On spectral stability of solitary waves of nonlinear Dirac equation on a line

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.Comment: 20 pages with figure

O(k)O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds

Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, $V_k(\mathbb{R}^N)$ and $Gr(k, \mathbb{R}^N)$ respectively, and benefit from projection onto lower-dimensional Stiefel (respectively, Grassmannian) manifolds. In this work, we propose an algorithm called Principal Stiefel Coordinates (PSC) to reduce data dimensionality from $V_k(\mathbb{R}^N)$ to $V_k(\mathbb{R}^n)$ in an $O(k)$-equivariant manner ($k \leq n \ll N$). We begin by observing that each element $\alpha \in V_n(\mathbb{R}^N)$ defines an isometric embedding of $V_k(\mathbb{R}^n)$ into $V_k(\mathbb{R}^N)$. Next, we optimize for such an embedding map that minimizes data fit error by warm-starting with the output of principal component analysis (PCA) and applying gradient descent. Then, we define a continuous and $O(k)$-equivariant map $\pi_\alpha$ that acts as a ``closest point operator'' to project the data onto the image of $V_k(\mathbb{R}^n)$ in $V_k(\mathbb{R}^N)$ under the embedding determined by $\alpha$, while minimizing distortion. Because this dimensionality reduction is $O(k)$-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that the PCA output globally minimizes projection error in a noiseless setting, but that our algorithm achieves a meaningfully different and improved outcome when the data does not lie exactly on the image of a linearly embedded lower-dimensional Stiefel manifold as above. Multiple numerical experiments using synthetic and real-world data are performed.Comment: 26 pages, 8 figures, comments welcome

Hybrid optical-thermal antennas for enhanced light focusing and local temperature control

Metal nanoantennas supporting localized surface plasmon resonances have become an indispensable tool in bio(chemical) sensing and nanoscale imaging applications. The high plasmon-enhanced electric field intensity in the visible or near-IR range that enables the above applications may also cause local heating of nanoantennas. We present a design of hybrid optical-thermal antennas that simultaneously enable intensity enhancement at the operating wavelength in the visible and nanoscale local temperature control. We demonstrate a possibility to reduce the hybrid antenna operating temperature via enhanced infrared thermal emission. We predict via rigorous numerical modeling that hybrid optical-thermal antennas that support high-quality-factor photonic-plasmonic modes enable up to two orders of magnitude enhancement of localized electric fields and of the optical power absorbed in the nanoscale metal volume. At the same time, the hybrid antenna temperature can be lowered by several hundred degrees with respect to its all-metal counterpart under continuous irradiance of 104-105 W/m2. The temperature reduction effect is attributed to the enhanced radiative cooling, which is mediated by the thermally-excited localized surface phonon polariton modes. We further show that temperature reduction under even higher irradiances can be achieved by a combination of enhanced radiative and convective cooling in hybrid antennas. Finally, we demonstrate how hybrid optical-thermal antennas can be used to achieve strong localized heating of nanoparticles while keeping the rest of the optical chip at low temperature.Comment: 24 pages, 11 figure

Exploring excited states of Pt(ii) diimine catecholates for photoinduced charge separation

The intense absorption in the red part of the visible range, and the presence of a lowest charge-transfer excited state, render Platinum(II) diimine catecholates potentially promising candidates for light-driven applications. Here, we test their potential as sensitisers in dye-sensitised solar cells and apply, for the first time, the sensitive method of photoacoustic calorimetry (PAC) to determine the efficiency of electron injection in the semiconductor from a photoexcited Pt(II) complex. Pt(II) catecholates containing 2,2′-bipyridine-4,4′-di-carboxylic acid (dcbpy) have been prepared from their parent iso-propyl ester derivatives, complexes of 2,2′-bipyridine-4,4′-di-C(O)OiPr, (COOiPr)2bpy, and their photophysical and electrochemical properties studied. Modifying diimine Pt(II) catecholates with carboxylic acid functionality has allowed for the anchoring of these complexes to thin film TiO2, where steric bulk of the complexes (3,5-ditBu-catechol vs. catechol) has been found to significantly influence the extent of monolayer surface coverage. Dye-sensitised solar cells using Pt(dcbpy)(tBu2Cat), 1a, and Pt(dcbpy)(pCat), 2a, as sensitisers, have been assembled, and photovoltaic measurements performed. The observed low, 0.02–0.07%, device efficiency of such DSSCs is attributed at least in part to the short excited state lifetime of the sensitisers, inherent to this class of complexes. The lifetime of the charge-transfer ML/LLCT excited state in Pt((COOiPr)2bpy)(3,5-di-tBu-catechol) was determined as 250 ps by picosecond time-resolved infrared spectroscopy, TRIR. The measured increase in device efficiency for 2a over 1a is consistent with a similar increase in the quantum yield of charge separation (where the complex acts as a donor and the semiconductor as an acceptor) determined by PAC, and is also proportional to the increased surface loading achieved with 2a. It is concluded that the relative efficiency of devices sensitised with these particular Pt(II) species is governed by the degree of surface coverage. Overall, this work demonstrates the use of Pt(diimine)(catecholate) complexes as potential photosensitizers in solar cells, and the first application of photoacoustic calorimetry to Pt(II) complexes in general

Ferromagnetic phase transition and Bose-Einstein condensation in spinor Bose gases

Phase transitions in spinor Bose gases with ferromagnetic (FM) couplings are studied via mean-field theory. We show that an infinitesimal value of the coupling can induce a FM phase transition at a finite temperature always above the critical temperature of Bose-Einstein condensation. This contrasts sharply with the case of Fermi gases, in which the Stoner coupling $I_s$ can not lead to a FM phase transition unless it is larger than a threshold value $I_0$. The FM coupling also increases the critical temperatures of both the ferromagnetic transition and the Bose-Einstein condensation.Comment: 4 pages, 4 figure

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

Hybrid Decays

The heavy quark expansion of Quantum Chromodynamics and the strong coupling flux tube picture of nonperturbative glue are employed to develop the phenomenology of hybrid meson decays. The decay mechanism explicitly couples gluonic degrees of freedom to the pair produced quarks and hence does not obey the well known, but model-dependent, selection rule which states that hybrids do not decay to pairs of L=0 mesons. However, the nonperturbative nature of gluonic excitations in the flux tube picture leads to a new selection rule: light hybrids do not decay to pairs of identical mesons. New features of the model are highlighted and partial widths are presented for several low lying hybrid states.Comment: 13 pages, 1 table, revte

An extension of the Marsden-Ratiu reduction for Poisson manifolds

We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.Comment: 15 pages. Final versio

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.Comment: 30 pages, 6 figure

Kernel Formula Approach to the Universal Whitham Hierarchy

We derive the dispersionless Hirota equations of the universal Whitham hierarchy from the kernel formula approach proposed by Carroll and Kodama. Besides, we also verify the associativity equations in this hierarchy from the dispersionless Hirota equations and give a realization of the associative algebra with structure constants expressed in terms of the residue formulas.Comment: 18 page