Einstein, $σ$-model and Ernst-type equations and non-isospectral GBDT version of Darboux transformation

Abstract

We present a non-isospectral GBDT version of Bäcklund-Darboux transformation for the gravitational and σ-model equations. New families of explicit solutions correspond to the case of GBDT with nondiagonal generalized matrix eigenvalues. An interesting integrable Ernst-type system, the auxiliary linear systems of which are nonisospectral canonical systems, is studied as well.

Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

“…3. Another (related to AQ = QA) commutation property is required for the explicit construction of solutions of sigma models and gravitational equations in [34]. Namely (after a change of notations for the ones used in this paper), it is required in [34, (B.1) and (B.2)] that…”

“…Bäcklund-Darboux transformation is a fruitful tool in the study of linear and nonlinear differential equations (see, e.g., [3,6,14,15,17,21,24,25,28,42] and numerous references therein). GBDT version of Bäcklund-Darboux transformation (with generalised matrix eigenvalues) was actively developed starting from our work [30] (see further references in [20,27,[34][35][36][37]).…”

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

“…More precisely, GBDT (generalised Bäcklund-Darboux transformation) was constructed for these systems. It is important that GBDT (see, e.g., [22,30,41,46,50] and references therein) is characterized by the generalised matrix eigenvalues (not necessarily diagonal) and the corresponding generalised eigenfunctions. In Section 3, the generalised matrix eigenvalues and the generalised eigenfunctions are denoted by A and Λ(x), respectively.…”

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

“…More precisely, GBDT (generalized Bäcklund-Darboux transformation) was constructed for these systems. It is important that GBDT (see, e.g., [20,27,38,44,45] and references therein) is characterized by the generalized matrix eigenvalues (not necessarily diagonal) and the corresponding generalized eigenfunctions. In Section 3, generalized matrix eigenvalues and generalized eigenfunctions are denoted by A and Λ(x), respectively.…”

“…We start with some initial systems (1.2), where initial Hamiltonians H(x) are comparatively simple, and construct explicitly the fundamental solutions and generalized eigenfunctions for these systems. (In particular, some considerations from [40,44] were helpful for this purpose.) Using generalized eigenfunctions, the transformed generalized Hamiltonians and so called Darboux matrices are constructed as well.…”