67,222 research outputs found
New anallytic approximations based on the Magnus expansion
The Magnus expansion is a frequently used tool to get approximate analytic solutions of time-dependent linear ordinary differential equations and in particular the Schrödinger equation in quantum mechanics. However, the complexity of the expansion restricts its use in practice only to the first terms. Here we introduce new and more accurate analytic approximations based on the Magnus expansion involving only univariate integrals which also shares with the exact solution its main qualitative and geometric propertie
Singular solutions for a class of traveling wave equations arising in hydrodynamics
We give an exhaustive characterization of singular weak solutions for
ordinary differential equations of the form , where is an analytic function. Our motivation stems from the
fact that in the context of hydrodynamics several prominent equations are
reducible to an equation of this form upon passing to a moving frame. We
construct peaked and cusped waves, fronts with finite-time decay and compact
solitary waves. We prove that one cannot obtain peaked and compactly supported
traveling waves for the same equation. In particular, a peaked traveling wave
cannot have compact support and vice versa. To exemplify the approach we apply
our results to the Camassa-Holm equation and the equation for surface waves of
moderate amplitude, and show how the different types of singular solutions can
be obtained varying the energy level of the corresponding planar Hamiltonian
systems.Comment: 24 pages, 5 figure
The radial-hedgehog solution in Landau-de Gennes' theory
We study the radial-hedgehog solution on a unit ball in three dimensions, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. We use a combination of Ginzburg-Landau techniques, perturbation methods and stability analyses to study the qualitative properties of the radial-hedgehog solution, the structure of its defect core, its stability and instability with respect to biaxial perturbations. Our results complement previous work in the field, are rigorous in nature, give information about the role of geometry, elastic constants and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities
Analytic Behaviour of Competition among Three Species
We analyse the classical model of competition between three species studied
by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the
approaches of singularity analysis and symmetry analysis to identify values of
the parameters for which the system is integrable. We observe some striking
relations between critical values arising from the approach of dynamical
systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic
The analytic value of a 3-loop sunrise graph in a particular kinematical configuration
We consider the scalar integral associated to the 3-loop sunrise graph with a
massless line, two massive lines of equal mass , a fourth line of mass equal
to , and the external invariant timelike and equal to the square of the
fourth mass. We write the differential equation in satisfied by the
integral, expand it in the continuous dimension around and solve the
system of the resulting chained differential equations in closed analytic form,
expressing the solutions in terms of Harmonic Polylogarithms. As a byproduct,
we give the limiting values of the coefficients of the expansion at
and .Comment: 9 pages, 3 figure
On the wave length of smooth periodic traveling waves of the Camassa-Holm equation
This paper is concerned with the wave length of smooth periodic
traveling wave solutions of the Camassa-Holm equation. The set of these
solutions can be parametrized using the wave height (or "peak-to-peak
amplitude"). Our main result establishes monotonicity properties of the map
, i.e., the wave length as a function of the wave
height. We obtain the explicit bifurcation values, in terms of the parameters
associated to the equation, which distinguish between the two possible
qualitative behaviours of , namely monotonicity and unimodality.
The key point is to relate to the period function of a planar
differential system with a quadratic-like first integral, and to apply a
criterion which bounds the number of critical periods for this type of systems.Comment: 14 pages, 5 figure
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