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Exactly and completely integrable nonlinear dynamical systems

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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 Schrödinger, Korteweg-de 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 Heisenberg 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.

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  1. Institute for High Energy Physics, 142284, Serpukhov, U.S.S.R.

    A. N. Leznov & M. V. Saveliev

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Leznov, A.N., Saveliev, M.V. Exactly and completely integrable nonlinear dynamical systems. Acta Appl Math 16, 1–74 (1989). https://doi.org/10.1007/BF00046886

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