Decay of bound states in a sine-Gordon equation with double-well potentials

Ali, Amir; Susanto, Hadi; Wattis, Jonathan A. D.

doi:10.1063/1.4917284

Cited by 9 publications

(7 citation statements)

References 25 publications

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“…The error bound that is of order O(ǫ 3/2 ) in Theorem 1 is linked to the choice of our rotating wave ansatz (3) that creates a residue of order O(ǫ 5/2 ). If we include a higher-order correction term in the ansatz (3), see Chapter 5.3 of [9] for the procedure to do it, we will obtain a smaller residue and in return a smaller error bound.…”

Section: Resultsmentioning

confidence: 99%

“…In this paper, we consider a Klein-Gordon equation with external damping and drive. Our present work will be relevant to models appearing in the study of, e.g., superconducting Josephson junctions [3], mechanical oscillators [18], electrical lattices [10], etc. Using the rotating-wave approximation, we will show that the corresponding modulation equation is a damped, driven discrete nonlinear Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

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Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

Muda

Akbar

Kusdiantara

et al. 2020

ASY

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We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

Muda

Akbar

Kusdiantara

et al. 2020

ASY

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“…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].…”

Section: Discussionmentioning

confidence: 99%

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

Saifullah

Ali

Irfan

et al. 2021

Mathematical Problems in Engineering

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In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

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“…We consider a stable state solution ψ 0 = 0 and linearizing about the ground state, one obtains the breathing mode [33] ψ 1 (X 0 , . .…”

Section: Driving Frequency Close To Natural Frequency Of the Systemmentioning

confidence: 99%

“…Recently, the localized modes excitation has been studied in 0 − π − 0 long Josephson junctions for single-mode and two mode oscillations [30][31][32][33] by considering an inhomogeneous sine-Gordon equation with a variety of direct and parametric drives. It has been reported that in 0 − π − 0 long Josephson junctions, the amplitudes of the breathing modes decay with time due to radiative damping and emission of high harmonic radiations, which can be re-balanced by applying appropriate external drives.…”

Section: Introductionmentioning

confidence: 99%

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

Gul

Ali

Ullah³

2018

J. Phys. A: Math. Theor.

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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.

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Decay of bound states in a sine-Gordon equation with double-well potentials

Cited by 9 publications

References 25 publications

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

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

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

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