First-order Darboux transformations for Dirac equations with arbitrary diagonal potential matrix in two dimensions

2021 ◽  
Vol 136 (7) ◽  
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Two Dimensions ◽  
Darboux Transformations ◽  
First Order ◽  
Dirac Equations ◽  
Potential Matrix

A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces

2017 ◽  
Vol 17 (4) ◽  
pp. 601-616 ◽  
Author(s):  
Zheng Li ◽  
Shuo Zhang
Keyword(s):  
Biharmonic Equation ◽  
Two Dimensions ◽  
Zero Order ◽  
Helmholtz Decomposition ◽  
Numerical Verification ◽  
Order Function ◽  
First Order ◽  
Mixed Element ◽  
New Formulation ◽  
Element Method

AbstractThis paper studies the mixed element method for the boundary value problem of the biharmonic equation {\Delta^{2}u=f} in two dimensions. We start from a {u\sim\nabla u\sim\nabla^{2}u\sim\operatorname{div}\nabla^{2}u} formulation that is discussed in [4] and construct its stability on {H^{1}_{0}(\Omega)\times\tilde{H}^{1}_{0}(\Omega)\times\bar{L}_{\mathrm{sym}}^% {2}(\Omega)\times H^{-1}(\operatorname{div},\Omega)}. Then we utilise the Helmholtz decomposition of {H^{-1}(\operatorname{div},\Omega)} and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.

Darboux transformation of the second-type nonlocal derivative nonlinear Schrödinger equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950123 ◽  
Author(s):  
De-Xin Meng ◽  
Kuang-Zhong Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Darboux Transformations ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation

The second-type nonlocal derivative nonlinear Schrödinger (NDNLSII) equation is studied in this paper. By constructing its [Formula: see text]-order Darboux transformations (DT) from the first-order DT, Vandermonde-type determinant solutions of the NDNLSII equation are obtained from zero seed solutions, which would be singular unless the square of eigenvalues are purely imaginary.

scholarly journals Darboux transformations for a generalized Dirac equation in two dimensions

2010 ◽  
Vol 51 (11) ◽  
pp. 113501 ◽  
Author(s):  
Ekaterina Pozdeeva ◽  
Axel Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Two Dimensions ◽  
Darboux Transformations ◽  
Generalized Dirac Equation

Closed-form representations of iterated Darboux transformations for the massless Dirac equation

2021 ◽  
Vol 36 (08n09) ◽  
pp. 2150064
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Closed Form ◽  
Scalar Potential ◽  
Higher Order ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
First Order ◽  
Dirac Systems ◽  
Form Representation ◽  
Exactly Solvable

It is shown that first-order Darboux transformations for the two-dimensional massless Dirac equation with scalar potential and for the Schrödinger equation are the same up to a change of coordinates. As a consequence, we obtain a closed-form representation of iterated, higher-order Darboux transformations for our Dirac equation. We use the formalism to generate several new exactly-solvable Dirac systems through higher-order Darboux transformations.

Melting in two dimensions: first-order versus continuous transition

2002 ◽  
Vol 314 (1-4) ◽  
pp. 396-404 ◽  
Author(s):  
V.N. Ryzhov ◽  
E.E. Tareyeva
Keyword(s):  
Two Dimensions ◽  
Continuous Transition ◽  
First Order

scholarly journals SEPARATION OF COUPLED SYSTEMS OF SCHRODINGER EQUATIONS BY DARBOUX TRANSFORMATIONS

2012 ◽  
Vol 24 (03) ◽  
pp. 1250006 ◽  
Author(s):  
MAYER HUMI
Keyword(s):  
Two Dimensions ◽  
Explicit Representation ◽  
Coupled Systems ◽  
Schrodinger Equations ◽  
Matrix Functions ◽  
Schrödinger Equations ◽  
Darboux Transformations ◽  
Complex Matrix ◽  
Independent Variable ◽  
Mathematics And Physics

Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper, we introduce a novel application of these transformations in two dimensions to decouple systems of Schrodinger equations. We derive explicit representation for three classes of such systems which can be decoupled by such transformations. We also show that there is an elegant relationship between these transformations and analytic complex matrix functions.

scholarly journals Equations for massless and massive spin-1/2 particles with varying speed and neutrino in matter

2014 ◽  
Vol 29 (06) ◽  
pp. 1450031 ◽  
Author(s):  
S. I. Kruglov
Keyword(s):  
Energy Spectrum ◽  
Density Matrix ◽  
Lorentz Invariance ◽  
First Order ◽  
Dirac Equations ◽  
Order Form ◽  
Invariance Violation ◽  
Massive Spin ◽  
Lorentz Invariance Violation ◽  
Massless Fermions

The modified Dirac equations describing massless and massive spin-1/2 particles violating the Lorentz invariance are considered. The equation for massless fermions with varying speed is formulated in the 16-component first-order form. The projection matrix, which is the density matrix, extracting solutions to the equation has been obtained. Exact solutions to the equation and energy spectrum for a massive neutrino are obtained in the presence of background matter. We have considered as subluminal as well as superluminal propagations of neutrinos. The bounds on the Lorentz invariance violation parameter from astrophysical observations by IceCube, OPERA, MINOS collaborations and by the SN1987A supernova are obtained for superluminal neutrinos.

DYNAMIC SCALING FOR FIRST-ORDER PHASE TRANSITIONS

2000 ◽  
Vol 11 (03) ◽  
pp. 553-559
Author(s):  
BANU EBRU ÖZOĞUZ ◽  
YIĞIT GÜNDÜÇ ◽  
MERAL AYDIN
Keyword(s):  
Phase Transitions ◽  
Finite Size ◽  
Two Dimensions ◽  
Dynamic Scaling ◽  
Finite Size Scaling ◽  
Potts Models ◽  
Time Dynamics ◽  
First Order ◽  
Short Time ◽  
First Order Phase

The critical behavior in short time dynamics for the q = 6 and 7 state Potts models in two-dimensions is investigated. It is shown that dynamic finite-size scaling exists for first-order phase transitions.

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