Integrable dynamical systems associated with the KdV equation

An isospectral deformation representation is constructed for a countable set of dynamical systems with a quadratic nonlinearity, which become the Korteweg–de Vries equation in the continuum limit. Integrable reductions of certain dynamical systems with an arbitrary degree of nonlinearity are obtained. The dynamics of the components of the scattering matrix are integrated for these infinitedimensional dynamical systems. An isospectral deformation representation is indicated for certain nonhomogeneous finite-dimensional dynamical systems. A new construction of integrable dynamical systems associated with simple Lie algebras and generalizing the discrete KdV equations is found.
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