Abstract
In this article, we model the current and voltage across the weak link between two superconductors. This gives us a nonhomogeneous, nonlinear parametricdriven sineGordon equation with phase shifts. This model equation cannot be solved directly but can be approximated. For the approximations, we use two methods, and analytic perturbation method and the numerical approximation method known as the Runge–Kutta method. For the analytic method, we construct a perturbation expansion method with multiplescale expansion. We discuss the parametricdriven in the sineGordon equation with phase shifts for the 0–π–0 junction. Further, we also describe the breathing modes for various order of perturbation. At the end, we compare the solutions obtained via perturbation and numerical methods of parametricdriven sineGordon equation with phase shifts. Finally, we concluded that the modes of the breathing decay to a constant in both cases. Also we found a good agreement between both approximate methods.
1 Introduction
Brian David was a British physicist who established the mathematical relationships for the voltage and current across the weak link between two superconductors. This phenomenon of super current is called Josephson effect, and the device is known as Josephson junction (JJ). A weak link can possibly consist of superconductor nonsuperconducting superconductor metal (S–N–S) and superconductor insulator superconductor junction (S–I–S), or a physical constriction, which weakens the superconductivity at the point of contact. The S–I–S is shown in Figure 1. Extremely fast switch is one of the most important applications of Josephson effects and two Josephson contacts jointly can be used to measure the magnetic flux.
The compensation of radiation loses along with additional dissipative loses were studied by resonant drive of kink. The resonance taking place when natural wobbling frequency becomes equal to the driving frequency. The emission of secondharmonic radiation becomes a cause of decay of oscillations amplitude in freely wobbling kink given in Oxtoby and Barashenkov [1].
The Josephson Junction having regimes with the phase shift was originated by a rapidly oscillating ACDrive. Furthermore, it is found that the critical length increases by (0–
Overlapped Josephson junction.
Graphenebased Josephson junction.
Electron in an NType GaAs crystal driven by fluctuated electric field.
A new phenomenon was revealed for longtime resonant energy exchange in carbon nanotubes (CNTs) with radial breathing mode. It was found to be stable wide excitation energy range. The modified nonlinear Schrödinger equation (NLS) describes the nonlinear dynamics of CNTs given in Smirnov and Manevitch [5]. An initial value problem of Hamilton’s principle applied to nonconservative systems, was proposed for complex partial differential equations of the NLS type equation. In the study by Rossi et al. [6], dynamics of coherent solitary wave structures of NCVA was examined, and also it is proved that the NLSr and linear perturbative variation equations are equivalent through nonconservative variation methods.
The dubbed oscillations built in the Minkowski background for some nonlinear potentials such as axionic sineGordon potential are a counterpart of the real scalar field. The bosonic fields with mass such as axions, axionlike candidates, and hidden photons. These fields form the breathing mode. The space–time geometry and the field oscillate can interact the cluster at the center of stars. In the study by Brito et al. [7], it was concluded that these stars are generically stable, and also the criteria for instability are provided. By using the sitter geometry, the stability of membranes was analyzed and found that the Jacobi equation specializes to a Klein–Gordon equation as explained in the study by Norma et al. [8].
The application of the magnetic field results in partially gaped spectrum, which is one of the main problems of Rashba spin–orbit coupling in nanowires such as helical gap, spectral function, structure factor, or the tunneling density of states. The form factors of emerging sineGordon model and besonization, a renormalization group analysis, were used to calculate dynamic response functions. It was shown how two types of helical gaps, can be distinguished in experiments given in the study by Pedder et al. [11]. The nontrivial dynamics was expected associated with soliton lattice oscillations and breathing as per sineGordon equation. It is concluded that rich postquench dynamics leads to thermalized and prethermalized stationery states that display strong dependence on the initial groundstate can be seen in the study by Yin and Radzihovsky [12].
The dynamics of a soliton in the generalized NLS with a small external potential
The analytical and numerical solutions of directdriven sineGordon (sG) equation modeled by the Josephen junction on a finite domain with
The source of coherent electromagnetic waves was investigated with unprecedented spectral range given in the study by Wright et al. [16]. The spatiotemporal nonhomogeneous direct drive type sineGordon equation with phase shift and a doublewell potential was considered. An interested study of the closely related model to our current work of sineGordon equation with phase shift was explained by Ali et al. [17]. They also used multiplescale expansion with the perturbation method. It was founded that the algebraic mode decay from discrete to continuous spectrum. In the paper they considered the direct driven vase, while ours is the parameter with extra phase shift.
The radial breathing modes (RBMs) of CNTs is a lowfrequency mode but used for the strongest features observed in the CNT Raman spectrum. The RBMs are modes of vibration characteristic of carbon nanotubes, which do not appear in any other structure (beams, plates, or shells), see, for example, She et al. [18]. Further, the RBMs arise only in the presence of freefree boundary conditions (when the CNTs are left free to vibrate without constraints imposed on the edges), while they do not appear in other boundary conditions (simply supported and clamped), see, for example, Strozzi et al. [19]. Also, RBMs are studied experimentally by resonant Raman spectroscopy and numerically by molecular dynamics simulations, see, for example, Araujo et al. [20], Batra and Gupta [21], and Rehman et al. [9], and Ahmad [10].
In this article, we considered a parametricdriven sineGordon equation with two phase shifts
The outline of this work is given as follows: In Section 2, we introduce the mathematical formulation, the undriven and driven cases, and the detail analytic calculation. While in Section 3, we derived an amplitude equation. The numerical computations are given in Section 4. In Section 5, we provide the results and discussion of this work. Finally, we present the conclusion and some future research work in Section 6. The mathematical derivation is presented in the Appendix section.
2 Mathematical formulation
In this article, we modeled a weak link between two superconductors and obtained the following inhomogeneous, nonlinear parametricdriven sineGordon equation with phase shifts. This is given as follows:
This model is a non linear partial differential equation, which represents the infinitely long Josephson junction. Here,
The unknown
Here,
Here,
Now we use the transformation in Eq. (1) and we obtained
The scaling parameters
The multiplescale spatial and temporal coordinates and their derivatives are given as follows:
with
A special case, i.e., for
This case has been explained in the study by Ali et al. [2] and no need to repeat again.
2.1 Driven case of the model equation
In this work, our main focus is to study the parametricdriven case, i.e., for
2.1.1
O
(
1
)
equation
After expansion and comparison on both sides, we obtain the following equation at first order:
The groundstate solution, i.e.,
2.1.2
O
(
ζ
)
equation
The equation at
By substituting
The breather solution of the aforementioned equation is given by
Here,
2.1.3
O
(
ζ
2
)
equations
We obtained the equation of order
For
Now using
These equations can be expressed with the help of an operator as follows:
Here,
Here,
Now we apply the aforementioned result and obtained the following condition:
The required solution is given as follows:
Now by applying the continuity conditions at the discontinuity points, i.e.,
also, if
2.1.4
O
(
ζ
3
)
equation
We obtained the equation of
Rearranging the aforementioned equation and inserting the known values of
where
The equations given in this Appendix are linear wave equations with forcing at frequencies
Equation for the first Harmonic
We obtain an equation for the first harmonic:
In the aforementioned equations, it should be noted that we have imposed
where
and
The numerical values of
Here,
Analgously,
Equation for the third Harmonic
For third harmonic, we obtain the following system:
By using the same procedure as for
where
Note that due to the assumption
2.1.5
O
(
ζ
4
)
equation
The equation for the
We substitute the values of
By using the solvability condition, we obtain
2.1.6
O
(
ζ
5
)
equation
We obtain the following equation at
By rearranging and inserting known values of
We use the known function and calculate the righthand side to split the solution proportional to the simple harmonic. It is represented as follows:
The equation for the first harmonic is given by
where
The solvability condition of Eq. (17) is given by
Here,
3 Amplitude equation
Now we defined an amplitude equation on the bases of the solvability conditions of each order of
By defining,
We obtain the final amplitude equation as follows:
Here, we also follow Ali et al. [2]. The energy emission in terms of radiation decrease in the amplitude of oscillations. This satisfied the following form of radiation:
Here,
4 Results and discussion of the analytic results
In this work, we study the parametricdriven sineGordon equation with phase shifts. For the numerical approximation, we use the Runge–Kutta method of order four (RK4) with the help of Laplacian operator using a central difference discretization. For the numerical computation, we take the spatial and temporal discretization
The values of the constants are obtained in the aforementioned derivation with the help of MAPLE software. The values are given in Table 1.








The comparison of analytic and numerical solutions for the breathing modes of oscillation is shown in Figure 7. This clearly shows that the modes decay to a constant.
For the analytic solution, we used the multiplescale expansion together with the perturbation method. For the perturbation method, we obtained the equations for the different order of the perturbed parameter
Furthermore, we discussed the driving case for the governing equation. In the driving case, the system oscillate and decay algebraically to a constant rate due to the presence of extra frequency and driving frequency terms. The derived numerical solution of the parametric sineGordon equation is also agreed with the analytical solution of amplitude equation as shown in Figure 7. Moreover, we tried to detect the resonance in the presence of external driving frequency when natural frequency is closed to driving frequency for the 0–
The solutions of the undriven case has been plotted in Figure 6. It is observed in Figure 7 that the mode in the two junction types does not oscillate with an unbounded or growing amplitude. After a while, there is a balance of energy input into the breathing mode due to the external drive and the radiative damping. This result shows the regular oscillation of the mode that the junction voltage disappears even at the condition when the driving frequency is same as the eigenfrequency. This raises for the question at which the breathing mode of a junction with a phase shift can be excited.
5 Conclusion
In this study, we considered inhomogeneous, nonlinear parametricdriven sineGordon equation with phase shifts, also known as nonlinear wave equation. This modeled an infinitely long Josephson junctions, driven by a microwave field. For the analytic solution, we constructed a perturbation series with multiplescale expansion. For a small amplitude of oscillations, we discussed the breathing modes. We studied slowly varying amplitude of the oscillation for the 0–

Funding information: The authors extend their appreciation to the Deans of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through research group program under grant number RGP217643.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
Appendix
A.1 Order
u
2
A.2 Order
u
3
A.3 Order
u
3
, first harmonic
A.4 Order
u
3
third harmonic
A.5 Order
u
5
first harmonic
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