A simple model of the integrable Hamiltonian equation

A method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Bäcklund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrödinger equation and the so-called Harry Dym equation turn out to be particular examples.