Quantized Vortex Dynamics of the Nonlinear Schr\"odinger Equation with Wave Operator on the Torus

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the nonlinear Schr\"odinger equation with wave operator on the torus when the core size of vortex $\varepsilon \to 0$. It is proved that the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator is a mixed state of the vortex motion laws for the nonlinear wave equation and the nonlinear Schr\"odinger equation. We will also investigate the convergence of the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator to the vortex motion law of the nonlinear Schr\"odinger equation via numerical simulation.Comment: 15 page

Systems of Points with Coulomb Interactions

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as "the mean-field limit" results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201

From chiral anomaly to two-fluid hydrodynamics for electronic vortices

Many recent experiments addressed manifestations of electronic crystals, particularly the charge density waves, in nano-junctions, under electric field effect, at high magnetic fields, together with real space visualizations by STM and micro X-ray diffraction. This activity returns the interest to stationary or transient states with static and dynamic topologically nontrivial configurations: electronic vortices as dislocations, instantons as phase slip centers, and ensembles of microscopic solitons. Describing and modeling these states and processes calls for an efficient phenomenological theory which should take into account the degenerate order parameter, various kinds of normal carriers and the electric field. Here we notice that the commonly employed time-depend Ginzburg-Landau approach suffers with violation of the charge conservation law resulting in unphysical generation of particles which is particularly strong for nucleating or moving electronic vortices. We present a consistent theory which exploits the chiral transformations taking into account the principle contribution of the fermionic chiral anomaly to the effective action. The resulting equations clarify partitions of charges, currents and rigidity among subsystems of the condensate and normal carriers. On this basis we perform the numerical modeling of a spontaneously generated coherent sequence of phase slips - the space-time vortices - serving for the conversion among the injected normal current and the collective one

Soliton generation and control in engineered materials

Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities. New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis. We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two components allows the stabilization of the whole soliton state. Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case. Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed. We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes. Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption

Fluid Dynamics with Incompressible Schrödinger Flow

This thesis introduces a new way of looking at incompressible fluid dynamics. Specifically, we formulate and simulate classical fluids using a Schrödinger equation subject to an incompressibility constraint. We call such a fluid flow an incompressible Schrödinger flow (ISF). The approach is motivated by Madelung's hydrodynamical form of quantum mechanics, and we show that it can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. The effective dynamics under an ISF is shown to be an Euler equation modified with a Landau-Lifshitz term. We show that the modifying term not only enhances the dynamics of vortex filaments, but also regularizes the potentially singular behavior of incompressible flows. Another contribution of this thesis is the elucidation of a general, geometric notion of Clebsch variables. A geometric Clebsch variable is useful for analyzing the dynamics of ISF, as well as representing vortical structures in a general flow field. We also develop an algorithm of approximating a "spherical" Clebsch map for an arbitrarily given flow field, which leads to a new tool for visualizing, analyzing, and processing the vortex structure in a fluid data.</p

Vortex motion for the lake equations

The lake equations $$\left\{\begin{aligned} \nabla \cdot \big( b \, \mathbf{u}\big) &= 0 & & \text{on}\ \mathbb{R}\times D,\\ \partial_t\mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} &= -\nabla h & & \text{on}\ \mathbb{R}\times D ,\\ \mathbf{u} \cdot \boldsymbol{\nu} &= 0 & & \text{on}\ \mathbb{R}\times\partial D . \end{aligned}\right.$$ model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b : D \to [0, + \infty)$ is small in comparison with the size of its two-dimensional projection $D \subset \mathbb{R}^2$. When the depth $b$ is positive everywhere in $D$ and constant on the boundary, we prove that the vorticity of solutions of the lake equations whose initial vorticity concentrates at an interior point is asympotically a multiple of a Dirac mass whose motion is governed by the depth function $b$.Comment: Minor revision, 43 page