A 2‐component Camassa‐Holm Equation, Euler‐Bernoulli Beam Problem, and Noncommutative Continued Fractions

Abstract

AbstractA new approach to the Euler‐Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed: motivated by an analogy with the Camassa‐Holm equation a class of isospectral deformations of the beam problem is formulated; 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; the inverse problem is solved for the special case of a discrete Euler‐Bernoulli beam. The solution involves a noncommutative generalization of Stieltjes’ continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel‐like determinants. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.