659 research outputs found
Higher Order Modulation Equations for a Boussinesq Equation
In order to investigate corrections to the common KdV approximation to long
waves, we derive modulation equations for the evolution of long wavelength
initial data for a Boussinesq equation. The equations governing the corrections
to the KdV approximation are explicitly solvable and we prove estimates showing
that they do indeed give a significantly better approximation than the KdV
equation alone. We also present the results of numerical experiments which show
that the error estimates we derive are essentially optimal
KP solitons and Mach reflection in shallow water
This gives a survey of our recent studies on soliton solutions of the
Kadomtsev-Petviashvili equation with an emphasis on the Mach reflection problem
in shallow water.Comment: Presented at the Autumn Conference of Mathematical Society of Japan,
September 20, 2012; 14 pages, 6 figure
Higher order terms in multiscale expansions: a linearized KdV hierarchy
We consider a wide class of model equations, able to describe wave
propagation in dispersive nonlinear media. The Korteweg-de Vries (KdV) equation
is derived in this general frame under some conditions, the physical meanings
of which are clarified. It is obtained as usual at leading order in some
multiscale expansion. The higher order terms in this expansion are studied
making use of a multi-time formalism and imposing the condition that the main
term satisfies the whole KdV hierarchy. The evolution of the higher order terms
with respect to the higher order time variables can be described through the
introduction of a linearized KdV hierarchy. This allows one to give an
expression of the higher order time derivatives that appear in the right hand
member of the perturbative expansion equations, to show that overall the higher
order terms do not produce any secularity and to prove that the formal
expansion contains only bounded terms.Comment: arxiv version is already officia
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