Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure

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

Eigenvalue correlations on Hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.Comment: latex 2e, seven pages, one figure. To appear in Journal of Physics

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

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

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

Entanglement entropy of two disjoint intervals in conformal field theory

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr\rho_A^n for any integer n is calculated as the four-point function of a particular type of twist fields and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.Comment: 34 pages, 7 figures. V2: Results for small x behavior added, typos corrected and refs adde

Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.Comment: 30 page

Evaporation from a small water reservoir: Direct measurements and estimates

Summary Knowing the rate of evaporation from surface water resources such as chan¬nels and reservoirs is essential for precise management of the water balance. However, evaporation is difficult to measure experimentally over water surfaces and several tech¬niques and models have been suggested and used in the past for its determination. In this research, evaporation from a small water reservoir in northern Israel was measured and estimated using several experimental techniques and models during the rainless summer. Evaporation was measured with an eddy covariance (EC) system consisting of a three-dimensional sonic anemometer and a Krypton hygrometer. Measurements of net radia¬tion, air temperature and humidity, and water temperature enabled estimation of other energy balance components. Several models and energy balance closure were evaluated. In addition, evaporation from a class-A pan was measured at the site. EC evaporation measurements for 21 days averaged 5.48 mm day�1. Best model predictions were obtained with two combined flux-gradient and energy balance models (Penman–Mon¬teith–Unsworth and Penman–Brutsaert), which with the water heat flux term, gave sim¬ilar daily average evaporation rates, that were up to 3% smaller than the corresponding EC values. The ratio between daily pan and EC evaporation varied from 0.96 to 1.94. The bulk mass transfer coefficient was estimated using a model based on measurements of water surface temperature, evaporation rate and absolute humidity at 0.9 and 2.9 m above the water surface, and using two theoretical approaches. The bulk transfer coefficient

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html