Asymptotic Analysis and Numerical Analysis of the Benjamin-Ono Equation

Abstract

The Benjamin-Ono equation is an integrable partial integro-differential equation which attracts much attention due to its application in modeling of the internal gravity waves in deep water and the ``morning glory cloud" in Northeastern Australia. In this dissertation, we first analyze the zero dispersion limit of the Cauchy problem of the Benjamin-Ono equation and give the first rigorous results. We demonstrate existence of the zero dispersion limit in the weak L2(R) sense and show this limit is equal to the signed sum of the branches of the multivalued solution of the inviscid Burgers equation with the same initial condition. Generalizations of these results are also given in this dissertation by using the formula of the $N$-soliton solutions of the higher order BO equations obtained by Matsuno. Moreover, we study three different numerical methods: the Fourier pseudospectral method, the Radial Basis Function method and the Christov method which are applied to solve the Benjamin-Ono equation. A comparison among the three methods is included. In the end, we also numerically illustrate and verify our theoretical results and study the traveling wave solutions of the cubic Benjamin-Ono equation.