Numerical and Analytical Tools for Localized Solutions of Generalized Wave Equations in Multidimension

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Boussinesq's equation (BE) was historically the first model for surface waves in shallow fluid layer that accounts for both nonlinearity and dispersion. The balance between the steepening effect of the nonlinearity and the flattening effect of the dispersion maintains the shape of the wave. In a coordinate frame moving with the center of the propagating wave, BE reduces to Korteweg-de Vries equation which is widely studied in 1D. Out of the mentioned balance are born solitary waves with quasi-particle behavior called also solitons under special conditions.


The overwhelming majority of the analytical and numerical results obtained so far is for one spatial dimension, while in multidimension, much less is possible to achieve analytically, and almost nothing is known about the unsteady solutions that involve interactions between solitons, especially for the full-fledged Boussinesq equations. Only sporadic papers with particular solutions appear in the literature, and even they are for some limiting cases, such as the Kadomtsev-Petviashvily equation.

At this stage, it is important to forward the concept of quasi-particle in multidimension by means of efficient numerical techniques. Since, no prior results are available, in order to reliably obtain the new solutions, the proposed activities aim at developing four different classes of algorithms for solving the Boussinesq equitation in two dimensions. These include: (i) Implicit difference schemes based on efficient iterative solvers for sparse linear systems; (ii) Implicit difference schemes based on coordinate operator splitting for bi-harmonic operators; (iii) Spectral method based on Fast Fourier Transform (FFT) and collocation approximation for the nonlinear terms; (iv) Galerkin spectral method based on a special complete orthonormal (CON) system of functions in L² over infinite interval.

 

Keywords:
Boussinesq Paradigm, Generalized Wave Equations, Solitary Waves and Solitons, Difference Methods, Galerkin Spectral Methods, Pseudospectral methods, Perfectly Matched Asysmptotic Boundary Conditions

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