Analysis of Nonlinear Dispersive Model Equations

• Main Author: Grey, Jacob
• Other Authors: Tom, Michael M.
• Format: Others
• Language: en
• Published: LSU 2015
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-07132015-161603/

In this work we begin with a brief survey of the classical fluid dynamics problem of water waves, and then proceed to derive well known evolution equations via a Hamiltonian Variational approach. This method was first introduced in the seminal work of Walter Craig, et al. \cite{CG}. The distinguishing feature of this scheme is that the Dirichlet-Neumann operator of the fluid domain appears explicitly in the Hamiltonian. In the second and third chapters, we utilize the Hamiltonian perturbation theory introduced in \cite{CG} to derive the Benjamin-Bona-Mahony (BBM) and Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP)equations. Finally, we briefly review the existence theory for their corresponding Cauchy problems. In the fourth chapter, I motivate and present my first result and demonstrate how it ties in with the literature and previous chapters. In particular, we show that the solution of the Cauchy problem for the BBM-KP equation converges to the solution of the Cauchy problem for the BBM equation in a suitable function space whenever the initial data for both equations are close as the transverse variable $y \rightarrow \pm \infty$. \\ \indent In the final chapter we introduce and analyze a new modified Kadomstev-Petviashvili equation. This model was introduced in an effort to remedy the "odd" behavior of the mass of a given solution to the Kadomstev-Petviashvili model. This results in a model which does not impose specified restrictions upon the initial data. After motivating and deriving the model we prove various linear estimates for the operator equation arising from the Duhamel formulation of system. To this end I discuss my future work.