A detailed analysis is given to the solution of the cubic Schrodinger equation iq,+qxx+2IqI2q=0 under the boundary conditions q(x,t)~Ae2iA2/ as Ixl~oo. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and "pulsates" in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x+ T,t)= q(x,t) as Ixl~oo] is discussed.

Solutions to the cubic nonlinear Schrödinger equation, periodic in space, are developed.

We give a complete analysis of the long-wave-short-wave resonance equations which appear in fluid mechanics as well as plasma physics. Using the inversescattering technique, these equations can be reduced to a pair of linear integral equations (Marchenko equations), with the N-soliton solutions intimately related to the asymptotic state of the evolution equations. The interaction of solitons and the conserved quantities are discussed. I. IntroductionEver since the development of the inverse-scattering technique for the nonlinear evolution equations [1-4], exact solutions have been available for a few very important nonlinear water-wave equations [1,3]. Among them are the Korteweg-deVries equation, which describes the propagation of unidirectional shallow-water waves (long waves) where the nonlinear steepening effect is counterbalanced by dispersion, and the nonlinear Schrodinger equation, which describes the long-time evolution of the envelope of a packet of plane finite-amplitude waves. In both cases it is shown that soliton (or envelope soliton) solutions are important and are intimately related to the asymptotic state of the evolution equations.Recently, Djordjevic and Redekopp [5] found that when the group velocity of the short wave matches the phase velocity of the long wave, the nonlinear Schrodinger equation breaks down, since the coefficient of the cubic nonlinear term becomes singular and there is a resonance between the long wave and the short wave. They presented an analysis and found that the evolution equations Dr. y.-c. Ma, Room 2-337, M.I.T .•

The evolution equations describing the interaction of long and short waves when they are resonantly coupled are discussed. Envelope-pulse soliton solutions of these equations are constructed using the method of Hirota and the existence of a breather state is shown. The wavetrain modulational characteristics and envelope-hole solitary wave solutions are also presented.