The generalization of the adiabatic perturbation approach of $N$-soliton interactions in the nonlinear Schrödinger equation (NLSE) has been shown to lead to the complex Toda lattice with $N$ nodes. This allows us to predict the asymptotic behavior of trains of $N$ solitonlike pulses with approximately equal amplitudes and velocities, but with arbitrary phase differences. These predictions agree very well with the numerical results.

We consider the critical Ising model on a randomly triangulated disk in the presence of a boundary magnetic field $h$, computing the partition function $Z$ and bulk magnetization $〈M〉$. These quantities depend only on an effective rescaled boundary length and bulk area, suggesting that the boundary field generates renormalization group flow towards $h=\infty$, just as in flat space. As $h>\mathrm{}0\mathrm{}$ increases, $〈M〉$ decreases, which we explain in terms of fluctuations of the geometry. We also derive an analytic expression for the boundary magnetization, which gives a finite nonzero boundary susceptibility at the critical temperature.

In this Letter, I present a precise quantum Monte Carlo calculation at finite temperature for a very large number (many thousands) of bosons in a harmonic trap, which may be anisotropic. The calculation applies directly to the recent experiments of Bose-Einstein condensation of atomic vapors in magnetic traps. I show that the critical temperature of the system decreases with the interaction. I also present profiles for the overall density and the one of condensed particles, and obtain excellent agreement with solutions of the Gross-Pitaevskii equation.

The distributions $P\left(X\right)$ of singular thermodynamic quantities, on an ensemble of $d$-dimensional quenched random samples of linear size $L$ near a critical point, are analyzed using the renormalization group. For $L$ much larger than the correlation length $\xi$, we recover strong self-averaging (SA): $P\left(X\right)$ approaches a Gaussian with relative squared width $R_X\sim (L/\xi {)}^{-d}$. For $L\ll \xi$ we show weak SA ( $R_X$ decays with a small power of $L$) or no SA [ $P\left(X\right)$ approaches a non-Gaussian, with universal $L$-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.

A “persistence exponent” $\theta$ is defined for nonequilibrium critical phenomena. It describes the probability, $p\left(t\right)\sim t^{-\theta }$, that the global order parameter has not changed sign in the time interval $t$ following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the $n=\infty$ limit of the $O\left(n\right)$ model, to first order in $\epsilon =4-d$, and for the 1D Ising model. Numerical results are obtained for the 2D Ising model. We argue that $\theta$ is a new independent exponent.

We discuss the quantum properties of the Bose-Einstein condensate of a dilute gas of atoms in a trap. We show that the phase of the condensate undergoes quantum diffusion which can be detected in far off-resonant light scattering experiments.

We prove stability for systems composed of arbitrarily many nonrelativistic Pauli electrons minimally coupled to the quantized, ultraviolet-cutoff electromagnetic field and of static nuclei interacting with each other through Coulomb forces.

It is shown that a resonant nonlinear interaction of waves may result in a variety of collective multiwave modes of coherent resonantly coupled waves. The equations of motion for the modes have a universal set of integrals (“collision constants”) reflecting a “particle balance” (in quantum-mechanical terms) in wave collisions. The existence of these integrals ensures that the collective modes are completely integrable. Any multiwave mode can be described by an effective single degree of freedom conservative Hamiltonian, similar to the one known for three-wave coherent modes (triads).

We consider a small system $s$ in a bath $b$, in the case that the state of $b$ and all Hamiltonians are possibly time dependent. We obtain for the reduced density matrix of $s$ an exact evolution equation ${\rho }_s\left(t\right)=\Lambda (t,\tau ){\rho }_s\left(\tau \right)$, with $(\partial /\partial t)\Lambda (t,\tau )=\gamma (t,\tau )\Lambda (t,\tau )$, where $\gamma (t,\tau )$ depends on the system-bath correlations at time $\tau$. The open evolutor $\Lambda (t,\tau )$ can (but need not) be chosen completely positive. It is argued that as $t-\tau$ increases, $\gamma (t,\tau )\to {\gamma }_0\left(t\right)$ forgets the initial correlations and tends to Lindblad form in time-coarse-grained weak coupling limits.

We show that binary mixtures of Bose condensates of alkali atoms have a great variety of ground state and vortex structures which can be accessed experimentally by varying the particle numbers of different alkalis. We have constructed a simple algorithm to determine the density profiles of the mixtures within Thomas-Fermi approximation. Many structures of the alkali binary contain a coexisting region, which is the analog of the long sought ${}^3\mathrm{He}$- ${}^4\mathrm{He}$ interpenetrating superfluids in ultralow temperature physics.

We present a procedure to systematically evaluate all the moments of the Fokker-Planck equation by expanding them in a power series in a given function of $t$. The expansion coefficients are easily determined in terms of algebraic recursion relations. Applications to a linear Fokker-Planck equation, as well as to a truly nonlinear mean-field model, whose drift coefficient exhibits a functional dependence on the distribution function, show this formalism to be advantageous over the standard time series expansion of the moments which is shown to be rather impractical.

A Comment on the Letter by Lior Goldenberg and Lev Vaidman, Phys. Rev. Lett. 75, 1239 (1995).

A Reply to the Comment by Asher Peres.

We consider the bifurcation problem $u^{\prime \prime }+\lambda u=N\left(u\right)$ with two point boundary conditions where $N\left(u\right)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the bifurcating branch $\lambda$ as a function of the amplitude of the solution. As an application we show how it can be used to obtain simple approximate closed formulas for the period of large amplitude oscillations.

An efficient geometric formulation of the problem of parameter estimation is developed, based on Hilbert space geometry. This theory, which allows for a transparent transition between classical and quantum statistical inference, is then applied to the analysis of exponential families of distributions (of relevance to statistical mechanics) and quantum mechanical evolutions. The extension to quantum theory is achieved by the introduction of a complex structure on the given real Hilbert space. We find a set of higher order corrections to the parameter estimation variance lower bound, which are potentially important in quantum mechanics, where these corrections appear as modifications to Heisenberg uncertainty relations for the determination of the parameter.

This Letter reports on three-dimensional simulations that follow exact, $z$ symmetric soliton solutions to an important equation of plasma physics and fluid dynamics. These solitons are seen to break up when perturbed slightly along $z$. Some fairly robust, three-dimensional entities are subsequently produced. However, they do not seem to resemble known, azimuthally symmetric solutions. An explanation of this interesting fact is offered.

A recently developed renormalized strong coupling expansion [E. J. Weniger, Ann. Phys. (N.Y.) 246, 133 (1996)] is employed to compute the coefficients of the standard strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator. This approach is very simple, both conceptually and technically, and produces more accurate results than previously used techniques which were in most cases applied to the quartic case only.

We present a class of systems for which the signal-to-noise ratio always increases when increasing the noise and diverges at infinite noise level. This new phenomenon is a direct consequence of the existence of a scaling law for the signal-to-noise ratio and implies the appearance of stochastic resonance in some monostable systems. We outline applications of our results to a wide variety of systems pertaining to different scientific areas. Two particular examples are discussed in detail.

The diffusion equation ${\partial }_t\phi ={▽}^2\phi$ is considered, with initial condition $\phi (\mathbf{x},0\mathrm{})$, a Gaussian random variable with zero mean. Using a simple approximate theory we show that the probability $p_n\left(t_1{,t}_2\right)$ that $\phi (\mathbf{x},t)$ (for a given space point $\mathbf{x}$) changes sign $n$ times between $t_1$ and $t_2$ has the asymptotic form $p_n\left(t_1{,t}_2\right)\sim c_n\left[\ln \right(t_2/t_1\left){]}^n\right(t_1/t_2{)}^{-\theta }$. The exponent $\theta$ has predicted values $0.1203$, $0.1862$, $0.2358$ in dimensions $d=1,2,3\mathrm{}$, in remarkably good agreement with simulation results.

The fraction $r\left(t\right)$ of spins which have never flipped up to time $t$ is studied within a linear diffusion approximation to phase ordering. Numerical simulations show that $r\left(t\right)$ decays with time like a power law with a nontrivial exponent $\theta$ which depends on the space dimension. The dynamics is a special case of a stationary Gaussian process of known correlation function. The exponent $\theta$ is given by the asymptotic decay of the probability distribution of intervals between consecutive zero crossings. An approximation based on the assumption that successive zero crossings are independent random variables gives values of $\theta$ in close agreement with the results of simulations.