On a class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl--Titchmarsh theory

Abstract

An important representation of the general-type fundamental solutions of the canonical systems corresponding to matrix string equations is established using linear similarity of a certain class of Volterra operators to the squared integration. Explicit fundamental solutions of these canonical systems are also constructed via the GBDT version of Darboux transformation. Examples and applications to dynamical canonical systems are given. Explicit solutions of the dynamical canonical systems are constructed as well. Three appendices are dedicated to the Weyl--Titchmarsh theory for canonical systems, transformation of a subclass of canonical systems into matrix string equations (and of a smaller subclass of canonical systems into matrix Schrödinger equations), and a linear similarity problem for Volterra operators.

Some results and ideas from our papers arXiv:nlin/0402055, arXiv:2002.04975, and arXiv:2003.13024 have been used in this work. Several references are added in the second version as well as small complements in the text

References in corpus (3)

• Dressing Method for the Degasperis-Procesi Equation
• Inverse problem for one-dimensional dynamical Dirac system (BC-method)
• Einstein, $\sigma$-model and Ernst-type equations and non-isospectral GBDT version of Darboux transformation

Cited by in corpus (3)

• On new classes of explicit solutions of Dirac, dynamical Dirac and Dirac--Weyl systems with non-vanishing at infinity potentials, their properties and applications
• On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations
• Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions