Hierarchy of symplectic forms for the Schrödinger and the Dirac equations on a line

A sequence of symplectic forms have been constructed, relative to each of which the Korteweg–de Vries equation and all its higher analogs are Hamiltonian. The well-known conservation laws serve as the Hamiltonians. An analogous system of forms has been constructed also for a family of equations solvable by use of the inverse scattering problem for the Dirac operator. The results are used in the investigation of the connection between various non-linear evolution equations.