Description
WIT Press Solitary Waves in Fluids 2007 Edition by R. H. J. Grimshaw
After the initial observation by John Scott Russell of a solitary wave in a canal, his insightful laboratory experiments and the subsequent theoretical work of Boussinesq, Rayleigh and Korteweg and de Vries, interest in solitary waves in fluids lapsed until the mid 1960s with the seminal paper of Zabusky and Kruskal, which described the discovery of the soliton. This was followed by the rapid development of the theory of solitons and integrable systems. At the same time came the realization that solitary waves occur naturally in many physical systems, and play a fundamental role in many circumstances. The aim of this text is to describe the role that soliton theory plays in fluids in several contexts. After an historical introduction, the book is divided into five chapters covering the basic theory of the Korteweg-de Vries equation, and the subsequent application to free-surface solitary waves in water, internal solitary waves in coastal ocean and the atmospheric boundary layer, solitary waves in rotating flows, and to planetary solitary waves with applications to the ocean and atmosphere.The final chapter examines the theory and application of envelope solitary waves and the nonlinear Schrodinger equation to water waves. Table of Contents : Chapter 1: Introduction Historical introduction; Korteweg-de Vries equation; Solitary waves: bifurcation from the linear spectrum Chapter 2: Korteweg-de Vries equation: solitons and undular bores Introduction; Periodic travelling wave solutions and solitary waves; Inverse scattering transform method and solitons; Whitham modulation equations; Undular bores; Propagation of KdV soliton through a variable environment Chapter 3: Solitary waves in water: numerical methods and results Introduction; Formulation; Linear solutions; Weakly nonlinear results; Numerical methods for periodic waves; Numerical methods for solitary waves; Extensions Chapter 4: Internal solitary waves Introduction; Approximate evolution equations; Solitons and their properties; Evolution of initial disturbances; Solitons in variable media; Conclusion Chapter 5: Solitary waves in rotating fluids Introduction; Axisymmetric inertial solitary waves; Internal gravity waves in a rotating channel Chapter 6: Planetary solitary waves Introduction; Coherent structures and solitons; Weakly nonlocal solitary waves: slow death by radiation; The dynamical isolation of the crests of cnoidal waves: Gauss' cosine-lemniscate function, imbricate series and all that; An exemplary model: equatorial solitary waves in the shallow-water equations; Equatorial Kelvin solitary waves; Vortices embedded in a shear zone and nonlinear critical latitude theory; Summary Chapter 7: Envelope solitary waves Linear waves and group velocity; Weakly nonlinear waves; Envelope solitons; Bifurcation to solitary waves; Two-dimensional solitary waves