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

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

Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

We analyze the case of a dense mKdV soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann-Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.Comment: 50 pages, 13 figure

Potential of sub-THz-wave generation in Li2B4O7 nonlinear crystal at room and cryogenic temperatures

Due to their high optical damage threshold, borate crystals can be used for the efficient nonlinear down-conversion of terawatt laser radiation into the terahertz (THz) frequency range of the electromagnetic spectrum. In this work, we carried out a thorough study of the terahertz optical properties of the lithium tetraborate crystal (Li2B4O7; LB4) at 295 and 77 K. Approximating the terahertz refractive index in the form of Sellmeier’s equations, we assessed the possibility of converting the radiation of widespread high-power laser sources with wavelengths of 1064 and 800 nm, as well as their second and third harmonics, into the THz range. It was found that four out of eight types of three-wave mixing processes are possible. The conditions for collinear phase matching were fulfilled only for the o - e -o type of interaction, while cooling the crystal to 77 K did not practically affect the phase-matching curves. However, a noticeable increase of birefringence in the THz range with cooling (from 0.12 to 0.16) led to an increase in the coherence length for o-o-e and e-e-e types of interaction, which are potentially attractive for the down-conversion of ultrashort laser pulses

Colloidal Stability of Guar-Based Hydraulic Fracturing Gels with the Addition of Nanoparticles

Выполнено изучение коллоидной устойчивости модифицированных гелей гидроразрыва пласта. Рассмотрены сшитые гели на основе гуара. В качестве модификаторов выступали сферические наночастицы оксида кремния и оксида алюминия разного размера, а также одностенные углеродные нанотрубки. Массовая концентрация наночастиц варьировалась от 0,01 до 0,4 %, а углеродных нанотрубок – от 0,01 до 0,1 %. Показано, что исследованные сшитые гели являются устойчивыми к агрегации и седиментации частиц. Установлено, что с увеличением концентрации наночастиц и их среднего размера коллоидная устойчивость гелей гидроразрыва пласта увеличивается. Получено, что гели с одностенными углеродными нанотрубками обладают более высокой коллоидной устойчивостьюA study of the colloidal stability of modified fracturing gels was performed. Guar-based cross-linked gels were considered. Spherical nanoparticles of silicon oxide and aluminum oxide of different sizes, as well as single-walled carbon nanotubes, were used as modifiers. The mass concentration of nanoparticles varied from 0.01 to 0.4 %, and carbon nanotubes – from 0.01 to 0.1 %. The cross-linked gels studied were shown to be resistant to particle aggregation and sedimentation. It was found that with increasing concentration of nanoparticles and their average size, the colloidal stability of hydraulic fracturing gels increases. Gels with single-walled carbon nanotubes were found to have higher colloidal stabilit