Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system

Lie-algebraic structure of (2 + 1)-dimensional Lax-type integrable nonlinear dynamical systems

“…Our approach devised in the present work is substantially based on the results obtained earlier in [11,13] and explains completely the computational properties of multidimensional flows delivered earlier in [5,15,16]. As the key points of our approach, we have used canonical Hamiltonian structures naturally existing on the extended phase space and the related Båcklund transformation that saves Casimir invariants of a chosen matrix integrodifferential Lie algebra.…”

“…As the key points of our approach, we have used canonical Hamiltonian structures naturally existing on the extended phase space and the related Båcklund transformation that saves Casimir invariants of a chosen matrix integrodifferential Lie algebra. The latter gives rise to some additional Hamiltonian properties of considered extended evolution flows studied earlier in [5,13], making use of the standard inverse scattering transform [2,3] and the formal symmetry reduction for the KP-hierarchy [15,16] of commuting operator flows.…”

“…As one can verify by analyzing the structure of the Båcklund-type transformation (16), it strongly depends on the type of an ad-invariant scalar product chosen on an operator Lie algebra Ᏻ and its Lie algebra decomposition of the type (2). Since, in general, there exist other possibilities of choosing such decompositions and ad-invariant scalar products on Ᏻ , they naturally give rise to other resulting types of the corresponding Båck-lund transformations, which can be a subject of another special investigation.…”

“…In fact, we have found above a triple linearization for a (2 + l)-dimensional dynamical system, which is a new important ingredient of the Lie-algebraic approach to Lax-type integrable flows based on the Båcklund-type transformation (16) developed in this work. It is clear that a similar construction of a triple linearization of the type (26) -(28) can be done for many other, both old and new, (2 + l)-dimensional dynamical systems, on which we plan to stop in detail in another work under preparation.…”

Lie-algebraic structure of (2 + 1)-dimensional lax-type integrable nonlinear dynamical systems

“…Our approach devised in the present work is substantially based on the results obtained earlier in [11,13] and explains completely the computational properties of multidimensional flows delivered earlier in [5,15,16]. The latter gives rise to some additional Hamiltonian properties of considered extended evolution flows studied earlier in [5,13], making use of the standard inverse scattering transform [2,3] and the formal symmetry reduction for the KP-hierarchy [15,16] of commuting operator flows. The latter gives rise to some additional Hamiltonian properties of considered extended evolution flows studied earlier in [5,13], making use of the standard inverse scattering transform [2,3] and the formal symmetry reduction for the KP-hierarchy [15,16] of commuting operator flows.…”