Closed geodesics on Riemannian manifolds via the heat flow

The main topic discussed in this paper is the following question: Given a Riemannian manifold M and a closed C¹ curve f: S¹ → M does there exist a (unique) solution of the heat equation ?_tf_t = τ(f_t) defined for all t ≧ 0 which is continuous at t = 0 along with its first S¹-derivative and which coincides with f at t = 0.     

Time boundedness of the energy for the charge transfer model

The energy behavior of the time-dependent Schrödinger equation $$i\frac{\partial }{{\partial t}}\psi = \frac{{ - 1}}{{2m}}\Delta \psi + \sum\limits_{j = 1}^N {V_j } (x - y_j (t))\psi $$ is discussed, where they _j (t) are trajectories of classical scattering. In particular, we prove that the energy cannot become arbitrarily large ast→∞.     

A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {a_i(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)?x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.     

The effect of an external field on an interface,entropic repulsion

We consider the effects of an external potential -h∑f(S _x ) withh>0,f increasing, on the equilibrium state of a system with a Hamiltonian of the form $$H^0 (S) = \sum\limits_{\left\langle {xy} \right\rangle } {\Phi (S_x - S_y )} ,S_x \in R,x \in Z^d ,d \geqslant 3$$ Φ even and convex, e.g.,Φ(t)=1/2t ² andf(t)=signt. This can be thought of as a model of the interactions between a random interface S _x and a “soft” wall. We show that the random surface is (entropically) repelled to infinity for allh>0, i.e., with probability one,S _x ≥K, for anyK ε R.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

On the field dependence of random walks in the presence of random fields

Numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one-dimensional system with a bias field on each site. The bias is taken randomly with equal probability to be +E or ?E. The probability density¯P(x, t) is found to scale asymptotically as $$\left\{ {[A(E)]^{\beta /2} /\ln ^2 t} \right\}\exp \left( { - \left\{ {x[A(E)]^{\beta /2} /\ln ^2 t} \right\}^\alpha } \right)$$ withA(E)=ln[(1+E)/(1-E)],β=4.25, and α=1.25. The mean square displacement scales as \(\langle x^2 \rangle \sim [A(E)]^{ - \beta } F[tA^\beta (E)]\) , where F(u)～ln⁴ u asymptotically.     

Symmetric random walks in random environments

We consider a random walk on thed-dimensional lattice ^d where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify–x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix.     

Mehrfache Dispersionsrelationen für kurzreichweitige Potentiale,die sich nicht durch eine Überlagerung von Yukawa-Potentialen darstellen lassen

A triple dispersion relation is derived for the functionf(s, t, u) which becomes the scattering amplitudef(s, t) foru=4s?t. Besides the usual conditions which are needed for deriving a dispersion relation ins, the potential must decrease faster than exponentially at infinity. For this class of potentialsf(s, t, u) has essential singularities fort→∞ andu→∞. It is shown thatf(s, t, u) is bounded in the physical sheets of two independent Riemannian surfaces which are constructed by conformal mappings of thet- and theu-plane. In the new variables the conditions for the existence of dispersion relations are fulfilled.     

Alzheimer random walk

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution \(b{\left( {\tfrac{a}{b}} \right)^{\tfrac{{a + 1}}{b}}}{\left[ {\Gamma \left( {\tfrac{{a + 1}}{b}} \right)} \right]^{ - 1}}{x^a}\exp \left( { - \tfrac{a}{b}{x^b}} \right)\) where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p ^{? α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) ? ν(0) ] / [ ν(1) ? ν(0) ] = p ^β with β = 0.388 when p ? 0.1 and β = 0.0822 when p ? 0.1.     

Non-stationary random response of a finite cable

Numerous problems of current concern involve the designs of aerodynamic systems which either travel at high speeds or contain structural elements which are excited by moving pressure fluctuations. In a number of recent papers responses of dynamic systems to random excitation have been considered. The appropriate theory for calculating the mean square response of linear systems to both stationary and non-stationary random excitation is well known [1–7]. In this paper, the mean square response of a finite cable to non-stationary random excitation is considered. The non-stationary random excitation is of the form s(t) = e(t)α(t), where e(t) is a well defined envelope function and α (t) is the Guassian, narrow band, stationary part of the excitation which has zero mean. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. The results obtained are shown to be a complete expression for the mean square response when checked for accuracy by reduction to expressions previously obtained by Lyon [4]. It is felt that these results will aid the design of both linear and two-dimensional aerodynamic systems excited by random pressure fluctuations.     

An approach to radially symmetrical solutions of the sine-Gordon equation

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2π-kink, in the expanding and in the shrinking phase, for R(t)j? R(0). It is shown that the parameterization φ(r, t) = 4 arcian exp[γ(r?R(0)] + x(r, t), where R(t) describes the exact propagation of the maximum of φ,(r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x(r = R(t), t) is found and tested numerically. A relationship between the fluctuations in x(r = R(t), t) and those in $\text{R}\text{?}\text{(t), t)}\text{and}\text{R(t)}$ explains why the solitary wave is almost stable. From x(r = R(t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ_r(r, t) with respect to r = R(t) is predicted. It also exhibits fluctuations corresponding to those in x(r = R(t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.     

Evolution of the density for a chaotic map

For the discrete-time map x_t+11 = 4x_t(1?x_t) an exact, explicit expression is given for the time-dependent density r_t (x) evolving from a uniform initial density on (0,1). As t → ∞, r_t(x) approaches the known invariant density $\text{r(x) = 1/[π}\text{x(1?x)}\text{]}$.     

The Inviscid Burgers Equation with Initial Value of Poissonian Type

We study the discontinuities (shocks) of the solution to the Burgers equation in the limit of vanishing viscosity (the inviscid limit) when the initial value is the opposite of the standard Poisson process p. We show that this solution is only defined for t ε (0, 1). Let T ₀ = 0 and T _n , n≧1, be the successive jumps of p. We prove that for all M > 0 the inviscid limit is characterized on the region x ε (-∞, M], t ε (0, 1) by the increasing process $N(t) = \sup \{ n \in \mathbb{N} {\text{| }}M + nt > T_n \} $ and the random set I(x) = {n ε {0,..., N(t)}‖T _n -nt≦x<T _n+1 - nt}. The positions of shocks are given in a precise manner. We give the distribution of N(t) and also the distribution of its first jump. We also prove similar results when the initial value is u _μ(y, 0) = -μp(y/μ²) + μ^-1 max(y, 0), μ ε (0, 1).     

Photoionisation of atoms and Möller operators

The motion of an hydrogenoïd atom in a laser field is usually given by the time-dependent hamiltonian H(t)=[p?A(t)]²/2+V(r) where V(r) is the atomic potential whileA(t) is to be connected with the laser field. The existence and unicity for the Cauchy problem of the solutions of the corresponding Schrödinger equation are established under mild conditions onA(t) and V(r). The existence of Möller operators is investigated in two cases, namely, when the laser field is a function of time only and when it vanishes asymptotically in time. Special attention is paid for the Coulomb case for which a “distorted” Möller operator is derived. Finally, when the laser field vanishes ast→∞, the photoionisation probability is properly defined by means of the Möller operator $$\Omega (H_{At} ,H) = s - \mathop {\lim }\limits_{t \to \infty } U_{At} (t)^{ - 1} U(t)$$ , whereU(t) is the evolution operator for the system whileU _Att (t) is the evolution operator for the atom.     

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}The initial value problem for an integrable system, such as the Nonlinear Schr?dinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(x,t){\mathcal {R} = \mathcal {R}(x,t)} in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of R(x,t){\mathcal {R}(x,t)}, are called breaking curves or nonlinear caustics. The genus of R(x,t){\mathcal {R}(x,t)} is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves. In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function á h(z;x,t){{\Im\,h(z;x,t)}}, which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of á h(z;x,t){\Im\,h(z;x,t)} (regular break), or through the interaction of zero level curves of á h(z;x,t){{\Im\,h(z;x,t)}} with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.     

Electromagnetic induction in conductors accelerated in magnetic fields amplified by flux compression

The initial boundary-value problem for the electromagnetic induction in a conducting slab ats(t)≦x≦s(t)+a resulting from its accelerated motionv={s(t), 0, 0} across a transverse magnetic fieldB={0,B(x,t), 0} is treated, when the latter is amplified by orders-of-magnitude with respect to its initial valueB(x,t=0)=B ₀(x) by flux compression in the gap between the moving conductor surfacex=s(t) and an ideal resting conductor atx=0. Two initial (t=0) configurations are considered, assuming that (I)B ₀ (step-shaped) has not yet and (II)B ₀ (uniform) has completely diffused into the conductor atx=s(t=0). By means of a time-dependent coordinate transformation ξ=[x ? s(t)]/a and Fourier series expansions, the electromagnetic fields in the moving conductor are represented as integralfunctionals of the magnetic fieldB ₁ (t) in the gap 0≦x≦s(t).B ₁ (t) is given analytically as solution of a singular Volterra integro-differential equation. The theory is valid for arbitrary (nonrelativistic) speeds.(t) and accelerationss(t)) of the moving conductor. Applications to explosion driven electric induction generators and magnetic flux experiments are discussed briefly.     

Theory of the decay process of metastable state

The short-time and long-time behavior of the distribution function P(x, t) are investigated in the laser model by using the generating function $\text{G(α,}\text{β}\text{;t) = σ(α-y(t)) Π}{}^{\mathrm{\infty }}{}_{\mathrm{n=2}}\text{σ(β}{}_{\mathrm{n}}\text{- M}{}_{\mathrm{n}}\text{(t)),}\text{where}\text{y(t)  ?xP(x, t)}\text{d}\text{x}\text{and}\text{M}{}_{\mathrm{n}}\text{(t)  ?(x - y(t))}{}^{\mathrm{n}}\text{P(x,t)}\text{d}\text{x}$.