Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov

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

MONITORING OF TECHNICAL STATE OF TECHNOGENIC UNSAFE PIPELINES

Ukraine has a wide network of main pipelines on its territory. For most of them pre-set operation time has already run out or is close to finishing. This fact together with a potential hazard of technogenic pollution of territory around the pipeline in case of accident rises a problem of monitoring of technical state of pipelines and, as one of possible outcomes, prolongation of operation life of the structure. Authors propose a system able to perform local measurements of mechanical stresses and deformations in the welded structure. Depending on the aim of measurement – single measurement, monitoring of an area, monitoring of the whole pipeline – system has a possibility to change its measuring and data processing sub-systems up to the case when its fully autonomous operation is needed.Украина имеет на своей территории разветвленную сеть магистральных трубопроводов, срок работы большинства из которых вышел или подходит к концу. В связи с этим, а также с потенциальной опасностью техногенного загрязнения территорий вокруг трубопроводов в случае аварии возникает вопрос мониторинга технического состояния труб и продления ресурса работы конструкции в целом. Авторы статьи предлагают систему, которая позволяет выполнять локальные измерения напряжений и деформаций на сварной конструкции. В зависимости от цели измерения - единичный контроль, мониторинг одного участка, мониторинг по всей длине трубопровода - система имеет возможность расширения измерительной и вычислительной частей с переходом к работе в полностью автономном режиме.Україна має на своїй території розгалужену мережу магістральних трубопроводів, термін роботи більшості з яких вийшов або добігає кінця. У зв’язку з цим, а також із потенційною небезпекою техногенного забруднення територій навколо трубопроводів у разі аварії постає питання моніторингу технічного стану труб та подовження ресурсу роботи конструкції в цілому. Автори статті пропонують систему, яка дозволяє виконувати локальні вимірювання напружень і деформацій на зварній конструкції. У залежності від мети вимірювання – одиничний контроль, моніторинг однієї ділянки, моніторинг по всій довжині трубопроводу – система має можливість розширення вимірювальної та обчислювальної частин із переходом до роботи у повністю автономному режимі

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

Asymptotics of Rarefaction Wave Solution to the mKdV Equation

I would like to express my gratitude to V.P. Kotlyarov for his valuable advices and attention paid to the work