The Complete Solution of the Long‐Wave–Short‐Wave Resonance Equations

Abstract

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 .•

On the stability of weakly-nonlinear gravity-capillary waves

“…Djordjevic and Redekopp [7] were concerned with similar instabilities in shallow water and Benney [8] with the class of long-wave short-wave systems. Ma [9] and Ma and Redekopp [10] studied a set of coupled equations similar to those derived in [7], but their main purpose was to consider envelope soliton solutions. In [10], it was shown that the growth rate of the triad interaction is…”

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

“…4) u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), which was introduced by Benney [2] (see also Yajima-Oikawa [12] and FunakoshiOikawa [4]) and both the inverse scattering method ( [12], [7]) and the theory of evolution equations ( [1], [6], [10]) have been applied. Zhang Fayong and Xiang Xinmin [14] investigated the pseudospectral method for (1.3)-(1.4).…”

Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions

“…The short wave is usually described by the Schrödinger type equation and the long wave is described by some sort of wave equation accompanied with a dispersive term. In the theory of capillary-gravity waves, Kawahara et al [13] studied the coupled system which was introduced by Benney [3] (see also Yajima-Oikawa [26] and FunakoshiOikawa [8]) and both the inverse scattering method ( [26], [19]) and the theory of evolution equations ( [24], [18], [1]) have been applied. Unlike the results for Benney's equation (1.3), the coupled Schrödinger-KdV equation (1.2) has been shown not to be a completely integrable system (BenirovBurtsev [2]).…”