Bäcklund transformations and Hamiltonian flows

Abstract

In this work we show that, under certain conditions, parametric Bäcklund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation leads to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, that indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An example with the periodic Toda lattice is given.

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

“…Here we applied the transformations found in [4] to the Gross-Pitaevskii equation in 1+1 dimensions. Indeed the transformations found are auto-Bäcklund transformations for the ordinary differential equation (1) and a reduction of the partial differential equation (26) was required to apply our result. The application to the Gross-Pitaevskii equation provided in this work can be considered a first approach to a more general research on this issue: the case of periodic potentials would be interesting for physical applications but also we believe that the analytical structures of the equations obtained deserve further efforts to fully understand the potentiality of the method.…”

“…The application to the Gross-Pitaevskii equation provided in this work can be considered a first approach to a more general research on this issue: the case of periodic potentials would be interesting for physical applications but also we believe that the analytical structures of the equations obtained deserve further efforts to fully understand the potentiality of the method. In particular equation (30) shows that the Schröder's functional equation Φ(f (x)) = sΦ(x) with an eigenvalue s equal to 1 plays a special role in the description of the solutions of equation (26). This and other issues will be further developed in future works.…”

On auto and hetero Bäcklund transformations for the Hénon–Heiles systems

“…The matrix V needs not to be unique because a dynamical system can have different auto BTs [8,17]. In contrast with [7] we do not require that the transformed Lax matrixL(λ) have the same structure in the spectral parameter λ as the original Lax matrix.…”

“…, H n [15]. The counterpart of the discretization for finite dimensional systems is also currently accepted: by viewing the new v-variables as the old u-variables, but computed at the next time step; then the Bäcklund transformation (1.2) defines an integrable symplectic map or discretization of the continuous model, see discussion in [8,17].…”

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

“…Usually, these transformations are obtained from the Lax representation of the model, by constructing an appropriate dressing matrix intertwining the Lax representation of the system (see e.g. [30] for details). These transformations turn out to be canonical in the phase space of the system.…”