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

TL;DR

This paper introduces a novel approach to the Euler-Bernoulli beam problem using inhomogeneous matrix strings, explores isospectral deformations inspired by the Camassa-Holm equation, and develops a non-commutative continued fraction method for solving inverse problems.

Contribution

It presents a new matrix string formulation for the beam problem, formulates isospectral deformations, and introduces non-commutative continued fractions for inverse spectral analysis.

Findings

Spectral properties of the inhomogeneous matrix string are characterized.

Inverse formulas are derived using non-commutative continued fractions.

A solution to the inverse problem for discrete beams is provided.

Abstract

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.