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The focus of this dissertation is the existence, stability, and resulting dynamical evolution of localized stationary solutions to Nonlinear Schr¨odinger (NLS) equations with periodic confining potent...
We provide a detailed treatment of Ruijsenaars-Toda (RT) hierarchy with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperellip...
We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting pe...
We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_x v+\partial_x^3 v+\delta(\partial_x^2 v +\partial_x...
We first establish local well-posedness for a periodic 2-component Camassa-Holm equation. We then present two global existence results for strong solutions to the equation. We finally obtain several b...
Abstract: We introduce a periodic two-dimensional $\mu$-$b$-equation and a periodic two-dimensional two-component $(\mu)$-Camassa-Holm equation which we study as geodesic flows on the diffeomorphism g...
Abstract: We study some geometric aspects of the periodic two-dimensional Camassa-Holm equation (2D-CH) which re-expresses geodesic motion on the diffeomorphism group of the torus $\T = S^1 \times S^1...
Abstract: We study the long-time behaviour of a nonlinear Fokker-Planck equation, which models the evolution of rigid polymers in a given flow, after a closure approximation. The aim of this work is t...
Abstract: Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient con...
Abstract: We consider weak solutions to dispersive partial differential equations with periodic boundary conditions and initial data with jump discontinuities. These are already known to be continuous...
We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier f...
In this paper, we study the Cauchy problem of a periodic 2-component μ-Hunter-Saxton system. We first establish the local well-posedness for the periodic 2-component μ-Hunter-Saxton system by Kato’s s...
The main objective of this paper is twofold. We first show that if the doublyweighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum o...
We consider the quadratic and cubic KP - I and NLS models in 1+2 dimensions with periodic boundary conditions. We show that the spatially periodic travelling waves (with period K) in the form u(t, x,...
We consider the one-dimensional propagation of electromagnetic waves in a weakly nonlinear and low-contrast spatially inhomogeneous medium with no energy dissipation. We focus on the case of a period...

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