Publication: Existence, Stability and Computation of Periodic Waves for a Generalized Benney-Luke Equation.
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Date
2006-12
Authors
Muñoz Grajales, Juan Carlos
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Abstract
In this work we consider a model which describes the propagation of surface long waves with small amplitude on a fluid of constant depth and including the effect of surface tension. We also analyze the interaction of a travelling wave with a forcing submersed topography. In this case, the model adopted is a generalization of the Benney-Luke model (gBL) which includes a forcing term coming from the topography named (fBL).
The gBL model has been considered by J. Quintero in the paper ¨Existence and Analyticity of Lumps Solutions for Generalized Benney - Luke Equations¨(Rev. Col. Mat.. 36, 2002) . Quintero proved the existence and analyticity of lump solutions for the gBL equation. The fBL model including the forcing topography was derived by Mileswki and Keller in the papers ¨ Three dimensional surface waves¨( Stud. Appl. Math., 37, pp. 149-166, 1996) and ¨the generation and evolution of lump solitary waves in surface-tension-dominated flows¨(SIAM J. Appl. Math., Vol. 61, No. 3, pp. 731-750, 2000). Berger and Milewski studied the propagation of a two-dimensional lump solitary wave over a localized topographical feature (a bump), when the surface tension is significant and near the critical shallow water speed. However there is a lack of mathematical theory for existence and orbital stability of Cnoidal and Dnoidal wave solutions to equation (gBL) in constant depth. Furthermore to our knowledge, periodic travelling waves propagating over more complicated topographical profiles have not been yet considered.
The first specific objective of this project is to prove existence of 1-D T-periodic travelling waves (Cnoidal and Dnoidal solutions) governed by the (gBL) equation. It is also expected to analyze the behavior of these periodic solutions as a function of the period T and to prove its orbital stability. The main theoretical results to be used are the Concentration-Compactness Principle, Grillakis-Shatah-Strauss' theory and the Lyapunov method to prove orbital stability.
The second specific objective is the numerical solution to the equation (gBL) in 1D and 2D, including the effect of a submersed rough topography in the wave scattering. To achieve this objective, we will implement an efficient pseudospectral solver to compute with good accuracy solutions of the gBL and fBL equations over a large time interval.
In addition to the theoretical results, we expect to generate a mathematical activity in applied analysis area in Colombia and inside of the Mathematics Department at Universidad del Valle by offering novel thesis problems to undergraduate and graduate students in our academic programs.