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A spectral method with window technique for the initial value<br/><br/>problems of the Kadomtsev-Petviashvili equation

Monday, March 23, 2009 - 11:00am - 11:45am
EE/CS 3-180
Chiu-Yen Kao (University of Minnesota, Twin Cities)
The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersive
wave equation
which was proposed to study the stability of one soliton solution of the
KdV equation
under the influence of weak transversal perturbations. It is well know
that some closed-form
solutions can be obtained by function which have a Wronskian determinant
form.
It is of interest to study KP with an arbitrary initial condition and see
whether the
solution converges to any closed-form solution asymptotically. To reveal
the answer to this
question both numerically and theoretically, we consider different types
of initial conditions,
including one-line soliton, V-shape wave and cross-shape wave, and
investigate the behavior
of solutions asymptotically. We provides a detail description of
classification on the results.

The challenge of numerical approach comes from the unbounded domain and
unvanished
solutions in the infinity. In order to do numerical computation on the
finite domain, boundary
conditions need to be imposed carefully. Due to the non-periodic boundary
conditions, the
standard spectral method with Fourier methods involving trigonometric
polynomials cannot
be used. We proposed a new spectral method with a window technique which
will make the
boundary condition periodic and allow the usage of the classical approach.
We demonstrate
the robustness and efficiency of our methods through numerous simulations.
MSC Code: 
35L05
Keywords: