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• W2118621474 abstract "This chapter presents a brief history and overview of nonlinear water waves. While water waves are one of the most common of all natural phenomena, they possess an extremely rich mathematical structure. Water waves belong to one of the most difficult areas of fluid dynamics and wave mechanics; Craik, 2005––namely, the study of nonlinear, dispersive waves in two-space and one-time dimensions. The power of the Fourier method for determining the exact solution of linear wave equations is often cast in terms of the Cauchy problem for one-space and one-time dimensions: given the wave profile as a function of space, x, at some initial value of time, t = 0, determine the solution of the surface wave dynamics for all values of x for all future (and past) times, t––that is, given the initial surface elevation η(x, 0) compute η(x, t) for all t. From data analysis and numerical modeling points of view, the inverse scattering transform (IST) plays a role in the study of nonlinear wave dynamics similar to the linear, periodic Fourier transform provided that the IST exists for periodic boundary conditions for a physically suitable nonlinear wave equation. One motivation for periodic boundary conditions for nonlinear equations rests with the fact that most applications of linear Fourier analysis are based upon the fast Fourier transform (FFT), a periodic algorithm. The systematic study of water waves was one of the first fluid-mechanical problems to be approached using the modern formulation of the Navier–Stokes type of equations. The chapter discusses a number of physically important nonlinear wave equations: the Korteweg–deVries (KdV), the Kadomtsev–Petviashvili (KP), and the nonlinear Schrödinger (NLS) equations." @default.
• W2118621474 created "2016-06-24" @default.
• W2118621474 creator A5062415557 @default.
• W2118621474 date "2010-01-01" @default.
• W2118621474 modified "2023-10-16" @default.
• W2118621474 title "Brief History and Overview of Nonlinear Water Waves" @default.
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• W2118621474 doi "https://doi.org/10.1016/s0074-6142(10)97001-0" @default.
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