On equations solvable by the inverse scattering method for the Dirac operator.

Consider the Dirac operator L with the potentials <f>u=u(x,t)</f> and <f>v=v(x,t)</f> satisfying an arbitrarily taken system of two nonlinear evolution equations of the form <f>∂u/∂t=f</f>, <f>∂v/∂t=g</f>, the right-hand sides of which are rather smooth functions x, u, v and their derivatives with respect to x up to a certain finite order <f>k0&ges;0</f>. It is shown that the evolution of the S-matrix of the operator L generated by the solutions of this system rapidly decreasing as <f>x→±∞</f> is described by ordinary differential equations if and only if the system of equations for the potentials u and v under consideration has the Lax representation with the operator A differential with respect to x. The method established in this paper can be applied to elucidate analogous problems concerning a possible application of other operators to solve nonlinear evolution equations by the inverse scattering method. [Copyright &y& Elsevier]