Steady water waves with multiple critical layers: interior dynamics

We study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat's-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function.Comment: 14 pages, 3 figures. As accepted for publication in J. Math. Fluid Mec

Modulational Instability in Equations of KdV Type

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic

On the particle paths and the stagnation points in small-amplitude deep-water waves

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.Comment: to appear in J. Math. Fluid Mech. arXiv admin note: text overlap with arXiv:1106.382

Politicising government engagement with corporate social responsibility: “CSR” as an empty signifier

Governments are widely viewed by academics and practitioners (and society more generally) as the key societal actors who are capable of compelling businesses to practice corporate social responsibility (CSR). Arguably, such government involvement could be seen as a technocratic device for encouraging ethical business behaviour. In this paper, we offer a more politicised interpretation of government engagement with CSR where “CSR” is not a desired form of business conduct but an element of discourse that governments can deploy in structuring their relationships with other social actors. We build our argument through a historical analysis of government CSR discourse in the Russian Federation. Laclau and Mouffe's (Hegemony and socialist strategy: Towards a radical democratic politics,Verso Books, London, 1985) social theory of hegemony underpins our research. We find that “CSR” in the Russian government’s discourse served to legitimise its power over large businesses. Using this case, we contribute to wider academic debates by providing fresh empirical evidence that allows the development of critical evaluation tools in relation to governments’ engagement with “CSR”. We find that governments are capable of hijacking CSR for their own self-interested gain. We close the paper by reflecting on the merit of exploring the case of the Russian Federation. As a “non-core”, non-western exemplar, it provides a useful “mirror” with which to reflect on the more widely used test-bed of Western industrial democracies when scrutinising CSR. Based on our findings, we invite other scholars to adopt a more critical, politicised stance when researching the role of governments in relation to CSR in other parts of the world

Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of 2π-periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x| 1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the H¨older and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities