Interaction of coupled higher order nonlinear Schrödinger equation solitons

The novel inelastic collision properties of two-soliton interaction for an n-component coupled higher order nonlinear Schr?dinger equation are studied. Some interesting features of three soliton interactions, related to the integrability of the n-component coupled higher order nonlinear Schr?dinger equation are also discussed. Received 17 April 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: abhijit@iitg.ernet.in RID="b" ID="b"e-mail: sasanka@iitg.ernet.in RID="c" ID="c"e-mail: sudipta@iitg.ernet.in     

Then-component cubic model and flows: Subgraph break-collapse method

We specialize to then-component cubic model the subgraph break-collapse method which we recently developed for theZ() model. The cubic model has less symmetry than the Potts model, for which the method was originally developed, but nevertheless it is still possible to reduce considerably the computational complexity of the generalZ() model. Our recursive algorithm is similar, forn=2, to the break-collapse method for theZ(4) model proposed by Mariz and co-workers. It allows the exact calculation for the partition function and correlation functions forn-component cubic clusters, withn as a variable, without the need to examine all of the spin configurations. An important application is therefore in real-space renormalization-group calculations.     

Discontinuity of the dynamic critical exponents ofO(n)-symmetric systems

Then-component model of Sasvári, Schwabl, and Szépfalusy is analyzed by means of renormalized field theory to two-loop order. Singular two-loop contributions have a significant effect on the nature of the stability boundary of the dynamic scaling region in then—d plane. Along this boundary the dynamic critical exponents exhibit unexpected discontinuities.     

On the 1/n expansion

The 1/n expansion is considered for then-component non-linear -model (classical Heisenberg model) on a lattice of arbitrary dimensions. We show that the expansion for correlation functions and free energy is asymptotic, for all temperatures above the spherical model critical temperature. Furthermore, the existence of a mass gap is established for these temperatures andn sufficiently large.Supported in part by the National Science Foundation under Grant PHY 79-16812     

Statistical mechanics of systems of unbounded spins

We develop the statistical mechanics of unboundedn-component spin systems on the latticeZ ^v interacting via potentials which are superstable and strongly tempered. We prove the existence and uniqueness of the infinite volume free energy density for a wide class of boundary conditions. The uniqueness of the equilibrium state (whose existence is established in general) is then proven for one component ferromagnetic spins whose free energy is differentiable with respect to the magnetic field.     

Axial and diagonal anisotropy crossover exponents for cubic systems

Cubic n-component spin systems exhibit two distinct quadratic spin anisotropy crossover exponents, appropriate for axial and diagonal anisotropies. Results for n→∞ and to order ?², and for nm-component systems are given, and relevance to multi-critical points in various systems is discussed.     

On the Integrability of (2+1)-Dimensional Quasilinear Systems

A (2+1)-dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of sufficiently many n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.     

Adsorption of gas mixtures on heterogeneous surfaces: The integral representation for a monolayer total adsorption isotherm

The authors extended the integral equation for a one-component adsorption isotherm on to n-component adsorption from gaseous mixtures. It appears that the n-component adsorption isotherm on heterogeneous surfaces can be presented as a multiple integral. Two methods were used for evaluation of the surface heterogeneity with regard to a gas mixture. The first method is based on the n-dimensional gaussian energy distribution. The second one concerns the extension of our method for studying adsorption systems of high surface heterogeneity. Two-component adsorption systems have been analyzed in details.     

On the multicomponent surface phase balance equations

The system of the field equations is formulated at the phenomenologically modelledn-component surface phase (e.g. interface, solidification front, shock wave, membrane etc.). The surface phase balance equations then constitute a closed set of the general boundary conditions of the appropriate balance equations of volume phases (as, for example, the heat conduction or diffusion equations etc.).     

Dynamic critical exponent of a Bose system to O(1n)

The dynamic critical exponent of an n-component Bose system is calculated by the 1/n expansion technique to order 1/n in 2 < d < 4 dimensions.     

Christoffel formula and geodesic motion in hyperspin manifolds

A hyperspin manifoldS _N constructed fromN-component hyperspinors is an alternative to Riemannian manifoldsR _n for Kaluza-Klein-type theories of higher dimensions. Hyperspin manifolds possess a fundamental Chronometric tensor withN n-valued indices, where alwaysn=N ². Some concepts of Riemannian geometry therefore have to be extended. A hyper-Christoffel formula is presented that expresses the connection in terms of the chronometric, assuming the Chronometric is covariantly constant and the connection is torsion-free. Thus, the chronometric can be used as sole dynamical variable. Extremals and selfparallel curves, which coincide in Riemannian manifolds, in general differ in hyperspin manifolds, but coincide again for nonnull curves.     

Reformulation of n→0 models using anticommuting scalar fields

We show that a class of models which are described by the n→0 limit of an n-component field theory can be reformulated so as to avoid this limit. Generalisations to a wider class of models are discussed.     

Correlation functions for n species of one-dimensional impenetrable bosons

The correlation functions for n species of one-dimensional impenetrable bosons are expanded in terms of correlation functions for identical free fermions. The expansion is obtained by exploiting the quantum inverse method for the n-component nonlinear Schrödinger model.     

Renormalized field theory of the critical dynamics ofO(n)-symmetric systems

The critical dynamics of an-component order parameter of anO(n)-symmetric system is analyzed with a Gell'Mann-Low-type renormalization group equation ind=4– dimensions. Dynamical scaling holds forn>3/2+O(). Besides the dynamical exponentz=d/2 the correction exponents to order are found. To the same order the shape function of the dynamical order parameter correlation function is calculated which exhibits fluctuation induced peaks forn3 and extrapolated tod=3.     

Correlation inequalities and contour estimates

We give a simple estimate on the probability of contours in classical ferromagnetic spin systems, based on Griffiths' or Ginibre's correlation inequalities. This includes quite general one- and two-component spin models. Some extension also holds for alln-component anisotropic or isotropic rotators.Supported by NSF grant No. MCS78-01885.On leave from: Institut de Physique Théorique, Université de Louvain, Belgium.Supported by NSF grant No. PHY78-15920.     

Local rings of singularities andN=2 supersymmetric quantum mechanics

We investigate the Kähler structure arising inn-component,N=2 supersymmetric quantum mechanics. We defineL ²-cohomology groups of a modified and relate them to the corresponding spaces of harmonic forms. We prove that the cohomology is concentrated in the middle dimension, and is isomorphic to the direct sum of the local rings of the singularities of the superpotential. In the physics language, this means that the number of ground states is equal to the absolute value of the index of the supercharge, and each ground state contains exactlyn fermions.Supported in part by the Department of Energy under Grant DE-F602-88ER25065     

The symplecton: A prototype for semi-simple graded Lie algebras

An elementary, physically motivated, example of a semi-simple graded Lie algebra (SSGLA) is shown to be given by the symplecton realization of angular momentum [an associative, involutive, inner-product algebra whose characteristic polynomials realize the symplectic group Sp(2)]. The Pais-Rittenberg result shows that the n-component symplecton realizes the most general, SSGLA, Sp(2n).Work supported by the Energy Research and Development AdministrationSummer visitor, Group T-7.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999