Exactly and completely integrable nonlinear dynamical systems

Abstract

This survey is devoted to a consistent exposition of the group-algebraic methods for the integration of systems of nonlinear partial differential equations possessing a nontrivial internal symmetry algebra. Samples of exactly and completely integrable wave and evolution equations are considered in detail, including the generalized (periodic and finite nonperiodic) Toda lattice, nonlinear Schr6dinger, Korteweg~le Vries, Lotka-Volterra equations, etc.). For exactly integrable systems, the general solutions of the Cauchy and Goursat problems are given in an explicit form, while for completely integrable systems an effective method for the construction of their soliton solutions is developed.Application of the developed methods to a differential geometry problem of the classification of integrable manifolds embeddings is discussed.Supersymmetric extensions are constructed for exactly integrable systems. Using an example of the generalized Toda lattice, a quantization scheme is developed. It includes an explicit derivation of the corresponding Pleisenberg operators and their description in terms of Hopf-type quantum algebras.Among multidimensional systems, the four-dimensional self-dual Yang-Mills equations are investigated with the aim of constructing their general solutions.

Spinor field realizations of non-critical strings

“…where f 3 [3], f 3 [5], f 3 [7] and f 3 [10] are arbitrary constants but do not vanish at the same time.…”

“…Furthermore, it appears in the quantum Hall effect, black holes, in lattice models of statistical mechanics at criticality, and in other physical models [3,4] and so on.…”

Bäcklund Transformations in 10d Susy Yang–mills Theories

“…For this, we closely follow an earlier review. 10 There are two bosonic complex coordinates z, y and their conjugate z, y. One may start from the equations (Indices mean derivatives) …”

Solvable models in quantum mechanics

“…About these models, see, e.g., [1,2] and references therein. About these models, see, e.g., [1,2] and references therein.…”