Graduate Thesis Or Dissertation
 

Construction and numerical simulation of a two-dimensional analogue to the KdV equation

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/nk322j45d

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  • Arising from an investigation in Hydrodynamics, the Korteweg-de Vries equation demonstrates existence of nonlinear waves that resume their profile after interaction. In this thesis, the classical equations governing wave motion are the starting point for the development of an analogue of the KdV that describes the evolution of a wave surface. The resulting partial differential equation is non-linear and third order in two spatial variables. The linear and and non-linear parts of this equation are analyzed separately. A variant of the method of stationary phase is used to study the linear third order terms, and it is found that the non-linear part equates to the non-viscous Burger's equation. Numerical methods are also used to investigate behavior of wave shapes. We find initial conditions that behave in a manner similar to those of the KdV in that the waves are nonlinear but retain their shape after interaction. These include all solutions of the KdV, but also some "lump" initial conditions.
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