243 research outputs found
On the long-time asymptotic behavior of the modified korteweg-de vries equation with step-like initial data
We study the long-time asymptotic behavior of the solution q(x; t), of the modified Korteweg-de Vries equation (MKdV) with step-like initial datum q(x, 0). For the exact step initial data q(x,0)=c_+ for x>0 and q(x,0)=c_- for x<0, the solution develops an oscillatory region called the dispersive shock wave region that connects the two constant regions c_- and c_+ at x=-infinity and x=+infinity. We show that the dispersive shock wave is described by a modulated periodic traveling wave solution of the MKdV equation where the modulation parameters evolve according to a Whitham modulation equation. The oscillatory region is expanding within a cone in the (x,t) plane. For step-like initial data we show that the solution decomposes for long times into three main regions: (1) a region where solitons and breathers travel with positive velocities on a constant background c_+, (2) an expanding oscillatory region (that generically contains breathers); (3) a region of breathers traveling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the exact step initial data. The phase shift depends on the solitons, the breathers, and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers, and radiation
Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit
We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics
Entanglement of Two Disjoint Intervals in Conformal Field Theory and the 2D Coulomb Gas on a Lattice
In the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant
Juvenile growth and frost damages of poplar clone OP42 in Latvia
ArticleShort rotation plantations in the northern Europe are commonly established using
poplar clone OP42 (Populus maximowiczii Henry × P. trichocarpa Torr. and Gray). We assessed
its growth and suitability to the climate in central part of Latvia at juvenile age. Trees that had
formed single stem were significantly higher (121 ± 2.5 cm), thicker (7.1 ± 0.48 mm) and had
longer branches (32 ± 1.5 cm) than trees that had formed multiple stems. In beginning of the
second growing season all trees had died stems and 19.6% of them formed new shoots from the
ground level. The sprouting trees had random spatial distribution in the field. Regardless of the
number of stems, the sprouting trees were significantly lower (110 ± 3.9 cm) than the dead trees
(119 ± 2.0 cm). During the repeated assessment about one month later, proportion of the
sprouting trees increased up to 44%, but the detected relations between measured traits of
sprouting and dead trees remained. Clone OP42 had serious frost induced damages also in autumn
phenology experiment (96% trees with severely damaged leaves). Our results suggest that frost
prone sites are not suitable for establishment of plantations of OP42 clone
Asymptotics for a special solution to the second member of the Painleve I hierarchy
We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page
The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation
We establish the existence of a real solution y(x,T) with no poles on the
real line of the following fourth order analogue of the Painleve I equation,
x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the
existence part of a conjecture posed by Dubrovin. We obtain our result by
proving the solvability of an associated Riemann-Hilbert problem through the
approach of a vanishing lemma. In addition, by applying the Deift/Zhou
steepest-descent method to this Riemann-Hilbert problem, we obtain the
asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure
Imaging in major salivary gland diseases
Most of the salivary glands diseases are characterized only by a few distinct clinical patterns. Medical history and clinical examination are still considered of great relevance. However, in order to obtaine a definite diagnosis, imaging techniques are required in most of the cases. Salivary glands ultrasonography (US) is the technique to be used as the first because US can easily differentiate calculosis, inflammatory diseases and tumors. Sonography is also frequently needed to perform needle aspiration or biopsy (FNAC). Sialography should be used essentially for assessing chronic sialoadenitis as well as Sjogren's syndrome. At present, Magnetic Resonance sialography should be preferred because of the greater sensibility in diagnosing inflammatory diseases of the salivary glands. It allows to evaluate both intraglandular oedema and nodules, so that incannulation of the salivary duct is not required. Computer Tomography (CT) and Magnetic Resonance imaging (MR) are useful when neoplasm are suspected, particularly if deep areas of the gland, which cannot be visualized by US, are involved. Sequential scintigraphy is currently employed for assessing the functional status of all the 4 major salivary glands and evaluating the chronic evolution of glandular damage
Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
The small dispersion limit of solutions to the Camassa-Holm (CH) equation is
characterized by the appearance of a zone of rapid modulated oscillations. An
asymptotic description of these oscillations is given, for short times, by the
one-phase solution to the CH equation, where the branch points of the
corresponding elliptic curve depend on the physical coordinates via the Whitham
equations. We present a conjecture for the phase of the asymptotic solution. A
numerical study of this limit for smooth hump-like initial data provides strong
evidence for the validity of this conjecture. We present a quantitative
numerical comparison between the CH and the asymptotic solution. The dependence
on the small dispersion parameter is studied in the interior and at
the boundaries of the Whitham zone. In the interior of the zone, the difference
between CH and asymptotic solution is of the order , at the trailing
edge of the order and at the leading edge of the order
. For the latter we present a multiscale expansion which
describes the amplitude of the oscillations in terms of the Hastings-McLeod
solution of the Painlev\'e II equation. We show numerically that this
multiscale solution provides an enhanced asymptotic description near the
leading edge.Comment: 25 pages, 15 figure
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