Universality and integrability of the nonlinear evolution PDE’s describing N-wave interactions

Abstract

The universality of the equations describing N-wave interactions is demonstrated by deriving them from a very large class of nonlinear evolution equations (essentially all those whose linear part is dispersive). Various forms of these equations are displayed. The fact that these ‘‘universal’’ nonlinear evolution equations obtain, by an appropriate asymptotic limit, from such a large class of nonlinear evolution equations, suggests that they should be integrable; since for this it is sufficient that the large class from which they are obtainable contain just one integrable equation. This expectation is validated in several cases, by deriving the equations from known integrable equations. In this manner an explanation may be provided of the (already known) integrable nature of certain equations; and new integrable equations may be obtained. Both S-integrable and C-integrable equations are discussed, namely both equations integrable via an appropriate spectral transform and solvable via an appropriate change of variables. In this paper the treatment is limited to equations in 1+1 dimensions.