On Curvelet based Multi-scale Analysis of Higher Dimensional Solitons

Abstract

In shallow water theory, completely integrable nonlinear partial differential equations arise at various levels of approximations/simulations. Such equations possess traveling wave solutions, called solitons, which themselves are self-phase-modulated signals. Wavelets are traditionally applied in signal processing but it has limitations to deal effectively with geometric features with line and surface singularities. Due to its poor orientation selectivity, wavelet can only capture limited directional information in signal processing. A multi-resolution geometric analysis, named curvelet transform overcomes the drawbacks of the conventional wavelet transform. Such a geometric characterization provides better understanding of cascade mechanism and dissipation range dynamics. In the present work, a multi-scale geometric analysis with curvelets as basis functions is proposed for analyzing higher dimensional lump-like solitons- that arise in the evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromigration equation. Moreover, the application of curvelet thresholding technique for de-noising the contaminated signals is presented.