On the integrable perturbations of the Camassa-Holm equation

We present an investigation of the nonlinear partial differential equations (PDE) which are asymptotically representable as a linear combination of the equations from the Camassa-Holm hierarchy. For this purpose we use the infinitesimal transformations of dependent and independent variables of the original PDE. This approach is helpful for the analysis of the systems of the PDE which can be asymptotically represented as the evolution equations of polynomial structure. © 2000 American Institute of Physics.

Numerical study of the spherically symmetric Gross-Pitaevskii equation in two space dimensions

The Gross-Pitaevskii equation for Bose-Einstein condensation (BEC) in two space dimensions under the action of a harmonic oscillator trap potential for bosonic atoms with attractive and repulsive interparticle interactions was numerically studied by using time-dependent and time-independent approaches. In both cases, numerical difficulty appeared for large nonlinearity. Nonetheless, the solution of the time-dependent approach exhibited intrinsic oscillation with time iteration which is independent of space and time steps used in discretization.

Galilean covariance and the Duffin-Kemmer-Petiau equation

We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6 × 6 matrices. We consider the harmonic oscillator as an example.

Exact solutions of the dirac equation for modified coulombic potentials

Exact solutions are found for the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions. The method works for the ground state or for the lowest orbital state with l = j - 1/2 , for any j.

Numerical study of the coupled time-dependent Gross-Pitaevskii equation: Application to Bose-Einstein condensation

The Bose-Einstein condensate of several types of trapped bosons at ultralow temperature was described using the coupled time dependent Gross-Pitaevskii equation. Both the stationary and time evolution problems were analyzed using this approach. The ground state stationary wave functions were found to be sharply peaked near the origin for attractive interatomic interaction for larger nonlinearity while for a repulsive interatomic interaction the wave function extends over a larger region of space.

Renormalization of the ladder light-front Bethe-Salpeter equation in the Yukawa model

The reduction of the two-fermion Bethe-Salpeter equation in the framework of light-front dynamics is studied for the Yukawa model. It yields auxiliary three-dimensional quantities for the transition matrix and the bound state. The arising effective interaction can be perturbatively expanded in powers of the coupling constant gs allowing a defined number of boson exchanges; it is divergent and needs renormalization; it also includes the instantaneous term of the Dirac propagator. One possible solution of the renormalization problem of the boson exchanges is shown to be provided by expanding the effective interaction beyond single boson exchange. The effective interaction in ladder approximation up to order g4 s is discussed in detail. It is shown that the effective interaction naturally yields the box counterterm required to be introduced ad hoc previously. The covariant results of the Bethe-Salpeter equation can be recovered from the corresponding auxiliary three-dimensional quantities.

O limiar de esforço percebido (LEP) corresponde à potência crítica e a um indicador de máximo estado estável de consumo de oxigěnio

The perceived exertion has been a target of several investigations, many times with association with objective physiological indicators in exercise. Recently, the identification of the perceived exertion threshold (PET) was proposed in the water running, which presented no difference in relation to the critical velocity. Theoretically, both parameters would be indicators of the maximum steady state of variables such as V̇O2 and blood lactate. The objective of this work was to verify the coincidence between PET, critical power (PCrit) and an indicator of maximum V̇O2 steady state (PCrit') in cycle ergometer. Eight male participants were submitted to progressive effort test in order to determine V̇O2peak (46.7 ± 8.5 ml/kg/min) and to four rectangular tests until exhaustion for the estimation of the critical power model parameters, PET and PCrit'. The hyperbolic relation between mechanical power and time spent for the V̇O2peak to be reached in each test was used for the PCrit' estimation, considered as the asymptote in the power axis, and the portion of the anaerobic work capacity (CTAnaer) depleted up to the establishment of the V̇O2peak (CTAnaer'). In order to identify PET, the straight lines angular coefficients of the perceived exertion in time (ordinate) and the powers used (abscissa) were adjusted to a linear function that provided a point in the power axis in which the perceived exertion would be kept indefinitely stable. The parameters PCrit and CTAnaer were estimated by means of the power-time non-linear equation. In order to compare the estimations of PET, PCrit and PCrit', the analysis of variance ANOVA for repeated measurements was employed, and the associations were established through the Pearson correlation. CTAnaer and CTAnaer' were compared through the t test. PET (180 W ± 61 W), PCrit (174 W ± 43 W) and PCrit' (176 W ± 48 W) were not significantly different and the correlations were of 0.92-0.98. CTAnaer' (14,080 ± 5,219 J) was lower than CTAnaer (22,093 ± 9,042 J). One concludes that the PET predicts the intensity of PCrit and PCrit' with accuracy.

Mapping deformed hyperbolic potentials into nondeformed ones

In this work we introduce a mapping between the so-called deformed hyperbolic potentials, which are presenting a continuous interest in the last few years, and the corresponding nondeformed ones. As a consequence, we conclude that these deformed potentials do not pertain to a new class of exactly solvable potentials, but to the same one of the corresponding nondeformed ones. Notwithstanding, we can reinterpret this type of deformation as a kind of symmetry of the nondeformed potentials. © 2005 Elsevier B.V. All rights reserved.

Hausdorff dimension for non-hyperbolic repellers II: Da diffeomorphisms

We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.

The Dirac-Hestenes equation for spherical symmetric potentials in the spherical and Cartesian gauges

In this paper, using the apparatus of the Clifford bundle formalism, we show how straightforwardly solve in Minkowski space-time the Dirac-Hestenes equation - which is an appropriate representative in the Clifford bundle of differential forms of the usual Dirac equation - by separation of variables for the case of a potential having spherical symmetry in the Cartesian and spherical gauges. We show that, contrary to what is expected at a first sight, the solution of the Dirac-Hestenes equation in both gauges has exactly the same mathematical difficulty. © World Scientific Publishing Company.

On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation?

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Antinucleon spectra in the Dirac equation with scalar and vector Wood-Saxon potentials

We analyze here the spin and pseudospin symmetry for the antinucleon spectra solving the Dirac equation with scalar and vector Wood-Saxon potentials. In relativistic nuclear mean field theories where these potentials have large magnitudes and opposite signs we show that contrary to the nucleon case where pseudospin interaction is never very small and cannot be treated perturbatively, for antinucleon systems this interaction is perturbative and an exact pseudospin symmetry is possible. This result manifests the relativistic nature of the nuclear pseudospin symmetry. © 2009 American Institute of Physics.

Darboux-Bäcklund transformations and rational solutions of the painlevé IV equation

Rational solutions of the Painlevé IV equation are constructed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy subject to the additional non-isospectral Virasoro symmetry constraint. Convenient Wronskian representations for rational solutions are obtained by successive actions of the Darboux-Bäcklund transformations. ©2010 American Institute of Physics.

Stability of the null solution of the equation ẋ(t) = -a(t)x(t) + b(t)x([t])

The asymptotic stability of the null solution of the equation ẋ(t) = -a(t)x(t)+b(t)x([t]) with argument [t], where [t] designates the greatest integer function, is studied by means of dichotomic maps. © 2010 Academic Publications.

Sharp Turán inequalities via very hyperbolic polynomials

We present new sharp inequalities for the Maclaurin coefficients of an entire function from the Laguerre-Pólya class. They are obtained by a new technique involving the so-called very hyperbolic polynomials. The results may be considered as extensions of the classical Turán inequalities. © 2010 Elsevier Inc.