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Abdon E. Choque-Rivero

Publications and source records attributed to Abdon E. Choque-Rivero.

11 recordsLinked to original sources
arXiv2026

On the Coefficients of Hurwitz-Type Matrix Polynomials

Consider the matrix polynomial $\mathbf f_n(z)=I_qz^n+A_1z^{\,n-1}+\cdots+A_n,$ where $A_j\in\mathbb C^{q\times q}$. Write $\mathbf f_n(z)=\mathbf h_n(z^2)+z\,\mathbf g_n(z^2).$ A matrix polynomial $\mathbf f_n$ is called a Hurwitz-type matrix polynomial if, for $n=2m$, $\mathbf g_n(z)\mathbf h_n(z)^{-1}$, and for $n=2m+1$, $\mathbf h_n(z)\bigl(z\mathbf g_n(z)\bigr)^{-1}$, admit finite continued fraction expansions with positive definite coefficients. We derive explicit formulas for the coefficients of Hurwitz-type matrix polynomials in terms of orthogonal matrix polynomials, Markov parameters, and Schur complements. We introduce the associated block Hurwitz matrix and establish determinant identities relating it to the corresponding block Hankel matrices. Finally, we settle a conjecture concerning the positivity of the determinants of the coefficient matrices of Hurwitz-type matrix polynomials. We prove that this property holds for degrees at most three, but fails in degree four by means of an explicit counterexample.

math.CA↗
arXiv2026

On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Every matrix polynomial ${\mathbf f}_n$ admits a decomposition of the form \[ {\mathbf f}_n(z)={\mathbf h}_n(z^2)+z\,{\mathbf g}_n(z^2). \] The matrix polynomial ${\mathbf f}_{2m}$ is said to be of Hurwitz type if the expression ${\mathbf g}_{2m}(z){\mathbf h}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial ${\mathbf f}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}{\mathbf h}_{2m+1}(z){\mathbf g}_{2m+1}^{-1}(z)$ has the same property. We derive an explicit representation of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this representation, we obtain a direct proof that every Hurwitz-type matrix polynomial is Hurwitz. The Hurwitz property of this class was also investigated in [52]; our approach is based on an explicit Bezoutian representation. This provides a constructive connection between continued-fraction representations, matrix Bezoutians, and Hurwitz stability. We also develop a completion procedure that associates with a given matrix polynomial a Hurwitz-type matrix polynomial of higher degree. As a consequence, whenever such a completion exists, the original polynomial is Hurwitz. The proposed construction is illustrated by examples.

math.CA↗
arXiv2026

Factorization for the matrix-valued general Jacobi system on the full-line lattice

The Jacobi system with matrix-valued coefficients and with the spectral parameter depending on a matrix-valued weight factor is considered on the full-line lattice. The scattering from the full-line lattice is expressed in terms of the scattering from the fragments of the whole lattice by developing a factorization formula for the corresponding transition matrices. In particular, the matrix-valued transmission and reflection coefficients for the full-line lattice are explicitly expressed in terms of the scattering coefficients for the left and right lattice fragments. Since the matrix-valued scattering coefficients are easier to determine for the fragments than for the full-line lattice, the factorization formula presented provides a method to determine the scattering coefficients for full-line lattices. The theory presented is illustrated with various explicit examples, including an example demonstrating that the matrix-valued left transmission coefficient in general is not equal to the matrix-valued right transmission coefficient for a lattice.

math-ph↗
arXiv2025

Soliton solutions to the Sawada--Kotera equation

We consider the direct and inverse scattering problems for the third-order differential equation in the reflectionless case. We formulate a corresponding Riemann--Hilbert problem using input consisting of the bound-state poles of a transmission coefficient and the bound-state dependency constants. With the time-evolved dependency constants, using the solution to the Riemann--Hilbert problem, we construct soliton solutions to an integrable system of fifth-order nonlinear partial differential equations. By imposing some appropriate restrictions on the dependency constants, we show that those soliton solutions yield soliton solutions to the Sawada--Kotera equation.

nlin.SI↗
arXiv2025

Soliton solutions associated with a class of third-order ordinary linear differential operators

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3ψ/dx^3+Q\,dψ/dx+Pψ=k^3ψ,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.

nlin.SI↗
arXiv2022

Discretization of the wave equation on a metric graph

The question of what conditions should be set at the nodes of a discrete graph for the wave equation with discrete time is investigated. The variational method for the derivation of these conditions is used. A parallel with the continuous case is also drawn. As an example the problem of shape controllability from the boundary is studied.

math.OC↗
arXiv2018

Darboux transformation for the discrete Schrödinger equation

The discrete Schrödinger equation on a half-line lattice with the Dirichlet boundary condition is considered when the potential is real valued, is summable, and has a finite first moment. The Darboux transformation formulas are derived from first principles showing how the potential and the wavefunction change when a bound state is added to or removed from the discrete spectrum of the corresponding Schrödinger operator without changing the continuous spectrum. This is done by explicitly evaluating the change in the spectral density when a bound state is added or removed and also by determining how the continuous part of the spectral density changes. The theory presented is illustrated with some explicit examples.

math-ph↗
arXiv2018

On the bound states of the discrete Schrödinger equation with compactly supported potentials

The discrete Schrödinger operator with the Dirichlet boundary condition is considered on the half-line lattice $n\in \{1,2,3,\dots\}.$ It is assumed that the potential belongs to class $\mathcal A_b,$ i.e. it is real valued, vanishes when $n>b$ with $b$ being a fixed positive integer, and is nonzero at $n=b.$ The proof is provided to show that the corresponding number of bound states, $N,$ must satisfy the inequality $0\le N\le b.$ It is shown that for each fixed nonnegative integer $k$ in the set $\{0,1,2,\dots,b\},$ there exist infinitely many potentials in class $\mathcal A_b$ for which the corresponding Schrödinger operator has exactly $k$ bound states. Some auxiliary results are presented to relate the number of bound states to the number of real resonances associated with the corresponding Schrödinger operator. The theory presented is illustrated with some explicit examples.

math-ph↗
arXiv2016

The Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem via New Dyukarev-Stieltjes Parameters and Extremal Solutions via Continued Fractions

We obtain a new multiplicative decomposition of the resolvent matrix of the truncated Hausdorff matrix moment (THMM) problem in the case of an odd and even number of moments via new Dyukarev-Stieltjes matrix (DSM) parameters. Explicit interrelations between new DSM parameters and orthogonal matrix polynomials on a finite interval $[a,b]$, as well as the Schur complements of the block Hankel matrices constructed through the moments of the THMM problem are given. Additionally, the extremal solutions of the THMM problem are represented via matrix continued fractions in terms of the DSM parameters.

math.CA↗
arXiv2016

Factorization of the transition matrix for the general Jacobi system

The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is done by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line.

math-ph↗