Continuations of the nonlinear Schr\"odinger equation beyond the singularity

We present four continuations of the critical nonlinear \schro equation (NLS) beyond the singularity: 1) a sub-threshold power continuation, 2) a shrinking-hole continuation for ring-type solutions, 3) a vanishing nonlinear-damping continuation, and 4) a complex Ginzburg-Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that leads to a loglog collapse, the sub-threshold power limit is a Bourgain-Wang solution, both before and after the singularity, and the vanishing nonlinear-damping and CGL limits are a loglog solution before the singularity, and have an infinite-velocity{\rev{expanding core}} after the singularity. Our results suggest that all NLS continuations share the universal feature that after the singularity time $T_c$, the phase of the singular core is only determined up to multiplication by $e^{i\theta}$. As a result, interactions between post-collapse beams (filaments) become chaotic. We also show that when the continuation model leads to a point singularity and preserves the NLS invariance under the transformation $t\rightarrow-t$ and $\psi\rightarrow\psi^\ast$, the singular core of the weak solution is symmetric with respect to $T_c$. Therefore, the sub-threshold power and the{\rev{shrinking}}-hole continuations are symmetric with respect to $T_c$, but continuations which are based on perturbations of the NLS equation are generically asymmetric

Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation

We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2\sigma+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^\alpha, where \alpha=(4-\sigma)/(\sigma(d-1)). The blowup rate of these solutions is 1/(3+\alpha) for 4/d\le\sigma<4, and slightly faster than 1/4 for \sigma=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.Comment: 21 pages, 13 figures, research articl

Absorbing boundary conditions for the Westervelt equation

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation

Enhanced Pulse Propagation in Non-Linear Arrays of Oscillators

The propagation of a pulse in a nonlinear array of oscillators is influenced by the nature of the array and by its coupling to a thermal environment. For example, in some arrays a pulse can be speeded up while in others a pulse can be slowed down by raising the temperature. We begin by showing that an energy pulse (1D) or energy front (2D) travels more rapidly and remains more localized over greater distances in an isolated array (microcanonical) of hard springs than in a harmonic array or in a soft-springed array. Increasing the pulse amplitude causes it to speed up in a hard chain, leaves the pulse speed unchanged in a harmonic system, and slows down the pulse in a soft chain. Connection of each site to a thermal environment (canonical) affects these results very differently in each type of array. In a hard chain the dissipative forces slow down the pulse while raising the temperature speeds it up. In a soft chain the opposite occurs: the dissipative forces actually speed up the pulse while raising the temperature slows it down. In a harmonic chain neither dissipation nor temperature changes affect the pulse speed. These and other results are explained on the basis of the frequency vs energy relations in the various arrays

How does neopatrimonialism affect the African state? The case of tax collection in Zambia

Following the neopatrimonialism paradigm, it can be hypothesised that in African states informal politics of the rulers infringe on the collection of taxes and in turn reduce state revenue. This article tests this proposition for the case of Zambia. Neopatrimonial continuity in the country is evidenced by three factors : the concentration of political power, the award of personal favours, and the misuse of state resources. Despite this continuity, the revenue performance increased considerably with the creation of the semi-autonomous Zambia Revenue Authority. Donor pressure has been the most important intervening variable accounting for this improvement. Yet, strengthening the collection of central state revenue has been consistent with a neopatrimonial rationale, and may even have fed neopatrimonialism overall, by providing increased resources for particularistic expenditure

The historical origins of corruption in the developing world: a comparative analysis of East Asia

A new approach has emerged in the literature on corruption in the developing world that breaks with the assumption that corruption is driven by individualistic self-interest and, instead, conceptualizes corruption as an informal system of norms and practices. While this emerging neo-institutionalist approach has done much to further our understanding of corruption in the developing world, one key question has received relatively little attention: how do we explain differences in the institutionalization of corruption between developing countries? The paper here addresses this question through a systematic comparison of seven developing and newly industrialized countries in East Asia. The argument that emerges through this analysis is that historical sequencing mattered: countries in which the "political marketplace" had gone through a process of concentration before universal suffrage was introduced are now marked by less harmful types of corruption than countries where mass voting rights where rolled out in a context of fragmented political marketplaces. The paper concludes by demonstrating that this argument can be generalized to the developing world as a whole

Contextualising Apartheid at the End of Empire: Repression, ‘Development’ and the Bantustans

This article examines the global dynamics of late colonialism and how these informed South African apartheid. More specifically, it locates the programmes of mass relocation and bantustan ‘self-government’ that characterised apartheid after 1959 in relation to three key dimensions. Firstly, the article explores the global circulation of idioms of ‘development’ and trusteeship in the first half of the twentieth century and its significance in shaping segregationist policy; secondly, it situates bantustan ‘selfgovernment’ in relation to the history of decolonisation and the partitions and federations that emerged as late colonial solutions; and, thirdly, it locates the tightening of rural village planning in the bantustans after 1960 in relation to the elaboration of anti-colonial liberation struggles, repressive southern African settler politics and the Cold War. It argues that, far from developing policies that were at odds with the global ‘wind of change’, South African apartheid during the 1960s and 1970s reflected much that was characteristic about late colonial strategy