DARBOUX TRANSFORMATIONS FOR EFFECTIVE MASS SCHRÖDINGER EQUATIONS WITH ENERGY-DEPENDENT POTENTIALS

2008 ◽  
Vol 23 (03n04) ◽  
pp. 537-546 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Effective Mass ◽  
Darboux Transformation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Constant Mass ◽  
Darboux Transformations ◽  
First Order ◽  
Dependent Potential ◽  
Energy Dependent

We construct first-order Darboux transformations for stationary Schrödinger equations with position-dependent (effective) mass and energy-dependent potential. We point out characteristic differences to the corresponding Darboux transformation for constant mass.14

DARBOUX TRANSFORMATIONS FOR N-DIMENSIONAL EFFECTIVE MASS SCHRÖDINGER EQUATIONS WITH HYPERSPHERICAL SYMMETRY

2007 ◽  
Vol 22 (19) ◽  
pp. 3293-3304 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Effective Mass ◽  
Darboux Transformation ◽  
Spherical Symmetry ◽  
Schrodinger Equation ◽  
Arbitrary Order ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations

We construct Darboux transformations of arbitrary order for stationary Schrödinger equations with effective mass that exhibit spherical symmetry in N dimensions (hyperspherical symmetry). The Darboux transformation and the associated transformed Schrödinger equation are obtained in closed form.

DARBOUX TRANSFORMATIONS FOR TIME-DEPENDENT SCHRÖDINGER EQUATIONS WITH EFFECTIVE MASS

2006 ◽  
Vol 21 (06) ◽  
pp. 1359-1377 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Wave Function ◽  
Darboux Transformation ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations ◽  
Explicit Formulas ◽  
The Difference ◽  
Dependent Mass ◽  
Position Dependent Mass

The formalism of Darboux transformations is established for time-dependent Schrödinger equations with an effective (position-dependent) mass. Explicit formulas are obtained for the transformed wave function and the difference between the original and the transformed potential. It is shown that for a noneffective mass our Darboux transformation reduces correctly to the well-known Darboux transformation.

REALITY CONDITION FOR DARBOUX TRANSFORMATIONS OF TIME-DEPENDENT SCHRÖDINGER EQUATIONS WITH EFFECTIVE MASS

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4853-4868 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Effective Mass ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations ◽  
Reality Condition ◽  
Set Up

We set up a reality condition for Darboux transformations of time-dependent Schrödinger equations (TDSE) with position-dependent (effective) mass. The reality condition guarantees the potential in the transformed TDSE to be real-valued and hence physical. This paper is a sequel of our former one.36

TRACE FORMULA FOR GREEN'S FUNCTIONS OF EFFECTIVE MASS SCHRÖDINGER EQUATIONS AND NTH-ORDER DARBOUX TRANSFORMATIONS

2008 ◽  
Vol 23 (16n17) ◽  
pp. 2635-2647 ◽  
Author(s):  
EKATERINA POZDEEVA ◽  
AXEL SCHULZE-HALBERG
Keyword(s):  
Effective Mass ◽  
Trace Formula ◽  
Green's Functions ◽  
Arbitrary Order ◽  
Finite Interval ◽  
Green’S Functions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations ◽  
Mass Case

We derive a trace formula for Green's functions of position-dependent (effective) mass Schrödinger equations that are defined on a real, finite interval and connected by a Darboux transformation of arbitrary order. Our findings generalize former results (J. Phys. A37, 10287 (2004)) on constant mass Schrödinger equations to the effective mass case.

Supersymmetry of time-dependent Schrödinger equations with effective mass

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Effective Mass ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Darboux Transformations

AbstractWe establish the supersymmetry formalism for time-dependent Schrödinger equations with effective mass and show that the corresponding supersymmetric transformations are equivalent to effective mass Darboux transformations

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Mixed interaction solutions for the coupled nonlinear Schrödinger equations

2021 ◽  
pp. 2150004
Author(s):  
Yaning Tang ◽  
Jiale Zhou
Keyword(s):  
Darboux Transformation ◽  
Rogue Waves ◽  
Transformation Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Interaction Solutions ◽  
Nonlinear Schrodinger Equations

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple.

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