Search results for "Inverse scattering"
showing 10 items of 34 documents
Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain
2002
The paper deals with the analytic theory of the quantum q-deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L. Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin–Barnes type. For the periodic chain the two dual Baxter equations are derived.
Soliton Statistical Mechanics: Statistical Mechanics of the Quantum and Classical Integrable Models
1988
It is shown how the Bethe Ansatz (BA) analysis for the quantum statistical mechanics of the Nonlinear Schrodinger Model generalises to the other quantum integrable models and to the classical statistical mechanics of the classical integrable models. The bose-fermi equivalence of these models plays a fundamental role even at classical level. Two methods for calculating the quantum or classical free energies are developed: one generalises the BA method the other uses functional integral methods. The familiar classical action-angle variables of the integrable models developed for the real line R are used throughout, but the crucial importance of periodic boundary conditions is recognized and t…
Asymptotic analysis of the form-factors of the quantum spin chains
2020
Since a long-time, the quantum integrable systems have remained an area where modern mathematical methods have given an access to interesting results in the study of physical systems. The exact computations, both numerical and asymptotic, of the correlation function is one of the most important subject of the theory of the quantum integrable models. In this context an approach based on the calculation of form factors has been proved to be a more effective one. In this thesis, we develop a new method based on the algebraic Bethe ansatz is proposed for the computation of the form-factors in thermodynamic limit. It is applied to and described in the context of isotropic XXX Heisenberg chain, w…
Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’
1995
Abstract The fundamental role played by the quantum repulsive Nonlinear Schrodinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) …
The fixed angle scattering problem and wave equation inverse problems with two measurements
2019
We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the far field pattern generated by plane waves coming from exactly two opposite directions. This implies that a reflection symmetric potential is uniquely determined by its fixed angle scattering data. We also prove a Lipschitz stability estimate for an associated problem. Motivated by the point source inverse problem in geophysics, we show that a compactly supported potential is uniquely determined from boundary measurements of the waves generated by exactl…
A sampling method for detecting buried objects using electromagnetic scattering
2005
We consider a simple (but fully three-dimensional) mathematical model for the electromagnetic exploration of buried, perfect electrically conducting objects within the soil underground. Moving an electric device parallel to the ground at constant height in order to generate a magnetic field, we measure the induced magnetic field within the device, and factor the underlying mathematics into a product of three operations which correspond to the primary excitation, some kind of reflection on the surface of the buried object(s) and the corresponding secondary excitation, respectively. Using this factorization we are able to give a justification of the so-called sampling method from inverse scat…
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
2017
International audience; We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the 'energy' parameter E. We show that as |E| -> infinity, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when |E| is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.
On the discrete linear ill‐posed problems
1999
An inverse problem of photo‐acoustic spectroscopy of semiconductors is investigated. The main problem is formulated as the integral equation of the first kind. Two different regularization methods are applied, the algorithms for defining regularization parameters are given. Diskrečiųjų blogai sąlygotų uždavinių klausimu Santrauka Darbe nagrinejamas foto‐akustines spektroskopijos puslaidininkiuose uždavinys, kuriame i vertinami nešeju difuzijos ir rekombinacijos procesai. Reikia atstatyti šaltinio funkcija f(x), jei žinoma antrosios eiles difuzijos lygtis ir atitinkamos kraštines salygos. Naudojantis matavimu, atliktu ivairiuose dažniuose, rezultatais sprendžiamas atvirkštinis uždavinys, kel…
Electromagnetic scattering solutions for digital signal processing
2010
Fixed Angle Inverse Scattering for Almost Symmetric or Controlled Perturbations
2020
We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally controlled potentials are uniquely determined by their fixed angle scattering data. This is done by establishing an equivalence between the frequency domain and the time domain formulations of the problem, and by solving the time domain problem by extending the methods of [RS19] which adapts the ideas introduced in [BK81] and [IY01] on the use of Carleman estimates for inverse problems.