Representation of solutions of the Burgers equation using functional integrals

In the paper, a representation of a solution of the Burgers equation in ℝ ⁿ is obtained by using integrals with respect to the Wiener measure on the space of trajectories in ℝ ⁿ . The Burgers equation is considered in a rigged Hilbert space. It is proved that, in the infinite-dimensional case, there is an analog of the Cole-Hopf transformation relating the Burgers equation and an analog of the heat equation with respect to measures. The Feynman-Kac formula for the heat equation (with potential) with respect to measures in a rigged Hilbert space is obtained.     

Limiting Dynamics for Spherical Models of Spin Glasses at High Temperature

We analyze the coupled non-linear integro-differential equations whose solution is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds. AMS (2000) Subject Classification: Primary: 82C44 Secondary: 82C31, 60H10, 60F15, 60K35     

Limiting Dynamics for Spherical Models of Spin Glasses at High Temperature

We analyze the coupled non-linear integro-differential equations whose solution is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p−spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds. AMS (2000) Subject Classification: Primary: 82C44, Secondery: 82C31, 60H10, 60F15, 60K35     

Exact solutions of the forced Burgers equation

Two new methods for obtaining exact solutions of the initial-value problem on an unbounded straight line (the Cauchy problem) for the inhomogeneous Burgers equation are considered. They are applied to the cases of a stationary and a transient external force. A self-similar solution and a solution which describes the localization (blocking) of solitary traveling waves are obtained as examples. Zh. Tekh. Fiz. 69, 10–14 (August 1999)     

Automorphism Groups of Orthomodular Lattices Obtained from Quadratic Spaces

We study the automorphism group of some orthomodular lattices, obtained from a quadratic space over a field K. We show how this group is linked to the semi-orthogonal group and with the group of all similarity transformations of the quadratic space. When the field K is finite, the cardinality of the automorphism group is given. AMS subject classification (1991): 06C15, 15A63, 20D45.     

Short-wavelength asymptotic behavior of the Burgers turbulence spectrum

Using a third-order structure function, an asymptotic expression is obtained for the stationary energy spectrum in the dissipative region of Burgers turbulence excited by a random external force. It is shown that in contrast to the case of turbulence described by a homogeneous Burgers equation, the spectrum contains a parameter characterizing energy transfer into the small-scale region. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 6, 383–386 (25 March 1998)     

An Alternative Approach to Obtain the Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x)p+p(1/x) Terms

By using the Lewis-Riesenfeld theory and algebraic method, we present an alternative approach to obtain the exact solution of time-dependent Hamiltonian systems involving quadratic, inverse quadratic and (1/x)p+p(1/x) terms. This solution is discussed and compared with that obtained by Choi, J. R. (2003). International Journal of Theoretical Physics 42, 853]. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

Convergence to diffusion waves of solutions for viscous conservation laws

We study the large-time behavior of solutions of viscous conservation laws. It is shown that solutions tend to diffusion waves, which are constructed based on the heat equation and Burgers equation. The convergence is in theL _p , 1p sense and is obtained as a consequence of theL ₂ decay of the difference between the solution and its asymptotic state of diffusion waves.Supported by the National Sciences Council of the Republic of China under the contract NSC-76-0208-M-001-09Supported by NSF under Grant No. DMS-84-01355     

Heat conduction in a random medium

The method of time-ordered cumulants is used to investigate the behavior of heat pulses in a one-dimensional medium in which the thermal conductivity is random. A partial differential equation is obtained for the average temperature profile; it is the heat equation modified by the addition of a fourth-order spatial derivative. A solution is obtained by asymptotic series. The first two spatial moments of the average temperature profile are evaluated and are shown to tend to those of a Gaussian whent is large. Finally, an equation is obtained for the covariance function.Alfred P. Sloan Fellow.     

Light Quark Hadron Mass Scale

With a symmetry procedure based on Noether's theorem, the field equation of motion is obtained from the Dirac Hamiltonian H(D_μ) of a massless quark interacting with a gluon. The equation of motion is the Yang-Mills equation with external current which is spin-dependent and follows from the group algebra. In addition to the pure gauge solution we find a gauge covariant solution which follows from current conservation and sets the mass scale m₀/M = g². This gluon field is due to the density of dipole moments squared and represents four harmonic oscillators with quadratic constraints; the gluon can be written as a string potential or as a 1/x potential with a sharp cutoff. The chiral symmetry group G^spin × G^D gives the light quark hadron degenerate multiplet mass spectrum in terms of m₀[SU(2) × SU(2)] with the spinorial decomposition and the multipole breaks into dipoles. Scaling from atomic lengths it is found that g = em₀/nM for light quarks is the quark charge e/3 renormalized by m₀/M and g is magnetic. Thus quarks occur at the ends of spinning magnetic strings with dipole lengths ∼m₀⁻¹. The mass scale is that of a degenerate magnetic multipole with charge n = 3, 4… .     

Effect of irradiation on ethyl 2‐amino‐4,5,6,7‐tetrahydrobenzo[b]thiophene‐3‐carboxylate: an ESR study

Ethyl 2‐amino‐4,5,6,7‐tetrahydrobenzo[b]thiophene‐3‐carboxylate [C₁₁H₁₅NO₂S] was synthesized by the Gewald method. Its single crystals were grown from an alcohol/ethyl acetate solution at 15 °C and characterized using IR and ¹H‐NMR. These single crystals were irradiated for 72 h at 298 K by a ⁶⁰Co gamma source with a dose speed of 0.864 kGy/h. After irradiation, electron spin resonance (ESR) measurements were carried out to study radiation‐induced radicals in the temperature range from 120 to 450 K. Additionally, for the single crystal, ESR angular dependencies were measured in the xy, xz and yz planes of the substance. This irradiated single crystal was analyzed based on the ESR spectra. Analysis of the spectra revealed that the radical was formed by a C–H bond fission at the carbon end of the substance. It was also observed that the color of the sample changed after irradiation. The hyperfine and g parameters were determined from the experimental spectra. It was inferred from these results that the hyperfine parameters and g value exhibited anisotropic behavior. The average values of these parameters were calculated as follows: g = 2.0088, A_H1=H2 = 20.70 G, A_H3=H4 = 10.80 G, A_Ha = 4.59 G, A_Hb = 3.24 G and, A_N = 6.10 G. Copyright © 2011 John Wiley & Sons, Ltd.     

Similarity Reductions of the Zabolotskaya-Khokhlov Equation with a Dissipative Term

Abstract

Similarity reductions of the Zabolotskaya-Khokhlov equation with a dissipative term to one-dimensional partial differential equations including Burgers’ equation are investigated by means of Lie’s method of infinitesimal transformation. Some similarity solutions of the Z-K equation are obtained.     

On the Zero Mass Limit of Tagged Particle Diffusion in the 1-d Rayleigh-Gas

We consider the M→0 limit for tagged particle diffusion in a 1-dimensional Rayleigh-gas, studied originaly by Sinai and Soloveichik [Ya. G. Sinai, M. R. Soloveichik, Commun. Math. Phys. 104:423–443 (1986)], and by Szász and Tóth [D. Szász, B. Tóth, Commun. Math. Phys. 104:445–457 (1986)], respectively. In this limit we derive a new type of model for tagged particle diffusion, for which the two central particles, in addition to elastic collisions with the rest of the gas, interact with Calogero-Moser-Sutherland (i.e. inverse quadratic) potential. Computer simulations on this new model reproduce exactly the numerical value of the limiting variance obtained by Boldrighini, Frigio and Tognetti in [C. Boldrighini, S. Frigio, D. Tognetti, J. Stat. Phys. 108:703–712 (2002)]. Dedicated to Domokos Szász on his 65th birthday.     

The stochastic heat equation: Feynman-Kac formula and intermittence

We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a space-time-dependent random medium. It has also been related to the distribution of two-dimensional directed polymers in a random environment, to the KPZ model of growing interfaces, and to the Burgers equation with conservative noise. We show how the solution can be expressed via a generalized Feynman-Kac formula. We then investigate the statistical properties: the two-point correlation function is explicitly computed and the intermittence of the solution is proven. This analysis is carried out showing how the statistical moments can be expressed through local times of independent Brownian motions.     

Long-Ranged Correlations in Bounded Nonequilibrium Fluids

In a recent paper, Liu and Oppenheim [J. Stat. Phys. 86:179 (1997)] solve the fluctuating heat diffusion equation for a bounded system with a temperature gradient. This note demonstrates that, contrary to their claims, their solution for the temperature correlation function is indeed long-ranged and reduces to that of Garcia et al.[J. Stat. Phys. 47:209 (1987)].     

Analytic Equation of State of a Quasi One-Dimensional Model Lipid Monolayer

The equation of state of a quasi one-dimensional model lipid monolayer is obtained in analytic form. The method used is the Laplace transform approach leading to a homogeneous Fredholm integral equation. Two cases are studied. The first considers a purely short range repulsive potential, when we recover the results previously obtained by Gianotti et al. (J. Phys. A.: Math. Gen. 25:2889 (1992)). The second incorporates the long range attractive Kac potential, and the equation of state is calculated in the van der Waals limit. This extends the approach originally developed by Kac et al. (J. Math. Phys. 4:216 (1963)).     

A Pregenerator for Burgers Equation¶Forced by Conservative Noise

We consider the stationarity of a Burgers equation with an external random force of gradient type in one space dimension. The expected stationary measure is the white noise measure on the space of tempered distributions. As a consequence, the nonlinearity of the formal equation u _t +λu u _x =νu _xx +η _x is ill-defined. Introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem. Received: 9 March 2001 / Accepted: 10 October 2001     

Kadomtsev-Petviashvili (KP) Burgers equation in a dusty plasmas with non-adiabatic dust charge fluctuation

The nonlinear dust acoustic waves in a dusty plasmas with the combined effects of non-adiabatic dust charge fluctuation and higher-order transverse perturbation are studied. Using the perturbation method, a Kadomtsev-Petviashvili (KP) Burgers equation that governing the dust acoustic waves is deduced for the first time. A particular solution of this KP Burgers equation is also obtained. It is show that the dust acoustic shock waves can exist in the KP Burgers equation.Received: 18 March 2003, Published online: 15 July 2003PACS: 52.35.Sb Solitons; BGK modes - 52.35.Mw Nonlinear phenomena: waves, and nonlinear wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.)     

Expanding and collapsing scalar field thin shell

This paper deals with the dynamics of scalar field thin shell in the Reissner-Nordstr?m geometry. The Israel junction conditions between Reissner-Nordstr?m spacetimes are derived, which lead to the equation of motion of scalar field shell and Klien–Gordon equation. These equations are solved numerically by taking scalar field model with the quadratic scalar potential. It is found that solution represents the expanding and collapsing scalar field shell. For the better understanding of this problem, we investigate the case of massless scalar field (by taking the scalar field potential zero). Also, we evaluate the scalar field potential when p is an explicit function of R. We conclude that both massless as well as massive scalar field shell can expand to infinity at constant rate or collapse to zero size forming a curvature singularity or bounce under suitable conditions.     

Global Uniqueness in the Inverse Scattering Problem¶for the Schrödinger Operator¶with External Yang–Mills Potentials

We consider the Schr?dinger equation in R ⁿ , n≥ 3, with external Yang–Mills potentials having compact supports. We prove the uniqueness modulo a gauge transformation of the solution of the inverse boundary value problem in a bounded convex domain. A similar uniqueness result holds for the inverse scattering problem at a fixed energy. Received: 11 August 2000 / Accepted: 24 May 2001