Complete integrability of coupled KdV and NLS equations

Painlevé analysis is performed for the coupled system of nonlinear partial differential equations consisting of the KdV equation and NLS equation initially studied by Nishikawa. Various possibilities for the constants occurring in the system are explored, paying attention to the integrability property. This equation occurring in the field of plasma physics satisfies all the requirements of Painlevé analysis and can be ascertained to be completely integrable, though no Lax pair is known for it.