Parametric envelope solitons in coupled Korteweg-de Vries equations

Abstract We demonstrate that a system of linearly coupled Korteweg-de Vries equations, which inter alia is a general model of resonantly coupled internal waves in a stratified fluid, can give rise to broad envelope solitons produced by a double phase-and group-velocity resonance between the fundamental and second harmonics for certain wavenumbers. We derive asymptotic equations for the amplitudes of the two harmonics, which are identical to the second-harmonic-generation equations in a diffractive medium, that have recently attracted a lot of attention in nonlinear optics and give rise to the so-called parametric solitons. To check if the predicted solitons are close to exact solutions of the coupled Korteweg-de Vries equations, we perform direct numerical simulations, with initial conditions suggested by the above-mentioned parametric-soliton solution to the asymptotic equations. Since the latter is known only in a numerical form, we use for them a recently developed analytical variational approximation. As a result, we observe very long-lived steadily propagating wave packets generated by these initial conditions. Thus we find a physical system that may allow experimental observation of propagating parametric solitons, while in nonlinear optics they are observed only as spatial solitons.