Ordinary second-order differential equations and soliton solutions connected with the algebra sl(2, ℂ)

For nonlinear equations describing chiral fields, explicit expressions for solutions are obtained in the case of an arbitrary number of poles and their multiplicities.     

Robertson-Walker Fluid Sources Endowed with Rotation Characterised by Quadratic Terms in Angular Velocity Parameter

Einstein's equations for a Robertson-Walker fluid source endowed with rotation are presented up to and including quadratic terms in angular velocity parameter. A family of analytic solutions are obtained for the case in which the source angular velocity is purely time-dependent. A subclass of solutions is presented which merge smoothly to homogeneous rotating and non-rotating central sources. The particular solution for dust endowed with rotation is presented. In all cases explicit expressions, depending sinusoidally on polar angle, are given for the density and internal supporting pressure of the rotating source. In addition to the non-zero axial velocity of the fluid particles it is shown that there is also a radial component of velocity which vanishes only at the poles. The velocity four-vector has a zero component between poles.     

Prescribing Zeros and Poles on a Compact Riemann Surface for a Gravitationally Coupled Abelian Gauge Field Theory

In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results.The authors research was supported in part by NSF grants DMS–9972300 and DMS–9729992 through the Institute for Advanced Study.     

Non-Markovian quantal Brownian motion model

A non-Markovian version of the quantal Brownian motion model is given. The integrodifferential equations of motion are solved, establishing the analytic form of the resolvent poles and analyzing their properties. An explicit investigation of the poles at zero temperature is performed. In this frame a rule can be found that relates the relevant poles of the non-Markovian resolvent to the eigenvalues of the associated Markovian generator of the motion.     

Two-loop anomalous-dimension matrix for soft-gluon exchange

The resummation of soft-gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions. We compute this matrix directly for arbitrary 2-->n massless processes for the first time at two loops. Using color-generator notation, we show that it is proportional to the one-loop matrix. This result reproduces all pole terms in dimensional regularization of the explicit calculations of massless 2-->2 amplitudes in the literature, and it predicts all poles at next-to-next-to-leading order in any 2-->n process that has been computed at next-to-leading order. The proportionality of the one- and two-loop matrices makes possible the resummation in closed form of the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2-->n processes.     

Review: Aspects of Solution-generating Techniques for Space-times with Two Commuting Killing Vectors

Certain aspects of solution-generatingtechniques for spacetimes with two commuting Killingvectors are reviewed. A brief historical introduction tostationary axisymmetric systems is given. The importance of the Homogeneous Hilbert problem associatedwith the equations, unifying the group-theoretic withthe soliton-theoretic approaches, is emphasized. Theformalism of generating functions is introduced, both for vacuum and electrovacuum.Sibgatullin's technique for electrovacuum solutions isrelated to the Hauser Ernst variables and a method byErnst is briefly discussed. The solitonic methods ofBelinsky-Zakharov and Alekseev are reviewed. Their relation isemphasized by an explicit proof, at the level ofgenerating techniques, that the BZ two soliton with twocomplex conjugate poles is isomorphic to the Alekseev one-soliton (restricted to vacuum) with trivialgauge. The Alekseev non-soliton technique is discussed.Some recent developments are brieflydiscussed.     

Bounds on Scattering Poles in One Dimension

For the class of super-exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated. In the case of nonnegative potentials, explicit estimates for the width of a pole-free strip are obtained. Received: 27 January 1999 / Accepted: 7 July 1999     

New explicit and exact solutions of the Benney--Kawahara--Lin equation

In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.     

广义Zakharov-Kuznetsov方程的对称及精确解

利用改进的直接方法给出了一类广义Zakharov-Kuznetsov方程ut auux bu2ux cuxxx duxyy=0新显式解与旧显式解之间的关系,并且得到了该方程的对称.这些对称推广了已有文献中应用Steinberg s相似方法获得的结果.利用广义Zakharov-Kuznetsov方程新旧显式解之间的关系,本文在已有显式解的基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.     

Stochastic model for the dynamics of a spin-oscillator coupled system

We construct a stochastic model for the dynamics of a one-dimensional system consisting of bilinearly coupled harmonic oscillators and spins. The spin dynamics is defined as a Glauber model where the spins are effectively coupled through their interaction with the oscillators. To maintain internal thermal equilibrium in the composite system, which does not exhibit Onsager symmetry, we introduce a phenomenological retarded friction in the oscillator equation of motion and relate it to the spin correlation function through a fluctuation-dissipation theorem. The oscillator susceptibility is derived and the behavior of its poles as functions of wavevector and temperature is studied. The results are compared to those obtained by other authors who have studied similar systems, using irreversible thermodynamics. In contrast to ours, these treatments do not give an explicit result for the wavevector dependence of the poles.     

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Instability of pole solutions for planar propagating flames in sufficiently large domains

It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front has exact pole solutions for which the PDE reduces to a set of ordinary differential equations (ODEs). The paradox, however, lies in the fact that the set of ODEs does not permit the appearance of new poles in the complex plane, or the formation of cusps in the physical space, as observed in experiments. The validity of the PDE itself has thus been questioned. We show here that the discrepancy between the PDE and the ODEs is due to the instability of exact pole solutions for the PDE. In previous work, we have reported that most exact pole solutions are indeed unstable for the PDE but, for each interval of relatively small length L, there remains one solution (up to translation symmetry) which is neutrally stable. The latter is a one-peak, coalescent solution for which the poles (whose number is maximal) are steady. The front undergoes bifurcations as the length of the domain considered increases: the one-pole, one-peak coalescent solution is first neutrally stable. As the length of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is, in turn, neutrally stable. This phenomenon occurs once again: as the two-pole solution becomes unstable, the three-pole solution becomes stable. The contribution of the present work is to show that subsequent bifurcations are of a different nature. As the interval length increases, the steady one-peak, coalescent solutions whose number of poles is maximal are no longer stable and bifurcations to unsteady states occur. In all cases, the appearance of new poles is observed in the unsteady dynamics. We also show analytically that such an instability is not permitted in the ODEs for which all steady one-peak, coalescent solutions are neutrally stable.     

The Existence of Non-Topological Multivortex Solutions in the Relativistic Self-Dual Chern–Simons Theory

We construct a general type of multivortex solutions of the self-duality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic Chern–Simons model with the non-topological boundary condition near infinity. For such construction we use a perturbation argument around the explicit solutions of the Liouville equation. Received: 6 July 1999 / Accepted: 14 June 2000     

Exact Solutions of the Matrix KP Equation

The matrix KP equation is a many component extension of the ordinary KP (Kodomtsev-Petvjshvilli) equation. Although the matrix KP equation is very important, however,its explicit exact solutions have not been reported up to now. In this letter we give a method to construct the matrix KP hierarchy and its exact solutions. Then we give here the explicit expressions of the exact solutions of the matrix KP equation with arbitrary soliton number.     

Exact periodic steady solutions for nonlinear wave equations: A new approach

In many practical physical problems, the nonlinear wave equations of interest are nonintegrable. In some cases they may be close to an integrable equation, and in this case perturbation techniques are available. However, even in these cases and certainly in general, it is useful to construct exact explicit solutions. Here we introduce a new method for finding exact and explicit periodic travelling wave solutions, from which solitary wave solutions can be extracted if they exist.     

The Gibbons–Tsarev equation: symmetries,invariant solutions,and applications

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation.     

Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials

In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schrödinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.     

Some properties of the S-matrix in a model coupled-channel calculation

In this paper we give the explicit solution to a coupled-channel problem consisting of two open channels coupled by a separable interaction. It is shown that the cross sections exhibit one rather narrow anomaly which displaces to higher energies as the coupling is increased and at the same time its width decreases and increases alternatively. This behavior is analyzed in terms of the eigenphaseshifts and also in terms of the motion of the poles of the S-matrix as a function of the coupling strength.     

Cut cancellation in the planar integral equation for the reggeon

Planar unitarity for the Reggeon, analyticity and the multi-Regge assumption with cluster production lead to integral equations of the Chew-Goldberger-Low type with separable self-consistent kernel. Contrary to common prejudice, we show the existence of solutions exhibiting moving poles and exact, non-perturbative cancellation of the cut. Previously studied consistency conditions are rederived.