Localized modes in parametrically driven long Josephson junctions with a double-well potential

Abstract

In this paper, we study, both analytically and numerically, the localized modes in long Josephson junctions in the presence of a variety of parametric drives. The phase-shift applied acts as a double-well potential which is known as the junction. The system is described by an inhomogeneous sine-Gordon equation that depicts the dynamics of long Josephson junctions with phase-shift. Using asymptotic analysis together with multiple scale expansion, we obtain the oscillation amplitudes for the junction, which show the decay behaviour with time due to radiative damping and the emission of high harmonic radiations. We found that the energy taken away from the internal modes by radiative waves can be balanced by some external drives. We also discuss in detail the excitation by parametric drives with frequencies to be either in the vicinity or double the natural frequency of the system. It is shown that the presence of externally applied drives stabilizes the nonlinear damping, producing stable breather modes in the junction. It is observed that, in the presence of external drives, the driving effect is stronger for a case of driving frequency nearly equal to the system’s natural frequency as compared to that of the driving frequency nearly equal to twice of the natural frequency of the oscillatory mode. In the numerical section, we calculate the Josephson voltage for the sine-Gordon equation with external drives. We observe that, an infinitely long Josephson junction with double well potential cannot be switched to a resistive state by an external drive with a frequency close to and double the systems eigenfrequency, provided that the driven amplitude is small enough. It is due to the fact that the higher harmonics with frequencies in the phonon band are excited in the form of continuous wave radiation.

Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

“…By considering u≃ sin u, the considered system becomes the Sine-Gordon equation. In future work, it will be interesting to investigate the Sine-Gordon model with nonlinear AC/DC drives with different fractional operators to study the solitonic behavior, localized modes in single and in stacked long Josephson junctions with a variety of potentials, parity time symmetry, the nonlinearity/dispersion effects, and evolution of the localized monotonic shocks [67][68][69][70][71][72][73].…”

BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer

“…Following the procedure outlined in Section II, we extract the residual operator and identify its action on U , 11) and use this to calculate higher-order approximants using B(z) in the iteration sequence (5).…”

“…The residual in the BLUES function and the residual functions for n = 0, 1 and 2 are shown in Fig. 3, together with the residual operator (11) applied to the numerically exact solution (red, full line). Note that the residual functions are negative in the domain around the global maximum of U (z), where U > 0, that they are zero (with a cubic dependence on z) where the curves of Fig.…”

“…which is an equation for the damped and driven Sine-Gordon model [10], often used to describe the dynamics of Josephson junctions in superconductors [11,12]. Another important application of equation ( 9) is the cubic-quintic Duffing oscillator, which is used to describe damped harmonic motion in a nontrivial potential and has become a paradigm for the study of chaos.…”

Analytic iteration procedure for solitons and traveling wavefronts with sources

“…When dealing with solitary waves on shallow water, calculating physically relevant profiles of tidal bores and exploring whether the BLUES function method can be used to refine the recent "minimal analytic model" for this problem [21], is a challenge. In the theory of superconductivity, we envisage applying the method to the sine-Gordon equation, which is used to describe the physics of fluxons in long Josephson junctions under the influence of a driving force [22,23]. In the area of interface growth, the Kardar-Parisi-Zhang equation and the related linear stochastic heat equation come to mind as candidates for application [24].…”