Harmonic Analysis Techniques in Nonlinear Dispersive Equations and Signal Processing

We consider the modified Zakharov-Kuznetsov (mZK) equation in two and three space dimensions in both focusing and defocusing cases. Using the $I$-method, for the 2D mZK equation, we prove the global well-posedness of the $H^s(\R^2)$ solutions for $s>\frac{3}{4}$ for any data in the defocusing case, and under the assumption that the mass of the initial data is less than the mass of the ground state solution of $\Delta \varphi - \varphi + \varphi^3 = 0$ in the focusing case. This improves the global well-posedness result of Linares and Pastor. We also prove that low-regularity solutions to the focusing 2D mZK equation which blow up in finite time have the property that the mass of the solutions concentrates inside a moving ball of shrinking radius. For focusing 3D mZK equation, we prove a sufficient condition on the initial data that guarantees global well-posedness in $ H^1(\R^{3}) $. We also consider the multichannel deconvolution problem in the presence of additive white noise. We propose a hybrid algorithm employing a regularized Fourier based approach followed by wavelet thresholding. Minimizing a proposed cost function, we determine the optimum regularization parameter that balances the Fourier and wavelet shrinkage. Our method extends the ForWaRD algorithm introduced by Neelamani, Baranuik, and Choi to the multichannel setup.

Author

• Bhattacharya, Debdeep

Language

• en

Keyword

• I-method
• Wavelets
• Multichannel lDeconvolution
• Zakharov-Kuznetsov
• Harmonic Analysis
• Signal Processing

Date created

• 2020

Type of Work

• Dissertation

License

• All rights reserved

Rights statement

• In Copyright

GW Unit

• Mathematics

Degree

• Ph.D.

Advisor

• Baginski, Frank

Committee Member(s)

• Zhao, Yanxiang
• Doroslovacki, Milos
• Conway, John
• Lewis, Joel
• Ren, Xiaofeng
• Robinson, E. Arthur

Persistent URL

•  https://scholarspace.library.gwu.edu/etd/6108vc03s