arXiv subjects

Richard Beals

Publications and source records attributed to Richard Beals.

6 recordsLinked to original sources

arXiv2025

A generalization of the beam problem: Connection to multi-component Camassa-Holm dynamics

We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete densities, we prove that the associated matrix Weyl function admits a Stieltjes-type continued fraction expansion with Clifford-valued coefficients. By mapping the problem from the finite interval to the real line, we uncover a direct link to a multi-component generalization of the Camassa-Holm equation. This yields a vectorized form of the Camassa-Holm equation invariant under arbitrary orthogonal group actions. As an illustration, we examine the dynamics of a two-atom (two-peakon) matrix measure in the special case of a Clifford algebra with two generators and Minkowski signature. Our analysis shows that, even when peakon waves remain spatially separated, they can engage in long-range, synchronized energy exchange.

nlin.SI↗

arXiv2020

A 2-component Camassa-Holm equation, Euler-Bernoulli Beam Problem and Non-Commutative Continued Fractions

A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed: (1) motivated by an analogy with the Camassa-Holm equation a class of isospectral deformations of the beam problem is formulated; (2) a reformulation of the matrix string problem in terms of a certain compact operator is used to obtain basic spectral properties of the inhomogeneous matrix string problem with Dirichlet boundary conditions; (3) the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a non-commutative generalization of Stieltjes' continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel-like determinants.

nlin.SI↗

arXiv2019

Constructing Solutions to Two-Way Diffusion Problems

A variety of boundary value problems in linear transport theory are expressed as a diffusion equation of the two-way, or forward-backward, type. In such problems boundary data are specified only on part of the boundary, which introduces several technical challenges. Existence and uniqueness theorems have been established in the literature under various assumptions; however, calculating solutions in practice has proven difficult. Here we present one possible means of practical calculation. By formulating the problem in terms of projection operators, we derive a formal sum for the solution whose terms are readily calculated. We demonstrate the validity of this approach for a variety of physical problems, with focus on a periodic problem from the field of active matter.

math-ph↗

arXiv1999

Multipeakons and the Classical Moment Problem

Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon solutions of the Camassa-Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon-antipeakon pairs, and the details of the collisions are analyzed using results {}from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly

solv-int↗

arXiv1998

Integrable Systems and Isomonodromy Deformations

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the $n\times n$ AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by $[(L^{k/n})_+,L]=I$ where $L$ is an $n^{th}$ order scalar differential operator. The monodromy data is constructed in each case; the inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple proof of the Painlevé property is given for the general case

solv-int↗

arXiv1997

Action-Angle variables for the Gel'fand-Dikii flows

Using the scattering transform for $n^{th}$ order linear scalar operators, the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real action-angle variables are constructed in the self-adjoint case.

math.AP↗