties. From Zabusky's precise computer simulations of the
elastic interactions of the soliton solutions of the Korteweg-de Vries (KdV) equation
and Kruskal's mathematical description of the dynamics observed in the computer
simulations emerged the paradigm for future studies of integrable nonlinear wave
equations. These studies led to the invention of the inverse scattering transform
AST) method for solving the initial value problem giving precisely the number of
solitons and the wave field components that arise from a given initial disturbance.
From this seminal method for integrating the KdV equation other nonlinear wave
equations, such as the cubic Schrodinger equation and the Kadomtsev-Petviashvili
equation, were found,to be integrable by extending the 1ST method developed for the
KdV equation. These methods are developed in Chapter 5. For a complete, modern
development of the analysis of integrable nonlinear wave equations the reader may
wish to consult Solitons, Nonlinear Evolution and Inverse Scattering by Ablowitz