Standing Waves in a Non-linear 1D Lattice : Floquet Multipliers, Krein Signatures, and Stability

We construct a class of exact commensurate and incommensurate standing wave (SW) solutions in a piecewise smooth analogue of the discrete non-linear Schr\"{o}dinger (DNLS) model and present their linear stability analysis. In the case of the commensurate SW solutions the analysis reduces to the eigenvalue problem of a transfer matrix depending parametrically on the eigenfrequency. The spectrum of eigenfrequencies and the corresponding eigenmodes can thereby be determined exactly. The spatial periodicity of a commensurate SW implies that the eigenmodes are of the Bloch form, characterised by an even number of Floquet multipliers. The spectrum is made up of bands that, in general, include a number of transition points corresponding to changes in the disposition of the Floquet multipliers. The latter characterise the different band segments. An alternative characterisation of the segments is in terms of the Krein signatures associated with the eigenfrequencies. When one or more parameters characterising the SW solution is made to vary, one occasionally encounters collisions between the band-edges or the intra-band transition points and, depending on the the Krein signatures of the colliding bands or segments, the spectrum may stretch out in the complex plane, leading to the onset of instability. We elucidate the correlation between the disposition of Floquet multipliers and the Krein signatures, presenting two specific examples where the SW possesses a definite window of stability, as distinct from the SW's obtained close to the anticontinuous and linear limits of the DNLS model.Comment: 31 pages, 11 figure

An Idiom for India: Hindustani and the Limits of the Language Concept

This essay explores the cultural legacy of Hindustani, which names the intimate overlap between two South Asian languages, Hindi and Urdu. Hindi and Urdu have distinct religious identities, national associations and scripts, yet they are nearly identical in syntax, diverging to some extent in their vocabulary. Hindi and Urdu speakers, consequently, understand each other most of the time, but not all of the time, though they can never read each other’s texts. Their shared space, Hindustani, finds no official recognition in India or in Pakistan, but it denotes, particularly in the early twentieth century, an aspiration for Hindu–Muslim unity: the dream of a shared, syncretic culture, crafted from the speech genres of everyday life. Beginning with the colonial project of Hindustani, the essay focuses on a discussion of the works of early twentieth-century writers like Nehru, Premchand and Sa’adat Hasan Manto. I argue that the aesthetic project of Hindustani attempted to produce, not a common language, but a common idiom: a set of shared conventions, phrases and forms of address, which would be legible to Indians from all religions and all regions. By theorizing Hindustani as an idiom, and not a language, I explain its persistence in Bollywood cinema well after its abandonment in all literary and official registers. Bollywood, I argue, is Hindustani cinema, not only because of its use of a mixed Hindi–Urdu language in its dialogues, but also because of its development of a set of clearly recognizable, easily repeatable conventions that can surmount linguistic differences

A new version of Brakke's local regularity theorem

Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's local regularity theorem yields that $(\mu_t)$ is actually smooth and graphical inside a smaller ball for times $t\in (C,T-C)$ for some constant $C$. Here we extend this result to times $t\in (C,T)$. The main idea is to prove that a Brakke flow that is initially locally graphical with small gradient will remain graphical for some time. Moreover we use the new local regularity theorem to generalise White's regularity theorem to Brakke flows.Comment: 40 pages, corrected an error in the former Lemma 4.4 (now Lemma 5.4), added a section about curvature estimates (used for the mentioned correction