5,808 research outputs found
On the Benjamin Ono equation in the half line
We consider the inhomogeneous Dirichlet initial boundary value problem for
the Benjamin-Ono equation formulated on the half line. We study the global in
time existence of solutions to the initial-boundary value problem. This work is
a continuation of the ones [14,15] by Hayashi and Kaikina where the global in
time existence and the asymptotic behaviour of solutions for large time were
considered.Comment: 24 Page
Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data
We compute the scattering data of the Benjamin-Ono equation for arbitrary
rational initial conditions with simple poles. Specifically, we obtain explicit
formulas for the Jost solutions and eigenfunctions of the associated spectral
problem, yielding an Evans function for the eigenvalues and formulas for the
phase constants and reflection coefficient.Comment: 16 Pages, 2 Figure
Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
In nonlinear dispersive evolution equations, the competing effects of
nonlinearity and dispersion make a number of interesting phenomena possible. In
the current work, the focus is on the numerical approximation of traveling-wave
solutions of such equations. We describe our efforts to write a dedicated
Python code which is able to compute traveling-wave solutions of nonlinear
dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} +
\mathcal{L} u_x = 0, \end{equation*} where is a self-adjoint
operator, and is a real-valued function with .
The SpectraVVave code uses a continuation method coupled with a spectral
projection to compute approximations of steady symmetric solutions of this
equation. The code is used in a number of situations to gain an understanding
of traveling-wave solutions. The first case is the Whitham equation, where
numerical evidence points to the conclusion that the main bifurcation branch
features three distinct points of interest, namely a turning point, a point of
stability inversion, and a terminal point which corresponds to a cusped wave.
The second case is the so-called modified Benjamin-Ono equation where the
interaction of two solitary waves is investigated. It is found that is possible
for two solitary waves to interact in such a way that the smaller wave is
annihilated. The third case concerns the Benjamin equation which features two
competing dispersive operators. In this case, it is found that bifurcation
curves of periodic traveling-wave solutions may cross and connect high up on
the branch in the nonlinear regime
On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data
We study the Cauchy initial-value problem for the Benjamin-Ono equation in
the zero-disperion limit, and we establish the existence of this limit in a
certain weak sense by developing an appropriate analogue of the method invented
by Lax and Levermore to analyze the corresponding limit for the Korteweg-de
Vries equation.Comment: 54 pages, 11 figure
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