On correlation functions of integrable models associated to the six-vertex R-matrix

We derive an analog of the master equation obtained recently for correlation functions of the XXZ chain for a wide class of quantum integrable systems described by the R-matrix of the six-vertex model, including in particular continuum models. This generalized master equation allows us to obtain multiple integral representations for the correlation functions of these models. We apply this method to derive the density-density correlation functions of the quantum non-linear Schrodinger model.Comment: 21 page

Power expansions for solution of the fourth-order analog to the first Painlev\'{e} equation

One of the fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlev\'{e} equation determines new transcendental functions.Comment: 28 pages, 5 figure

Integrable viscous conservation laws

We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven. We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe

Short Distance Modification of a Gravitational System and its Optical Analog

Motivated by developments in string theory, such as T-duality, it has been proposed that the geometry of spacetime should have an intrinsic minimal length associated with it. This would modify the short distance behavior of quantum systems studied on such a geometry, and an optical analog for such a short distance modification of quantum system has also been realized by using non-paraxial nonlinear optics. As general relativity can be viewed as an effective field theory obtained from string, it is expected that this would also modify the short distance behavior of general relativity. Now the Newtonian approximation is a valid short distance approximation to general relativity, and Schrodinger-Newton equation can be obtained as a non-relativistic semi-classical limit of such a theory, we will analyze the short distance modification of Schrodinger-Newton equation from an intrinsic minimal length in the geometry of spacetime. As an optical analog of the Schrodinger-Newton equation has been constructed, it is possible to optically realize this system. So, this system is important, and we will numerical analyze the solutions for this system. It will be observed that the usual Runge-Kutta method cannot be used to analyze this system. However, we will use a propose and use a new numerical method, which we will call as the two step Runge-Kutta method, for analyzing this system.Comment: 21 pages, 3 figures, 2 table

A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data

In the present paper we are concerned with the Novikov--Veselov equation at negative energy, i.e. with the $(2 + 1)$--dimensional analog of the KdV equation integrable by the method of inverse scattering for the two--dimensional Schr\"odinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non--singular scattering data behaves asymptotically as \frac{\const}{t^{3/4}} in the uniform norm at large times $t$. We also present some arguments which indicate that this asymptotics is optimal

A review of nonequilibrium effects and surface catalysis on shuttle heating

A review is given of the nonequilibrium calculation techniques by various authors over the past decade to predict heat fluxes to the windward side of the Space Shuttle orbiter. The results of these techniques are compared with measurements made on the first few flights of the Space Shuttle. The calculations attempt to account for finite rate chemistry in the shock layer around the vehicle and for finite rate catalytic atom recombination on the thermal protection materials. The techniques considered are the axisymmetric viscous shock layer method, three dimensional reacting Euler equation solutions coupled with axisymmetric analog boundary layer method, and a recently developed nonequilibrium 3-D viscous shock layer method