On the Quantum Kinetic Equation in Weak Turbulence

The quantum kinetic equation used in the study of weak turbulence is reconsidered in the context of a theory with a generic quartic interaction. The expectation value of the time derivative of the mode number operators is computed in a perturbation expansion which places the large diagonal component of the quartic term in the unperturbed Hamiltonian. Although one is not perturbing around a free field theory, the calculation is easily tractable owing to the fact that the unperturbed Hamiltonian can be written solely in terms of the mode number operators.Comment: 12 pages, LATEX, no figures, to appear in Phys. Rev.

A constructive approach to the soliton solutions of integrable quadrilateral lattice equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Backlund transformation, and the method of solving a Riccati map by exploiting two kn own particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler's lattice equation (or Q4).Comment: 22 page

Variational principle for frozen-in vortex structures interacting with sound waves

General properties of conservative hydrodynamic-type models are treated from positions of the canonical formalism adopted for liquid continuous media, with applications to the compressible Eulerian hydrodynamics, special- and general-relativistic fluid dynamics, and two-fluid plasma model including the Hall-magnetohydrodynamics. A variational formulation is found for motion and interaction of frozen-in localized vortex structures and acoustic waves in a special description where dynamical variables are, besides the Eulerian fields of the fluid density and the potential component of the canonical momentum, also the shapes of frozen-in lines of the generalized vorticity. This variational principle can serve as a basis for approximate dynamical models with reduced number of degrees of freedom.Comment: 7 pages, revtex4, no figure

On the analytical approach to the N-fold B\"acklund transformation of Davey-Stewartson equation

N-fold B\"acklund transformation for the Davey-Stewartson equation is constructed by using the analytic structure of the Lax eigenfunction in the complex eigenvalue plane. Explicit formulae can be obtained for a specified value of N. Lastly it is shown how generalized soliton solutions are generated from the trivial ones

On the asymptotic expansion of the solutions of the separated nonlinear Schroedinger equation

Nonlinear Schr\"odinger equation (with the Schwarzian initial data) is important in nonlinear optics, Bose condensation and in the theory of strongly correlated electrons. The asymptotic solutions in the region $x/t={\cal O}(1)$, $t\to\infty$, can be represented as a double series in $t^{-1}$ and $\ln t$. Our current purpose is the description of the asymptotics of the coefficients of the series.Comment: 11 pages, LaTe

Generic solutions for some integrable lattice equations

We derive the expressions for $\psi$-functions and generic solutions of lattice principal chiral equations, lattice KP hierarchy and hierarchy including lattice N-wave type equations. $\tau$-function of $n$ free fermions plays fundamental role in this context. Miwa's coordinates in our case appear as the lattice parameters.Comment: The text of the talk at NEEDS-93 conference, Gallipoli, Italy, September-93, LaTeX, 8 pages. Several typos and minor errors are correcte

Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions

Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schr\"odinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification

Why the general Zakharov-Shabat equations form a hierarchy?

The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature equations with rational dependence on a spectral parameter, if properly defined, can be considered as a hierarchy. The latter means a collection of commuting vector fields in the same phase space. Further properties of the hierarchy are discussed, such as additional symmetries, an analogue to the string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian definition of soliton solutions.Comment: 13p

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^\theta$ as $\theta \to \pm \pi /2$. It turns out that $T_{max}^\theta$ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /2$ in the case where the solution of the limiting nonlinear Schr\"odinger equation blows up in finite time (respectively, is global).Comment: 22 page

Correlations in the impenetrable electron gas

We consider non-relativistic electrons in one dimension with infinitely strong repulsive delta function interaction. We calculate the long-time, large-distance asymptotics of field-field correlators in the gas phase. The gas phase at low temperatures is characterized by the ideal gas law. We calculate the exponential decay, the power law corrections and the constant factor of the asymptotics. Our results are valid at any temperature. They simplify at low temperatures, where they are easily recognized as products of free fermionic correlation functions with corrections arising due to the interaction.Comment: 17 pages, Late