Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNegative index materials: some mathematical perspectives
98
0
0.0
(
0
)
Authors
Hoai-Minh Nguyen
Ask ChatGPT about the research
No Arabic abstract
Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones.
rate research
Read More
Starting from the construction of the free quantum scalar field of mass $mgeq 0$ we give mathematically precise and rigoro
The Boussinesq equations are known since the end of the XIXst century. However, the proliferation of various textsc{Boussinesq}-type systems started only in the second half of the XXst century. Today they come under various flavours depending on the goals of the modeller. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the Hamiltonian. In the present paper a family of Boussinesq-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known Boussinesq models, the identification of those systems with additional Hamiltonian structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full Euler equations is also discussed.
This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence and stability of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, the associated Zak phase (or Berry phase) may not be quantized. We investigate the existence of an interface mode which is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both time-reversal and inversion symmetry, the Zak phase is quantized, taking only two values $0, pi$. We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm-Liouville operators. The methods and results can be extended to general topological Sturm-Liouville systems in one dimension.
In this article, we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics (GFD) and climate dynamics. The main focus is on 1) the primitive equations (PEs) models and their related mathematical and computational issues, 2) climate variability, predictability and successive bifurcation, and 3) a new dynamical systems theory and its applications to GFD and climate dynamics.
In this paper, we begin by reviewing a certain number of mathematical challenges posed by the modelling of collective dynamics and self-organization. Then, we focus on two specific problems, first, the derivation of fluid equations from particle dynamics of collective motion and second, the study of phase transitions and the stability of the associated equilibria.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
