Articles

Phase Transitions in Materials with Thermal Memory: the Case of Unequal Conductivities

Murrough Golden — Fri, 12 Apr 2013 06:41:36 PDT

A model for thermally induced phase transitions in materials with thermal memory was recently proposed, where the equations determining heatflow were assumed to be the same in both phases. In this work, the model is generalized to the case of phase dependent heatflow relations. The temperature (or coldness) gradient is decomposed into two parts, each zero on one phase and equal to the temperature (or coldness) gradient on the other. However, they vary smoothly over the transition zone. These are treated as separate independent quantities in the derivation of field equations from thermodynamics. Heat flux is given by an integral over the history of the temperature gradient, with different kernels on each phase. Asymptotic analysis is carried out to obtain generalizations of previous results. These involve the jump in temperature across the transition zone and the normal derivatives of the temperature on each phase boundary, which are related to the velocity of the transition zone and a latent heat dependent on this velocity, as well as the speeds of thermal disturbances in the two phases.

G-Strands and Peakon Collisions on Diff(R)

Darryl Holm et al. — Tue, 09 Apr 2013 07:17:12 PDT

A G-strand is a map g : R x R --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G = Diff(R) corresponding to a harmonic map g : C --> Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well.

Second Gradient Viscoelastic Fluids: Dissipation Principle and Free Energies

G. Amendola et al. — Thu, 04 Apr 2013 08:31:51 PDT

We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature.

Integrable Models for Shallow Water With Energy Dependent Spectral Problems

Rossen Ivanov et al. — Thu, 07 Mar 2013 09:51:49 PST

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse problem as a Riemann–Hilbert problem with a Z₂ reduction group. The soliton solutions are explicitly obtained.

G-Strands

Darryl Holm et al. — Wed, 12 Dec 2012 01:37:01 PST

A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3)_K-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(R)-strand equations on the diffeomorphism group G=Diff(R) are also introduced and shown to admit solutions with singular support (e.g., peakons).

The 2-Component Dispersionless Burger's Equation Arising In The Modelling Of Bloodflow

Tony Lyons — Tue, 27 Nov 2012 01:06:59 PST

This article investigates the properties of the solutions of the dispersionless two-component Burgers (B2) equation, derived as a model for blood- ow in arteries with elastic walls. The phenomenon of wave breaking is investigated as well as applications of the model to clinical conditions.

Dark Solitons of the Qiao's Hierarchy

Rossen Ivanov et al. — Mon, 26 Nov 2012 02:33:03 PST

We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Z. Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called 'dark solitons'.

The Generalised Zakharov-Shabat System and the Gauge Group Action

Georgi Grahovski — Tue, 06 Nov 2012 01:41:44 PST

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type models, related to so(5,C) algebra.

On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space

Vladimir Gerdjikov et al. — Mon, 15 Oct 2012 03:06:35 PDT

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves; their velocities and their amplitudes are time dependent. Calculating the asymptotics of the N-soliton solutions for t \rightarrow \pm \infty we analyze the interactions of quadruplet solitons.

Cyclic Universe with an Inflationary Phase from a Cosmological Model with Real Gas Quintessence

Rossen Ivanov et al. — Mon, 08 Oct 2012 02:16:15 PDT

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

Holomorphic Liftings from Infinite Dimensional Spaces

Sean Dineen et al. — Mon, 11 Jun 2012 10:07:12 PDT

We obtain a number of positive solutions to a holomorphic lifting problem on a domain in a locally convex space.

On the N-Wave Equations and Soliton Interactions in Two and Three Dimensions

Vladimir S. Gerdjikov et al. — Fri, 25 May 2012 01:11:00 PDT

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann–Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave _{equations with}Z₂xZ₂ reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.

Euler Equations on a Semi-Direct Product of the Diffeomorphisms Group by Itself

Joachim Escher et al. — Mon, 21 May 2012 00:57:35 PDT

The geodesic equations of a class of right invariant metrics on the semi-direct product of two Diff(S) groups are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra are found.

Multiple Solutions of the Quasi Relativistic Choquard Equation

Michael Melgaard et al. — Wed, 16 May 2012 06:35:48 PDT

We prove existence of multiple solutions to the quasi relativistic Choquard equations with a scalar potential.

Existence of a Minimizer for the Quasi-Relativistic Kohn-Sham Model

Carlos Argaez et al. — Wed, 16 May 2012 06:35:46 PDT

We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator (see article) . For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge (see article).

Stiefel and Grassmann Manifolds in Quantum Chemistry

Eduardo Chiumiento et al. — Wed, 16 May 2012 06:35:45 PDT

We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slatertype variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove thatthey are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.

Smooth and Peaked Solitons of the Camassa-Holm Equation and Applications

Darryl Holm et al. — Thu, 08 Mar 2012 01:48:44 PST

The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T^∗(T^N) to the flow momentum (X^∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. The peakon solutions of the dispersionless CH equation in one dimension are shown to generalize in higher dimensions to peakon wave-front solutions of the EPDiff equation whose associated momentum is supported on smoothly embedded subspaces. The Eulerian representations of the singular solutions of both CH and EPDiff are given by the (cotangent-lift) momentum maps arising from the left action of the diffeomorphisms on smoothly embedded subspaces.

On Maximal Relatively Divisible Submodules

Brendan Goldsmith et al. — Tue, 20 Dec 2011 05:24:05 PST

In recent work Goebel and Goldsmith investigated the spectrum of maximal pure subgroups of certain Abelian groups. Here the situation relating to maximal pure submodules of a torsion-free module over an integral domain R is investigated. Connections to the level of coherency are established along with a detailed investigation of the situation where all maximal pure submodules are isomorphic to a product of copies of the ring R.

On Projection-Invariant Subgroups of Abelian P-Groups

Brendan Goldsmith — Tue, 20 Dec 2011 05:24:03 PST

A subgroup P of an Abelian p-group G is said to be projection-invariant in G if Pf is contained in P for all idempotent endomorphisms f. Clearly fully invariant subgroups are projection invariant, but the converse is not true in general. Hausen and Megibben have shown that in many familiar situations these two concepts coincide. In a different direction, the authors have previously introduced the notions of socle-regular and strongly socle-regular groups by focussing on the socles of fully invariant and characteristic subgroups of p-groups. In the present work the authors examine the socles of projection-invariant subgroups of Abelian p-groups.

Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry

Carlos Argaez et al. — Fri, 16 Dec 2011 01:15:22 PST

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N-electron Coulomb systems with quasi-relativistic kinetic energy for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.