A course on nonlinear waves

de Samuel S. Shen

This volume presents a carefully written introduction to nonlinear waves in the natural sciences and engineering. It not only contains many classical results but also includes more recent results, dealing with topics such as the forced Korteweg-de Vries equation and material relating to X-ray crystallography. The book contains nine chapters. Chapter 1 concerns asymptotics and nonlinear ordinary differential equations. Conservation laws are discussed in Chapter 2, and Chapter 3 considers water waves. The scattering and inverse scattering method is described in Chapter 4, which also contains a full explanation of using the inverse scattering method for finding 1-, 2- and 3-soliton solutions of the Korteweg-de Vries equation. Chapter 5 studies the Burgers' equation and Chapter 6 discusses the forced Korteweg-de Vries equations. Here the emphasis is on steady-state bifurcations and unsteady-state periodic soliton generation. The Sine-Gordon and nonlinear Schrodinger equations are the subject of Chapter 7. The final two chapters consider wave instability and resonance. Every chapter contains problems and exercises, together with guidance for their solution. The volume concludes with some appendices which describe symbolic derivations of certain results on solitons. Several user-friendly MATHEMATICA packages are included. The prerequisite for using this book is a background knowledge of basic physics, linear algebra and differential equations.