scholarly journals Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

2021 ◽  
Vol 2090 (1) ◽  
pp. 012038
Author(s):  
A Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Explicit Form ◽  
Higher Order ◽  
Diagonal Matrix ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
Dirac Systems ◽  
Matrix Potential ◽  
De Vries ◽  
The Matrix

Abstract We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

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.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

Numerical Simulation of a Soil Filtration Treatment Based on Balance

1986 ◽  
Vol 18 (7-8) ◽  
pp. 391-396
Author(s):  
M. Suzuki ◽  
H. Kawashima ◽  
T. Kawanishi
Keyword(s):  
Oxygen Diffusion ◽  
Infiltration Rate ◽  
Distribution Patterns ◽  
Operating Conditions ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Matrix Potential ◽  
Two Dimensional Model ◽  
The Matrix ◽  
Soil Filtration

A sophisticated two-dimensional model to simulate the concentration distributions of BOD, oxygen and biomass in a soil filtration treatment system is presented. The model incorporates, in addition to infiltration of wastewater, oxygen diffusion in a soil and bacterial growth and respiration. The matrix potential concept was used to describe two-dimensional infiltration from the source. The model showed that the distribution patterns of biomass in a soil are governed both by the rate of oxygen diffusion and by the wastewater infiltration rate. Applying the model to a trench type plant under ordinary operating conditions, it became apparent that only the soil adjacent to the trench is utilized for the treatment.

Homogenization of Rough Two-Dimensional Interfaces Separating Two Anisotropic Solids

2011 ◽  
Vol 78 (4) ◽  
Author(s):  
Pham Chi Vinh ◽  
Do Xuan Tung
Keyword(s):  
Explicit Form ◽  
Symmetry Plane ◽  
Homogenization Method ◽  
Two Dimensional ◽  
Linear Elasticity Theory ◽  
Practical Applications ◽  
Component Form ◽  
The Matrix ◽  
Straight Lines ◽  
Dimensional Domain

In this paper we have derived homogenized equations in explicit form of the linear elasticity theory in a two-dimensional domain with an interface highly oscillating between two straight lines, by using the homogenization method. First, the homogenized equation in the matrix form for generally anisotropic materials is obtained. Then, it is written down in the component form for specific cases when the materials are orthotropic, monoclinic with the symmetry plane at X1=0 and X2=0. Since these equations are in explicit form, they are significant in practical applications.

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