Search results for "Separation of variables"
showing 7 items of 7 documents
Fractional hereditariness of lipid membranes: Instabilities and linearized evolution
2016
In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elas- ticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is studied by minimizing the Gibbs free energy which penalizes perturbations of the changes of areal stretch and their gradients only [1]. As material instabilities arise whenever areal stretches characterizing homogeneous configurations lie inside the spinoidal zone of the free energy density, bifurcations from such configurations are shown to occur as oscillatory perturbations of the in-plane displacement. Experimental observations [2] show a power-law in-pl…
Analytic solutions of the diffusion-deposition equation for fluids heavir than atmospheric air
2008
A steady-state bi-dimensional turbulent diffusion equation was studied to find the concentration distribution of a pollutant near the ground. We have considered the air pollutant emitted from an elevated point source in the lower atmosphere in adiabatic conditions. The wind velocity and diffusion coefficient are given by power laws. We have found analytical solutions using or the Lie Group Analysis or the Method of Separation of Variables. The classical diffusion equation has been modified introducing the falling term with non-zero deposition velocity. Analytical solutions are essential to test numerical models for the great difficulty in validating with experiments.
Unitarity of the SoV Transform for the Toda Chain
2014
The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one should construct the map explicitly and then show that it is unitary. In the present paper, we develop a technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the …
Solution of XXZ and XYZ spin chains with boundaries by separation of variables
2014
In this thesis we give accounts on the solution of the open XXZ and XYZ quantum spin-1/2 chains with the most generic integrable boundary terms. By using the the Separation of Variables method (SoV), due to Sklyanin, we are able, in the inhomogeneous case, to build the complete set of eigenstates and the associated eigenvalues. The characterization of these quantities is made through a maximal system of N quadratic equations, where N is the size of the chain. Different methods, like the Algebraic Bethe ansatz (ABA) or other generalized Bethe ansatz techniques, have been used, in the past, in order to tackle these problems. None of them resulted effective in the reproduction of the full set …
Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms
2013
We use the quantum separation of variable (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates in the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables
2014
28 pages; International audience; We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to…
Mathematical Models and their Solutions for Domains of Compex Form
2014
Promocijas darbā tiek apskatīti dažādi oriģināli modeļi un to risinājumi sarežģītas formas apgabaliem. Intensīvās tērauda rūdīšanas procesi sistēmām ar ribām tiek aprakstīti ar 3D hiperbolisko, kā arī ar klasisko siltuma vadīšanas vienādojumu. Precīzā atrisinājuma iegūšanai izmantota Grīna funkciju metode un tās vispārinājums. Modernajos datoros sastopamajām sistēmām ar dubulsieniņu un dubultribu dota stacionārā un nestacionārā siltumvadīšanas problēma 2D gadījumā. Tās risinājums tiek iegūts ar konservatīvās viduvēšanas metodi, galīgo diferenču metodi un tās modifikāciju robežnosacījumiem. Piedāvāts jauns matemātiskais modelis vītola flautai, problēmas formulējumā izmantojot 1D lineāru viļņ…