On a criterion of integrability by the inverse scattering transform (Q1374980)

Several nonlinear PDEs which are integrable by the method of the inverse scattering transform are considered in this note. The cases of (1) the Korteweg- de Vries equation, (2) the nonlinear Schrödinger equation and (3) the equations of self-induced transparency are under consideration. The author writes down these equations using the dependent variable \(E(x, t)\) and its inverse \(r=1/E\). The linear terms of the equations in this form are not varied from the usual form, and the nonlinear part in the right-hand side contains ordinary derivatives. The author constructs the solution of prescribed equations by seeking a simultaneous solution of the two equations. One of these equations is linear with respect to \(r\) and coincides with the linearized initial equation, the second equation is an ordinary differential equation for the function \(E\) or its module where the second independent variable serves as parameter. The solutions to equations (1)-(3) thus constructed coincide with the single-soliton solutions to these equations obtained by the inverse scattering transform.