777 research outputs found
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
Nonlinear Evolution Equations: Analysis and Numerics
The qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations
Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation
In this paper we study the convergence of a fully discrete Crank-Nicolson
Galerkin scheme for the initial value problem associated with the fractional
Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and
non-linear convection terms. Our proof relies on the Kato type local smoothing
effect to estimate the localized -norm of the approximated
solution, where . We demonstrate that the scheme converges
strongly in to a weak solution of the
fractional KdV equation provided the initial data in .
Assuming the initial data is sufficiently regular, we obtain the rate of
convergence for the numerical scheme. Finally, the theoretical convergence
rates are justified numerically through various numerical illustrations
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
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