Hierarchy Of Partial Differential Equations Research Articles

Heat Transport on Ultrashort Time and Space Scales in Nanosized Systems: Diffusive or Wave-like?

The non-Fourier effects, such as wave-like temperature propagation and boundary temperature jumps, arise in nanosized systems due to the multiple time and space scales nature of out-of-equilibrium heat transport. The relaxation to equilibrium occurs in successive time and space scales due to couplings between different excitations, whose relaxation times have different physical meanings and may differ significantly in magnitude. The out-of-equilibrium temperature evolution is described by a hierarchy of partial differential equations of a higher order, which includes both the diffusive and wave modes of heat transport. The critical conditions of transition from wave to diffusive modes are identified. We demonstrate that the answer to the question concerning which of these modes would be detected by experimental measurements may also depend on the accuracy of the experimental setup. Comparisons between the proposed approach and other non-Fourier models, such as the Guyer–Krumhansl and Jeffreys type, are carried out. The results presented here are expected to be useful for the theoretical and experimental treatment of non-Fourier effects and particularly heat wave phenomena in complex nanosized systems and metamaterials.

Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber

We consider the hierarchy of partial differential equations with arbitrary power-low nonlinearity which can be used for description of the propagation pulse in optical fiber. The Cauchy problem for these partial differential equations cannot be solved by the inverse scattering transform and we look for exact solutions of differential equations using the traveling wave reduction. It is proven that all equations of the hierarchy have exact solutions in the form of periodic and solitary waves that are determined by means of the elliptic functions. A more detailed study of the hierarchy is presented for the equations of the second, fourth and sixth order. The parameter values for existence of exact solutions for these equations are given. Exact solutions of differential equations are expressed in terms of the Weierstrass elliptic function. A formula for describing solitary waves is also given. Exact solutions in the form of periodic and solitary waves for differential equations are illustrated.

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.

Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2 + 1 dimensions

We present two hierarchies of partial differential equations in 2 + 1 dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified version of the other. The connection between them can be achieved by means of a combination of reciprocal and Miura transformations.

Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

The ℓ-conformal Galilei algebra, denoted by is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of and vector field representations for d = 1, 2.

Integrable [formula omitted] dimensional hierarchies arising from reduction of a non-isospectral problem in [formula omitted] dimensions

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in 2+1 dimensions, which can be considered as a modified version of the Camassa–Holm hierarchy in 2+1 dimensions. A classification of reductions for this spectral problem is performed. Non-isospectral reductions in 1+1 dimensions are considered of remarkable interest.

On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental–Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit.

Isometric embeddings of infinite-dimensional Grassmannians

We investigate geometric properties of the (Sato-Segal-Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers-Stormer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations. We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds.

A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations

AbstractThis paper presents a heterogeneous multiscale method with efficient interscale coupling for scale‐dependent phenomena modeled via a hierarchy of partial differential equations. Physics at the global level is governed by one set of partial differential equations, whereas features in the solution that are beyond the resolution capability of the coarser models are accounted for by the next refined set of differential equations. The proposed method seamlessly integrates different sets of equations governing physics at various levels, and represents a consistent top–down and bottom–up approach to multi‐model modeling problems. For the top–down coupling of equations, this method provides a variational residual‐based embedding of the response from the coarser or global system equations, into the corresponding local or refined system equations. To account for the effects of local phenomena on the global response of the system, the method also accommodates bottom–up embedding of the response from the local or refined mathematical models into the global or coarser model equations. The resulting framework thus provides a consistent way of coupling physics between disparate partial differential equations by means of up‐scaling and down‐scaling of the mathematical models. An integral aspect of the proposed framework is an uncertainty quantification and error estimation module. The structure of this error estimator is investigated and its mathematical implications are delineated. Copyright © 2010 John Wiley & Sons, Ltd.

Characterizations of Jacobians of curves with automorphisms

We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal T -functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the τ-function of a p.p.a.v. Finally, since Baker-Akhiezer and T functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.

Tautological relations and the $r$-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations.

The higher flows of harmonic maps and an application to the Virasoro action

One characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of λ which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a −1-flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.

Local evolution rules and the diffusion equation

A partial differential equation describing the diffusion of a set of interacting particles on a lattice is obtained using the so-called local evolution rules approach. This equation is derived after truncating a hierarchy of partial differential equations using a mean-field ( m, n) closure. Both, the partial differential equations and the diffusion coefficients, are derived in a variety of cases. The description of the noninteracting set of particles is obtained always as a limiting case. The results are compared to those previously obtained by other method.

Toda and KdV

The main object of this paper is to produce a deformation of the KdV hierarchy of partial differential equations. We construct this deformation by taking a certain limit of the Toda hierarchy. This construction also provides a deformation of the Virasoro algebra.

Sine-Gordon Equation and Representations of Affine Kac–Moody Algebra [formula omitted

We give a representation-theoretic interpretation of the sine-Gordon equation using the action of affine Kac–Moody algebra sl 2 on the Weyl algebra. In this setup τ-functions become functions in non-commuting variables. The skew Casimir operators that we introduce here, give rise to a hierarchy of partial differential equations that includes the sine-Gordon equation and two copies of the Korteweg–de Vries equation.

An Extension of the Korteweg–de Vries Hierarchy Arising from a Representation of a Toroidal Lie Algebra

In this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal Lie algebras. In the smallest example—rank 2 toroidal cover of sl2—we obtain an extension of the KdV hierarchy. We use the action of the corresponding infinite-dimensional group to construct solutions for these non-linear PDEs.

Topographical scattering of gravity waves

A systematic hierarchy of partial differential equations for linear gravity waves in water of variable depth is developed through the expansion of the average Lagrangian in powers of [mid ]∇[mid ] (h=depth, ∇h=slope). The first and second members of this hierarchy, the Helmholtz and conventional mild-slope equations, are second order. The third member is fourth order but may be approximated by Chamberlain & Porter's (1995) ‘modified mild-slope’ equation, which is second order and comprises terms in ∇2h and (∇h)2 that are absent from the mild-slope equation. Approximate solutions of the mild-slope and modified mild-slope equations for topographical scattering are determined through an iterative sequence, starting from a geometrical-optics approximation (which neglects reflection), then a quasi-geometrical-optics approximation, and on to higher-order results. The resulting reflection coefficient for a ramp of uniform slope is compared with the results of numerical integrations of each of the mild-slope equation (Booij 1983), the modified mild-slope equation (Porter & Staziker 1995), and the full linear equations (Booij 1983). Also considered is a sequence of sinusoidal sandbars, for which Bragg resonance may yield rather strong reflection and for which the modified mild-slope approximation is in close agreement with Mei's (1985) asymptotic approximation.

Tau-functions and generalized intergrable hierarchies

The tau-function formalism for a class of generalized “zero-curvature” integrable hierarchies of partial differential equations is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables of the zero-curvature formalism and the tau-functions is established. The formalism also clarifies the connection between the zero-curvature hierarchies and the Hirotatype hierarchies of Kac and Wakimoto.

Generalized integrability and two-dimensional gravitation

We review the construction of generalized integrable hierarchies of partial differential equations, associated to affine Kac-Moody algebras, that include those considered by Drinfel'd and Sokolov. These hierarchies can be used to construct new models of 2D quantum or topological gravity, as well as new $\cal W$-algebras.