564 research outputs found
Fundamental connections in differential geometry : quantum field theory, electromagnetism, chemistry and fluid mechanics
This work presents novel hydrodynamic formulations that reconcile the continuum hypothesis with the emergence of electromagnetic interactions among molecules from fundamental principles. Two models are proposed: a relativistic version of the Navier-Stokes equations derived from commutation relations, and a Helmholtz-like system obtained by applying the Hodge operator to the extended Navier-Stokes equations. Preliminary analysis suggests that the second model, with its nonlinear terms serving as a generalized current, can reproduce microscopic quantum effects. It shows promise for generating self-consistent field equations via BĂ€cklund transformations, remaining valid across all scales despite the breakdown of the continuum hypothesis
Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation
The paper deals with the construction of the asymptotic soliton-like and the
asymptotic peakon-like solutions to the modified Camassa-Holm equation with
variable coefficicents and a singular perturbation. This equation is a
generalization of the well known modified Camassa-Holm equation
(\ref{CHE_cons_mod}) which is integrable system and in addition to the soliton
solutions the equation has the peakon solutions. The novelty of the ideas of
this paper lies in the development of a technique for constructing asymptotic
peakon-like solutions. In the paper a general scheme of finding asymptotic
approximation of any order is presented and accuracy of the asymptotic
approximation is found.
The obtained results are illustrated by examples both the soliton-like and
the peakon-like solutions. For the examples the equations for the phase
function as well as the main and the first terms of the soliton-like and
peakon-like solutions are found. Moreover, for different values of a small
parameter the graphs that demonstrate kind of the solutions are presented. The
considered examples demonstrate that for an adequate description of the wave
process it is enough obtain the main and the first terms of correspond
asymptotic solutions.
The results also confirm that the proposed technique can be used for
constructing asymptotic wave-like solutions of other equations.Comment: 33 pages and 12 figure
Computation and Physics in Algebraic Geometry
Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra.
First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case.
Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature.
Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry
Decay of solitary waves of fractional Korteweg-de Vries type equations
We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1- dimensional semi-linear fractional equations: |D|αu + u â f (u) = 0, with α â (0, 2), a prescribed coefficient pâ(α), and a non-linearity f (u) = |u|pâ1 u for p â (1,pâ(α)), or f (u) = up with an integer p â [2;pâ(α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.publishedVersio
VC-PINN: Variable Coefficient Physical Information Neural Network For Forward And Inverse PDE Problems with Variable Coefficient
The paper proposes a deep learning method specifically dealing with the
forward and inverse problem of variable coefficient partial differential
equations -- Variable Coefficient Physical Information Neural Network
(VC-PINN). The shortcut connections (ResNet structure) introduced into the
network alleviates the "Vanishing gradient" and unifies the linear and
nonlinear coefficients. The developed method was applied to four equations
including the variable coefficient Sine-Gordon (vSG), the generalized variable
coefficient Kadomtsev-Petviashvili equation (gvKP), the variable coefficient
Korteweg-de Vries equation (vKdV), the variable coefficient Sawada-Kotera
equation (vSK). Numerical results show that VC-PINN is successful in the case
of high dimensionality, various variable coefficients (polynomials,
trigonometric functions, fractions, oscillation attenuation coefficients), and
the coexistence of multiple variable coefficients. We also conducted an
in-depth analysis of VC-PINN in a combination of theory and numerical
experiments, including four aspects, the necessity of ResNet, the relationship
between the convexity of variable coefficients and learning, anti-noise
analysis, the unity of forward and inverse problems/relationship with standard
PINN
Elastic shape analysis of geometric objects with complex structures and partial correspondences
In this dissertation, we address the development of elastic shape analysis frameworks for the registration, comparison and statistical shape analysis of geometric objects with complex topological structures and partial correspondences. In particular, we introduce a variational framework and several numerical algorithms for the estimation of geodesics and distances induced by higher-order elastic Sobolev metrics on the space of parametrized and unparametrized curves and surfaces. We extend our framework to the setting of shape graphs (i.e., geometric objects with branching structures where each branch is a curve) and surfaces with complex topological structures and partial correspondences. To do so, we leverage the flexibility of varifold fidelity metrics in order to augment our geometric objects with a spatially-varying weight function, which in turn enables us to indirectly model topological changes and handle partial matching constraints via the estimation of vanishing weights within the registration process. In the setting of shape graphs, we prove the existence of solutions to the relaxed registration problem with weights, which is the main theoretical contribution of this thesis. In the setting of surfaces, we leverage our surface matching algorithms to develop a comprehensive collection of numerical routines for the statistical shape analysis of sets of 3D surfaces, which includes algorithms to compute Karcher means, perform dimensionality reduction via multidimensional scaling and tangent principal component analysis, and estimate parallel transport across surfaces (possibly with partial matching constraints).
Moreover, we also address the development of numerical shape analysis pipelines for large-scale data-driven applications with geometric objects. Towards this end, we introduce a supervised deep learning framework to compute the square-root velocity (SRV) distance for curves. Our trained network provides fast and accurate estimates of the SRV distance between pairs of geometric curves, without the need to find optimal reparametrizations. As a proof of concept for the suitability of such approaches in practical contexts, we use it to perform optical character recognition (OCR), achieving comparable performance in terms of computational speed and accuracy to other existing OCR methods.
Lastly, we address the difficulty of extracting high quality shape structures from imaging data in the field of astronomy. To do so, we present a state-of-the-art expectation-maximization approach for the challenging task of multi-frame astronomical image deconvolution and super-resolution. We leverage our approach to obtain a high-fidelity reconstruction of the night sky, from which high quality shape data can be extracted using appropriate segmentation and photometric techniques
Achievable information rates for nonlinear frequency division multiplexed fibre optic systems
Fibre optic infrastructure is critical to meet the high data rate and long-distance communication requirements of modern networks. Recent developments in wireless communication technologies, such as 5G and 6G, offer the potential for ultra-high data rates and low-latency communication within a single cell. However, to extend this high performance to the backbone network, the data rate of the fibre optics connection between wireless base stations may become a bottleneck due to the capacity crunch phenomena induced by the signal dependent Kerr nonlinear effect. To address this, the nonlinear Fourier transform (NFT) is proposed as a solution to resolve the Kerr nonlinearity and linearise the nonlinear evolution of time domain pulses in the nonlinear frequency domain (NFD) for a lossless and noiseless fibre. Nonlinear frequency division multiplexing (NFDM), which encodes information on NFD using the discrete and continuous spectra revealed by NFT, is also proposed. However, implementing such signalling in an optical amplifier noise-perturbed fibre results in complicated, signal-dependent noise in NFD, the signal-dependent statistics and unknown model of which make estimating the capacity of such a system an open problem.
In this thesis, the solitonic part of the NFD, the discrete spectrum is first studied. Modulating the information in the amplitude of soliton pulse, the maximum time-scaled mutual information is estimated. Such a definition allows us to directly incorporate the dependence of soliton pulse width to its amplitude into capacity formulation. The commonly used memoryless channel model based on noncentral chi-squared distribution is initially considered. Applying a variance normalising transform, this channel is approximated by a unit-variance additive white Gaussian noise (AWGN) model. Based on a numerical capacity analysis of the approximated AWGN channel, a general form of capacity-approaching input distributions is determined. These optimal distributions are discrete comprising a mass point at zero (off symbol) and a finite number of mass points almost uniformly distributed away from zero. Using this general form of input distributions, a novel closed-form approximation of the capacity is determined showing a good match to numerical results. A mismatch capacity bounds are developed based on split-step simulations of the nonlinear Schrdinger equation considering both single soliton and soliton sequence transmissions. This relaxes the initial assumption of memoryless channel to show the impact of both inter-soliton interaction and Gordon-Haus effects. Our results show that the inter-soliton interaction effect becomes increasingly significant at higher soliton amplitudes and would be the dominant impairment compared to the timing jitter induced by the Gordon-Haus effect.
Next, the intrinsic soliton interaction, Gordon Haus effect and their coupled perturbation on the soliton system are visualised. The feasibility of employing an artificial neural network to resolve the inter-soliton interaction, which is the dominant impairment in higher power regimes, is investigated. A method is suggested to improve the achievable information rate of an amplitude modulated soliton communication system using a classification neural network against the inter-soliton interaction. Significant gain is demonstrated not only over the eigenvalue estimation of nonlinear Fourier transform, but also the continuous spectrum and eigenvalue correlation assisted detection scheme in the literature.
Lastly, for the nonsolitonic radiation of the NFT, the continuous spectrum is exploited. An approximate channel model is proposed for direct signalling on the continuous spectrum of a NFDM communication system, describing the effect of noise and nonlinearity at the receiver. The optimal input distribution that maximises the mutual information of the proposed approximated channel under peak amplitude constraint is then studied. We present that, considering the input-dependency of the noise, the conventional amplitude-constrained constellation designs can be shaped geometrically to provide significant mutual information gains. However, it is observed that further probabilistic shaping and constellation size optimisation can only provide limited additional gains beyond the best geometrically shaped counterparts, the 64 amplitude phase shift keying. Then, an approximated channel model that neglects the correlation between subcarriers is proposed for the matched filtered signalling system, based on which the input constellation is shaped geometrically. We demonstrate that, although the inter-subcarrier interference in the filtered system is not included in the channel model, shaping of the matched filtered system can provide promising gains in mismatch capacity over the unfiltered scenario
Symbolic computation of solitary wave solutions and solitons through homogenization of degree
A simplified version of Hirota's method for the computation of solitary waves
and solitons of nonlinear PDEs is presented. A change of dependent variable
transforms the PDE into an equation that is homogeneous of degree. Solitons are
then computed using a perturbation-like scheme involving linear and nonlinear
operators in a finite number of steps.
The method is applied to a class of fifth-order KdV equations due to Lax,
Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic
homogeneous equations for which the bilinear form might not be known.
Furthermore, homogenization of degree allows one to compute solitary wave
solutions of nonlinear PDEs that do not have solitons. Examples include the
Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When
applied to a wave equation with a cubic source term, one gets a bi-soliton
solution describing the coalescence of two wavefronts. The method is largely
algorithmic and is implemented in Mathematica.Comment: Proceedings Conference on Nonlinear and Modern Mathematical Physics
(NMMP-2022) Springer Proceedings in Mathematics and Statistics, 60pp,
Springer-Verlag, New York, 202
Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse instability of its periodic traveling wave solutions
We derive the Whitham modulation equations for the Zakharov-Kuznetsov
equation via a multiple scales expansion and averaging two conservation laws
over one oscillation period of its periodic traveling wave solutions. We then
use the Whitham modulation equations to study the transverse stability of the
periodic traveling wave solutions. We find that all such solutions are linearly
unstable, and we obtain an explicit expression for the growth rate of the most
unstable wave numbers. We validate these predictions by linearizing the
equation around its periodic solutions and solving the resulting eigenvalue
problem numerically. Finally, we calculate the growth rate of the solitary
waves analytically. The predictions of Whitham modulation theory are in
excellent agreement with both of these approaches.Comment: 15 pages, 2 figure
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