The presence of exciton-phonon interactions is shown to play a key role in the exciton superfluidity. We apply the Landau criterion for an exciton-phonon condensate moving uniformly at zero temperature. It turns out that there are essentially two critical velocities in the theory. Within the range of these velocities the condensate can exist only as a bright soliton. The excitation spectrum and differential equations for the wave function of this condensate are derived. [S0031-9007(97)02937-2] PACS numbers: 71.35.Lk, 05.30.Jp, 63.20.Ls, 64.60.Ht The problem of critical velocities in the theory of superfluidity arose a long time ago when the experiments with the liquid He showed a substantial discrepancy with quantum-mechanical predictions. Later, the effect was analyzed, and its phenomenological description was given (e.g., see [1]). The fact that the liquid He could not be treated as a weakly nonideal Bose gas was believed to be the main reason for the inconsistency of microscopic theory with experimental data.For a long time, He has been the only substance where the superfluidity can be observed. The recent experiments with the dilute gas of excitons [2,3] provide new possibilities for studying different types of superfluidity.In this series of experiments the Cu 2 O crystal was irradiated with laser light pulses of several ns duration. At low intensities of the laser beam (low concentration of excitons), the system revealed a typical diffusive behavior of exciton gas. Once the intensity of the beam exceeds some value, the majority of particles move together in the packet. Their common propagation velocity is close to the longitudinal sound velocity, and the packet evolves as a bright soliton.Some alternative explanations of the phenomena are known. One of them [2] implies that the bright soliton is a one-dimensional traveling wave which satisfies the Gross-Pitaevsky (nonlinear Schrödinger) equation [4] for the Bose-condensate wave function C͑x, t͒with attractive potential of exciton-exciton interaction n , 0.A quantitative treatment given in [5] provides an iterative solution for the Heisenberg equation with the use of perturbational methods. In this picture the second order interactions, neglected in the Bogoliubov approximation, contribute to the negative value of n. However, the influence of exciton-phonon interactions on the dynamics of the condensed excitons is not treated [5].Another interpretation is based on a classical model [6] where the normal exciton gas is pushed towards the interior of a sample by the phonon wind emanating from the surface. Such an explanation has a discrepancy with the experiment because the signal observed is one order of magnitude longer than the excitation pulse duration [3].In this Letter we give an alternative and, in our opinion, more intrinsic interpretation of these phenomena. We argue that it is a propagation of a superfluid exciton-phonon condensate which is observed experimentally. The presence of exciton-phonon interactions is crucial for a "solitonlike superfluid...

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We apply the method of correlation functions to the coefficient problem in stochastic geometry. In particular, we give a proof for some universal patterns conjectured by M. Zinsmeister for the second moments of the Taylor coefficients for special values of κ in the whole-plane Schramm-Loewner evolution (SLE κ ). We propose to use multipoint correlation functions for the study of higher moments in coefficient problem. Generalizations related to the Levy-type processes are also considered. The exact integral means β-spectrum of this version of the whole-plane SLE κ is discussed.1 The coefficient problem for the whole plane SLE κ : main resultsThe Loewner evolution first appeared as a way for attacking the Bieberbach conjecture for the Taylor coefficients of injective conformal mappings of the open disc to the plane (for review see eg [4]) and since the beginning of the 20th century became an indispensable tool in related theory of complex variables. The years following Loewners' initial work also showed many applications of complex analytic methods to the study of problems involving fractal objects in the plane. In particular, the area of two-dimensional critical phenomena has enjoyed a breakthrough due to a radical development by O. Schramm in stochastic Loewner evolution, as an approach to description of boundaries of critical clusters (see for a review e.g. [3] and [8]).On the other hand it is interesting to return to the initial motivation of studying Loewner evolution and to review the coefficient problem for stochastic conformal mappings described by Loewner chains. In particular, an idea of considering statistical properties of the Taylor coefficients of conformal mappings for stochastic Loewner evolution driven by a wide class of Levy processes has been recently proposed by M. Zinmeister. Some moments of coefficients have been estimated and some universal patterns have been conjectured by Bertrand Duplantier, Thi Phuong Chi Nguyen, Thi Thuy Nga Nguyen and Michel Zinsmeister [1]. In the present article we prove the conjectures of the above authors as well as propose new approaches to the coefficient problem in stochastic geometry.We start the paper with a brief description of the problem and fixing the notations.Radial stochastic Schramm-Loewner Evolution with parameter κ (see e.g. [3], [7]) describes dynamics of a slit domain in the z-plane that can be represented by growth of a random planar curve Γ = Γ(t) starting from a point on a unit circle |z| = 1 at t = 0. The principal idea of the theory of SLE is that the growth of the curve can be given by the time dependent conformal 1

We examine connections between rationality of certain indefinite integrals and equilibrium of Coulomb charges in the complex plane .Polynomiality of all solutions is assured by the rationality of (1).

We consider the unbounded version of the whole-plane Schramm–Loewner evolution. Using exact solutions of differential equations for moments of derivatives of conformal mappings, we determine its average integral means beta spectrum.