INTEGRABLE AND NONINTEGRABLE CASES OF THE LAX EQUATIONS WITH A SOURCE

The Korteweg-de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that, depending on the choice of the basis of the eigenfunctions, we have the following three possibilities: (I) evolution equations for the scattering data are nonintegrable; (2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg-de Vries equation with a source at some t' > t(0) leaves the considered class of functions decreasing rapidly enough as x --> +/-infinity; (3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg-de Vries equation with a source exists at all t > t(0). All these possibilities are widespread and occur in other Lax equations with a source.