## Solitary Waves in FluidsR. 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 1960's with the seminal paper of Zabusky and Kruskal describing 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 solition 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 remaining chapter examines the theory and application of envelope solitary waves and the nonlinear Schrodinger equation to water waves. |

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Akylas amplitude approximation asymptotic Atmos bifurcation Boussinesq Boyd capillary waves Chapter cnoidal wave coefficient coherent structures compute constant crest decay derived described Dias dispersion relation dynamics envelope solitary wave equatorial evolution equation expansion finite Fluid Mech free surface function Gardner equation gravity waves Grimshaw group velocity initial conditions interaction internal solitary waves internal waves Iooss KdV equation Kelvin wave Korteweg–de Vries equation layer leading order linear wave long waves long-wave Math method mode modons modulation NLS equation nonlinear and dispersive nonlinear term numerical obtained ocean Ostrovsky parameter Pelinovsky perturbation phase speed Phys potential problem radiation Riemann invariants Rossby solitons Rossby waves rotating small-amplitude solitary wave solutions solitary waves stratified surface tension theory transformation travelling wave two-dimensional undular bore Vanden-Broeck variables vortex vortices water waves wave propagation wavelength wavenumber weakly nonlocal Whitham Whitham equations