Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

Fingering instability in particle systems

In many multiphase systems, material interfaces can be destabilized by shocks. Small disturbances at these interfaces can grow in size to form large-scale fingers. We consider a shock propagating through a system that consists of two types of particles, of different mass, that are initially separated by an interface, but are free to mix. In the classical case of immiscible fluids, the finger of heavy fluid propagating into the light fluid grows faster and becomes much thinner than the finger of light fluid propagating into the heavy fluid. We show that collisions between particles of different types lead to shock focusing that causes a secondary flow that is initially similar to the fluid case. However, the particle system can exhibit completely different qualitative behavior in the nonlinear-growth phase and can give rise to the situation where the finger of heavy material is actually wider than the finger of the light material. We show that this qualitative change is due to a strong decompression that occurs in the heavy material. We also show that microscopic mixing can have an important impact on finger growth.     

Exponential instability in the inverse scattering problem on the energy interval

We consider the inverse scattering problem on the energy interval in three dimensions. We focus on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows the optimality, up to the value of the exponent, of the logarithmic stability result obtained by P. Stefanov in 1990 with the use of some special norm for the scattering amplitude at fixed energy.     

Raman scattering of nanocrystalline silicon embedded in SiO2

Raman scattering of nanocrystalline silicon embedded in SiO2 matrix is systematically investigated. It is found that the Raman spectra can be well fitted by 5 Lorentzian lines in the Raman shift range of 100-600 cm-1. The two-phonon scattering is also observed in the range of 600-1100 cm-1. The experimental results indicate that the silicon crystallites in the films consist of nanocrystalline phase and amorphous phase; both can contribute to the Raman scattering. Besides the red-shift of the first order optical phonon modes with the decreasing size of silicon nanocrystallites, we have also found an enhancement effect on the second order Raman scattering, and the size effect on their Raman shift.     

Analytic properties of the scattering matrix of many particle systems

We generalize the previous results on a meromorphic continuation of the scattering matrix for N particle systems. Our proofs seem to be much simpler than those given before.Dedicated to the dear memory of david Milman     

Spatiotemporal instability in nonlinear dispersive media in the presence of space-time focusing effect

Spatiotemporal instability in nonlinear dispersive media is investigated on the basis of the nonlinear envelope equation. A general expression for instability gain which includes the effects of space-time focusing, arbitrarily higher-order dispersions and self-steepening is obtained. It is found that, for both normal and anomalous group-velocity dispersions, space-time focusing may lead to the appearance of new instability regions and influence the original instability gain spectra mainly by shrinking their regions. The region of the original instability gain spectrum shrinks much more in normal dispersion case than in anomalous one. In the former case, space-time focusing completely suppresses the growing of higher frequency components. In addition, we find that all the oddth-order dispersions contribute none to instability, while all the eventh-order dispersions influence instability region and do not influence the maximum instability gain, therein the fourth-order dispersion plays the same role as space-time focusing in spatiotemporal instability. The main role played by self-steepening in spatiotemporal instability is that it reduces the instability gain and exerts much more significant influence on the new instability regions resulting from space-time focusing.     

Thermal instability in compressible fluids in the presence of rotation and magnetic field

The thermal instability of compressible fluids pervaded by a uniform rotation and a uniform magnetic field, separately, is considered. For $\text{(}\text{Cp}\text{g}\text{)β < 1}$, with Cp, g, and β denoting the specific heat at constant pressure, the acceleration due to gravity, and the uniform temperature gradient, respectively, the system is shown to be stable. The magnetic field as well as rotation introduces oscillatory modes in thermal instability of compressible fluids, which are completely missing for $\text{(}\text{Cp}\text{g}\text{)β > 1}$ in the absence of rotation or magnetic field. The sufficient conditions which do not allow overstable modes are obtained.     

The scattering of light in diamond and its Raman spectrum

A detailed study of the scattering of light in diamond and its Raman spectrum has been made using the λ 2536·5 mercury resonance radiation as exciter. The seattered spectrum exhibits two pairs of Doppler-shifted components, one pair due to the longitudinal sound waves, and the other due to the two sets of transverse sound waves which have very nearly the same velocity. The velocities of the longitudinal and transverse sound waves estimated from the observed frequency shifts of the displaced components are in agreement with those calculated from the elastic constants of diamond. The directional dependence of sound velocity in diamond has been quantitatively verified. Contrary to expectation, the longitudinal Doppler components are found to be less intense than the transverse Doppler eomponents. The second-order spectrum of diamond has been examined with a quartz spectrograph of high dispersion and resolution. It exhibits a whole series of sharply defined Raman lines the frequency shifts of which have been tabulated. The prominent ones which are 15 in number have been satisfactorily explained as octaves of combinations of six of the eight fundamental frequencies of vibration of the diamond lattice to be expected on the basis of the Raman theory, some of which are split due to the removal of degeneracy by anharmonicity and due to resonance effect.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

Computation of three-dimensional magnetic fields and particle trajectories in the presence of ferromagnets

Translated from Programmnoe Oborudovanie i Voprosy Prinyatiya Reshenii, pp. 201–206, 1989.     

Two-dimensional nuclear Coulomb scattering of a slow quantum particle

We study two-dimensional scattering of a quantum particle by the superposition of a Coulomb potential and a central short-range potential. We analyze the low-energy asymptotic behavior of all radial wave functions, partial phases, and scattering cross sections of such a particle. We propose two approaches for evaluating the scattering length and the effective radius.