scholarly journals Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 271
Author(s):  
Hadi Susanto ◽  
Natanael Karjanto ◽  
Zulkarnain ◽  
Toto Nusantara ◽  
Taufiq Widjanarko
Keyword(s):  
Schrödinger Equation ◽  
Josephson Junctions ◽  
Small Amplitude ◽  
Schrodinger Equation ◽  
Gordon Equation ◽  
Complex Dynamics ◽  
Transmission Rate ◽  
Plane Waves ◽  
Star Graph ◽  
Sine Gordon Equation

We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary conditions that are determined by the continuity of the magnetic field and vanishing total fluxes. When one considers small-amplitude breather solutions, the system can be reduced into the nonlinear Schrödinger equation posed on a star graph. Using the equation, we show that a high-velocity incoming soliton is split into a transmitted component and a reflected one. The transmission is shown to be in good agreement with the transmission rate of plane waves in the linear Schrödinger equation on the same graph (i.e., a quantum graph). In the context of the sine-Gordon equation, small-amplitude breathers show similar qualitative behaviors, while large-amplitude ones produce complex dynamics.

Numerical modeling of long Josephson junctions in the frame of double sine-gordon equation

2011 ◽  
Vol 3 (3) ◽  
pp. 389-398 ◽  
Author(s):  
P. Kh. Atanasova ◽  
T. L. Boyadjiev ◽  
Yu. M. Shukrinov ◽  
E. V. Zemlyanaya
Keyword(s):  
Numerical Modeling ◽  
Josephson Junctions ◽  
Gordon Equation ◽  
Sine Gordon Equation

scholarly journals Exact eigenstates of a nanometric paraboloidal emitter and field emission quantities

2018 ◽  
Vol 474 (2214) ◽  
pp. 20170692 ◽  
Author(s):  
A. Chatziafratis ◽  
G. Fikioris ◽  
J. P. Xanthakis
Keyword(s):  
Schrödinger Equation ◽  
Field Emission ◽  
Schrodinger Equation ◽  
Near Field ◽  
Plane Waves ◽  
Spot Size ◽  
Emission Angle ◽  
Electron Velocity ◽  
Cartesian Geometry ◽  
Tip Radius

The progress in field emission theory from its initial Fowler–Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1–5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.

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