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EN
In the present paper, the fractional-order cubic nonlinear Schrödinger equation is considered. The Schrödinger equation with time and space fractional derivative is studied at the same time. Different types of travelling wave solutions including the kink solution, soliton solution, periodic solution, and singular solution for the mentioned equation are obtained by using the Jacobi elliptic functions expansion method. It is shown that the solutions turn into the exact solutions when the fractional orders go to 1. This method can be relied on gaining the solutions to time or space fractional order partial differential equations as well as ordinary ones. Throughout this work, the fractional derivative is given in a conformable sense.
EN
In this research, our purpose is to investigate some types of solutions to a simplified modified form of the Camassa-Holm equation. The Jacobi elliptic function expansion method is applied to this equation. Then, a lot of travelling wave solutions are obtained. The derived solutions are in the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric functions. Graphics of solutions are drawn in order to determine the types of the solutions. Furthermore, different kinds of solutions such as the singular kink wave solution, the kink wave solution, and the periodic solution are achieved.
3
Fluxon dynamics in the curved Josephson junction
EN
Purpose: The purpose of this report is to present the similarities and differences between modified sine-Gordon models used in description of the curved Josephson junctions. The leading dynamical variable in this system is a gauge invariant phase difference of the macroscopic wave functions of the superconducting electrodes that form the junction. Findings: The main finding of this article is the observation that in the model used in description of junctions with quickly varying curvatures the significant part of the kink energy is confined in the curved regions of the junction. Research limitations/implications: The paper is limited to the description of the dynamics of fluxions in the long Josephson junctions. These junctions due to small transverse sizes (smaller than the Josephson penetration depth) can be considered as a one dimensional systems. Practical implications: It seems that junctions with appropriate geometry will find applications in future electronic devices. It is expected that curved Josephson junctions can be used in order to store a binary data. Originality/value: The main idea of the paper is to use a Riemann geometry in order to describe the influence of the curvature on the kink motion in the junction.
EN
Purpose: Purpose of this report is to present the impact of the spherical geometry on the kink motion in large area Josephson junction. Design/methodology/approach: The effective Lagrangian method is used in order to obtain the dynamics of the sine-Gordon kink on a curved manifold. Findings: The main finding of this article is an effective potential that describes the influence of the curvature on the kink motion in the spherical Josephson junction. Research limitations/implications: The paper is limited to the description of the dynamics of kinks that move with non relativistic speeds in narrow Josephson junctions. Practical implications: It seems that junctions with appropriate geometry will find applications in future electronic devices. It is expected that curved Josephson junctions can be used in order to store a binary data. Originality/value: The main idea of the paper is to use a Riemann geometry in order to describe the influence of the curvature on the kink motion in the spherical junction.
5
Precise tuning of the kink width in the long Josephson junction
EN
Purpose: The purpose of this report is to present how the external magnetic field can be employed in order to precise control the kink width in the long Josephson junction. Design/methodology/approach: In this paper we concentrate on construction of the analytical kink solutions of the sine-Gordon model with the profile modified by the external magnetic field. Findings: The main findings of this article are exact solutions of the sine-Gordon model which describe the squeezed or stretched kinks. Research limitations/implications: The paper is limited to the description of the dynamics of the long Josephson junctions which are one dimensional systems with stable kink structures. Practical implications: It is expected that the possibility of control the width of the kink will find applications in future electronic devices. Originality/value: The main idea of the paper is to use some special magnetic field configurations to modulate (precisely) the properties of the Josephson junction.
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