Roberto André Kraenkel

Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length

We consider, by means of the variational approximation (VA) and direct numerical simulations of the Gross-Pitaevskii (GP) equation, the dynamics of two-dimensional (2D) and 3D condensates with a scattering length containing constant and harmonically varying parts, which can be achieved with an ac magnetic field tuned to the Feshbach resonance. For a rapid time modulation, we develop an approach based on the direct averaging of the GP equation, without using the VA. In the 2D case, both VA and direct simulations, as well as the averaging method, reveal the existence of stable self-confined condensates without an external trap, in agreement with qualitatively similar results recently reported for spatial solitons in nonlinear optics. In the 3D case, the VA again predicts the existence of a stable self-confined condensate without a trap. In this case, direct simulations demonstrate that the stability is limited in time, eventually switching into collapse, even though the constant part of the scattering length is positive (but not too large). Thus a spatially uniform ac magnetic field, resonantly tuned to control the scattering length, may play the role of an effective trap confining the condensate, and sometimes causing its collapse.

Coherent atomic oscillations and resonances between coupled Bose-Einstein condensates with time-dependent trapping potential

We study the quantum coherent tunneling between two Bose-Einstein condensates separated through an oscillating trap potential. The cases of slow and rapid varying in the time trap potential are considered. In the case of a slowly varying trap, we study the nonlinear resonances and chaos in the oscillations of the relative atomic population. Using the Melnikov function approach, we find the conditions for chaotic macroscopic quantum-tunneling phenomena to exist. Criteria for the onset of chaos are also given. We find the values of frequency and modulation amplitude which lead to chaos on oscillations in the relative population, for any given damping and the nonlinear atomic interaction. In the case of a rapidly varying trap, we use the multiscale expansion method in the parameter «51/V, where V is the frequency of modulations, and we derive the averaged system of equations for the modes. The analysis of this system shows that new macroscopic quantum self-trapping regions, in comparison with the constant trap case, exist.

Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms

We consider formation of dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms. It is shown that big enough initial inhomogeneity of density leads to wave breaking phenomenon followed by generation of a train of dark solitons. Analytical theory is confirmed by numerical simulations.

The Korteweg-de Vries hierarchy and long water-waves

By using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface‐waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion, we show that the Korteweg–de Vries hierarchy equations do play a role in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.

Biodiversity Can Help Prevent Malaria Outbreaks in Tropical Forests

Background Plasmodium vivax is a widely distributed, neglected parasite that can cause malaria and death in tropical areas. It is associated with an estimated 80–300 million cases of malaria worldwide. Brazilian tropical rain forests encompass host- and vector-rich communities, in which two hypothetical mechanisms could play a role in the dynamics of malaria transmission. The first mechanism is the dilution effect caused by presence of wild warm-blooded animals, which can act as dead-end hosts to Plasmodium parasites. The second is diffuse mosquito vector competition, in which vector and non-vector mosquito species compete for blood feeding upon a defensive host. Considering that the World Health Organization Malaria Eradication Research Agenda calls for novel strategies to eliminate malaria transmission locally, we used mathematical modeling to assess those two mechanisms in a pristine tropical rain forest, where the primary vector is present but malaria is absent. Methodology/Principal Findings The Ross–Macdonald model and a biodiversity-oriented model were parameterized using newly collected data and data from the literature. The basic reproduction number () estimated employing Ross–Macdonald model indicated that malaria cases occur in the study location. However, no malaria cases have been reported since 1980. In contrast, the biodiversity-oriented model corroborated the absence of malaria transmission. In addition, the diffuse competition mechanism was negatively correlated with the risk of malaria transmission, which suggests a protective effect provided by the forest ecosystem. There is a non-linear, unimodal correlation between the mechanism of dead-end transmission of parasites and the risk of malaria transmission, suggesting a protective effect only under certain circumstances (e.g., a high abundance of wild warm-blooded animals). Conclusions/Significance To achieve biological conservation and to eliminate Plasmodium parasites in human populations, the World Health Organization Malaria Eradication Research Agenda should take biodiversity issues into consideration.

Camassa-Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions

In this paper a relation between the Camassa-Holm equation and the non-local deformations of the sinh-Gordon equation is used to study some properties of the former equation. We will show that cuspon and soliton solutions can be obtained from soliton solutions of the deformed sinh-Gordon equation.

Asymptotic soliton train solutions of Kaup-Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h0(x) and u0(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u0(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.

Two-dimensional integrable generalization of the Camassa-Holm equation

Abstract In this letter we discuss the (2+1)-dimensional generalization of the Camassa–Holm equation. We require that this generalization be, at the same time, integrable and physically derivable under the same asymptotic analysis as the original Camassa–Holm equation. First, we find the equation in a perturbative calculation in shallow-water theory. We then demonstrate its integrability and find several particular solutions describing (2+1) solitary-wave like solutions.

Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation

Abstract In this Letter we investigate Lie symmetries of a (2+1)-dimensional integrable generalization of the Camassa–Holm (CH) equation. Through the similarity reductions we obtain four different (1+1)-dimensional systems of partial differential equations in which one of them turns out to be a (1+1)-dimensional CH equation. We establish their integrability by providing the Lax pair for all of them. Further, we present a brief analysis for some types of particular solutions which include the cuspon, peakon and soliton solutions for the two-dimensional generalization of the CH equation.

Catastrophic Regime Shift in Water Reservoirs and São Paulo Water Supply Crisis

The relation between rainfall and water accumulated in reservoirs comprises nonlinear feedbacks. Here we show that they may generate alternative equilibrium regimes, one of high water-volume, the other of low water-volume. Reservoirs can be seen as socio-environmental systems at risk of regime shifts, characteristic of tipping point transitions. We analyze data from stored water, rainfall, and water inflow and outflow in the main reservoir serving the metropolitan area of São Paulo, Brazil, by means of indicators of critical regime shifts, and find a strong signal of a transition. We furthermore build a mathematical model that gives a mechanistic view of the dynamics and demonstrates that alternative stable states are an expected property of water reservoirs. We also build a stochastic version of this model that fits well to the data. These results highlight the broader aspect that reservoir management must account for their intrinsic bistability, and should benefit from dynamical systems theory. Our case study illustrates the catastrophic consequences of failing to do so.