Step-initial function to the MKdV equation: Hyper-elliptic long-time asymptotics of the solution

The modified Korteweg-de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. $q(x,0) = c_r$ for $x > 0$ and $q(x,0) = c_l$ for $x c_r> 0.$ The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as $t\to+\infty.$ Using the steepest descent method we deform the original oscillatory matrix Riemann-Hilbert problem to explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the $xt$ plane. In the regions $x 4c_l^2 t + 2c_r^2 t$ the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $(-6c_l^2+ 12c_r^2)t < x < (4c_l^2+ 2c_r^2)t$ the asymptotics of the solution takes the form of a modulated hyper-elliptic wave generated by an algebraic curve of genus 2. V. Kotlyarov and A. Minakov, 2012

Proton beam self-modulation seeded by electron bunch in plasma with density ramp

Seeded self-modulation in a plasma can transform a long proton beam into a train of micro-bunches that can excite a strong wakefield over long distances, but this needs the plasma to have a certain density profile with a short-scale ramp up. For the parameters of the AWAKE experiment at CERN, we numerically study which density profiles are optimal if the self-modulation is seeded by a short electron bunch. With the optimal profiles, it is possible to "freeze" the wakefield at approximately half the wavebreaking level. High-energy electron bunches (160 MeV) are less efficient seeds than low-energy ones (18 MeV), because the wakefield of the former lasts longer than necessary for efficient seeding.Comment: 11 pages, 15 figure

Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schr\"odinger Equation

We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use a finite-gap potential of the Dirac operator given in an explicit form through hyper-elliptic theta-functions. The paper aim is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse scattering technics. We investigate this problem exceptionally using the Riemann-Hilbert problems technics that allow us to obtain explicit formulas for the asymptotic solitons themselves that in contrast with the cited papers where asymptotic formulas are obtained only for the square of absolute value of solution. Using transformations of the main RH problems we arrive to a model problem corresponding to the parametrix at the end points of continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials which are naturally appeared after appropriate scaling of the Riemann-Hilbert problem in a small neighborhoods of the end points of continuous spectrum. Further asymptotic analysis give an explicit formula for solitons at the edge of dispersive wave. Thus, we give the complete description of the train of asymptotic solitons: not only bearing envelope of each asymptotic soliton, but its oscillating structure are found explicitly. Besides the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock wave.Comment: 36 pages, 5 figure

Step-Initial Function to the MKdV Equation: Hyper-Elliptic Long-Time Asymptotics of the Solution

The modified Korteveg{de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. q(x, 0) = cr for x ≥ 0 and q(x, 0) = cl for x cr > 0. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t → ∞.Рассматривается модифицированное уравнение КдФ на всей прямой с начальным условием типа ступеньки, которая равна константе cl при x 0. При этом выполняется условие cl > cr > 0, что обеспечивает режим "гидродинамической волны сжатия" при t → ∞. Цель статьи - изучение асимптотического поведения решения начально-краевой задачи, когда t → ∞

The main principles and objectives of the national transport system

Стаття присвячена проблемам побудови ефективних транспортних систем. Запропоновані принципи та представлені задачі, які повинні служити орієнтирами та базовими ідеями формування та розвитку сучасних національних транспортних систем

Spontaneous imbibition experiments for enhanced oil recovery with silica nanosols

Experimental oil displacement as a result of spontaneous imbibition of silica nanosols has been carried out using two types of sandstone as the reservoir rock. The permeability of the cores ranged from 0.34 to 333 mD, while the porosity was 11% and 22%, respectively. During the research, the influence of the concentration and nanoparticle size, as well as the permeability of the rock, on the process of spontaneous imbibition, was studied. Silica nanosols were considered as an object of study. The nanoparticle size ranged from 10 to 35 nm. The mass concentration of nanoparticles varied from 0.01% to 0.25%. It was found that the use of silica nanosols significantly increases the rate of the spontaneous imbibition process. It was established that a silica nanosol with a nanoparticle size of 10 nm and a concentration of 0.25% allows to displace more than six times oil compared to the reservoir water model in the same time. As a result, it was shown that the oil displacement efficiency and the efficiency of spontaneous imbibition increase along with an increase in the nanoparticle concentration and a decrease in the nanoparticle size.Document Type: Original articleCited as: Pryazhnikov, M. I., Zhigarev, V. A., Minakov, A. V., Nemtsev, I. V. Spontaneous imbibition experiments for enhanced oil recovery with silica nanosols. Capillarity, 2024, 10(3): 73-86. https://doi.org/10.46690/capi.2024.03.0