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

A new reading of the economic system of the Grand Duchy of Tuscany through the georeferencing of shops and factories at the time of the setting up of the Lorraine Land Registry (1835)

This paper focuses on the reconstruction of the social and economic conditions of the Grand Duchy of Tuscany in the mid eighteen thirties on the basis of recent geohistorical information. To perform this reconstruction, we georeferenced information relating to the nineteenth-century factories and shops recorded in the Land Registry. To date, this aspect has received little attention when considering the historiographic traditions of this area, firmly rooted in a sharecropping system based on the three main Mediterranean agricultural products.The use of geostatistical tools allowed us to determine the height and slope of each of the geometries in the two geodatabases. We were also able to create positive spatial data autocorrelations, determine specific production and trading areas and thus determine the anthropisation levels of these territories in the outlying areas of the State.The online publication of these geographical databases on the cartographic portal of the Tuscany Region through a dedicated WebGIS was the last phase of our study. Considering both the number and density of geometries surveyed, our work is an extraordinary example of how geohistorical research may be combined with new technologies for the purpose of studying geohistorical content and disseminating it to the general public

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

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 $\epsilon$ 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 $\epsilon$, at the trailing edge of the order $\sqrt{\epsilon}$ and at the leading edge of the order $\epsilon^{1/3}$. 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