Introducing an Unexplored South Asian Treatise on I‘jāz Al-Qur’an

The present paper explores and introduces a manuscript on i‘jaz al-Qur’ān (inimitability of the Qur’ān) named Nūr al- Īqān bi i‘jaz al-Qur’ān written by a profound South Asian Sindhi Scholar, Maulana Muhammad Ismā‘īl al-Ūdvī. The field of i‘jaz al-Qur’ān has been and is the field of interest of many Muslim as well as non-Muslim theologians and scholars. Many Muslim scholars of the Middle East and the West have written books and research papers from third/ninth Century up to present times. The main and significant reason for this is that the doctrine of i‘jaz al- Qur’ān proves the prophecy of Prophet Muhammad peace be upon him. This paper argues that Shaikh al-Ūdvī is the first South Asian Sindhi Muslim Scholar, who wrote a separate treatise in Arabic on the doctrine of i‘jaz al-Qur’ān, which is not introduced properly in academic circles of the day. In this treatise, al-Ūdvī has advanced some new and original arguments to support the doctrine and the prophecy of Prophet Muhammad peace be upon him.

Localized Defect Modes in Parametically Driven Long Josephson Junctions With Phase Shifts

In this work, we investigate localized defect modes in long Josephson junctions (LJJs) with phase shifts. To study LJJs, we consider inhomogeneous sine-Gordon equations modeling infinitely long Josephson junctions and are driven by a variety of ac-direct andparametricdrives. Perturbationtechniquetogetherwithmultiplescaleexpansions is applied to attain the amplitude equations for slowly oscillations. The obtained oscillation amplitudes decay with time due to radiative damping and emission of high harmonicradiations. Theappropriateexternaldrivesareappliedtore-balancethedissipative and radiative losses. The excitation by direct and parametric drives with frequencies to be either in the vicinity or double of the natural frequency are discussed. It is observed that the external applied drives stabilize the non-linear damping, producing a stable breather mode oscillations. It is also confirmed that the driving frequency lies in the vicinity of the natural mode frequency is much more effective than that of twice the natural mode frequency. Particularly, the phase discontinuities, socalled0−π−0and0−κ junctionsarestudied. Numericalsimulationsareperformed, where novel results are obtained. Next, we consider the external driving force to be rapidly oscillating and applying technique of perturbation together with method of averaging to obtain an average non-linear dynamics, which illustrate the slowly varying dynamics of the considered equations. Theobtainedequationsaredoublesin-Gordonequationswhichshowsome novel characteristics for the junctions under the influence of the considered external ac-drives. Weanalyzethethresholddistanceof0−π−0junctionsandthecriticalbias current in 0−κ junctions in the presence of direct and parametric ac-drives. We also use the Euler-Lagrange approximation to discuss the localized modes in long Josephson junctions for both the junctions in an infinite domain, in the presence of a variety of dynamics. WefurtherstudythelocalizedmodesinlongJosephsonjunctionswithdouble-wellpotential, so-called 0−π−0−π−0 junctions. The system is illustrated by an inhomogeneoussine-Gordonequationwithparametricdrives. Itisobservedthat,theobtained amplitude equations decay with time due to radiative damping and emission of high harmonic radiations in the presence of double well. It is concluded that the energy taken away from the internal mode by radiation can be balanced by either applying direct or parametric drives. The external drives are applied to re-balance the dissipative and radiative losses. We discuss in detail the excitation by parametric drives with frequenciestobeeitherinthevicinityordoubleofthenaturalfrequencyofthesystem. Itisnotedthatthepresenceofexternallyapplieddrivesstabilizesthenonlineardamping,producingstablebreathermodes. Furthermore,inthepresenceofexternaldriving, the driving effect is more stronger for the case of driving frequency nearly equal to the system natural frequency as compared to that of the driving frequency nearly equal to twice the frequency of the oscillatory mode.