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

Abstract

We study matrix roots with certain commutation properties and their application to the explicit construction of Darboux matrices in the framework of the GBDT version of Bäcklund-Darboux transformation. The approach is demonstrated on the important case of generalised canonical systems depending rationally on the spectral parameter.

Hamiltonian systems with several space variables: dressing, explicit solutions and energy relations

“…This system coincides with the system [22, (3.1), (3.4)] if we put j = I m in [22]. The strong restrictions on H k , which appear in [22], are non-essential for the facts derived below similar to [22,Section 3]. Note that spectral theory of the important particular cases of (2.1) have been studied in [13] and more thoroughly in [20].…”

“…This system coincides with the system [22, (3.1), (3.4)] if we put j = I m in [22]. The strong restrictions on H k , which appear in [22], are non-essential for the facts derived below similar to [22,Section 3].…”

“…, ζ r ) may be a vector or an m × n matrix, H k (t) are m × m matrix-valued functions (matrix functions) and H k (t) * stands for the * e-mail address: oleksandr.sakhnovych@univie.ac.at conjugate transpose of the matrix H k (t). The paper may be considered as a continuation of [22,Section 4], where systems ∂ψ ∂t = J ψ with symmetric operators J appear. Here, J is skew-symmetric, which is of essential interest in the control theory and in the theory of port-Hamiltonian systems in particular.…”

“…Here, we use our GBDT (generalised Bäcklund-Darboux transformation) approach, which was initially introduced in [19]. For further references on GBDT, see, for instance, [1,8,9,[21][22][23].…”

Explicit solutions of Schrödinger and KdV equations in terms of square roots of the generalised matrix eigenvalues

“…If det A = 0 square roots Q of A always exist (see, e.g., [4, Chapter VIII, §6] with further details and references in [19,Section 2]). Clearly, Q is invertible in this case.…”