Solitary water wave interactions

Abstract

This article concerns the pairwise nonlinear interaction of solitary waves in the free surface of a body of water lying over a horizontal bottom. Unlike solitary waves in many completely integrable model systems, solitary waves for the full Euler equations do not collide elastically; after interactions, there is a nonzero residual wave that trails the post-collision solitary waves. In this report on new numerical and experimental studies of such solitary wave interactions, we verify that this is the case, both in head-on collisions (the counterpropagating case) and overtaking collisions (the copropagating case), quantifying the degree to which interactions are inelastic. In the situation in which two identical solitary waves undergo a head-on collision, we compare the asymptotic predictions of Su and Mirie [J. Fluid Mech. 98, 509 (1980)] and Byatt-Smith [J. Fluid Mech. 49, 625 (1971)], the wavetank experiments of Maxworthy [J. Fluid Mech. 76, 177 (1976)], and the numerical results of Cooker, Weidman, and Bale [J. Fluid Mech. 342, 141 (1997)] with independent numerical simulations, in which we quantify the phase change, the run-up, and the form of the residual wave and its Fourier signature in both small- and large-amplitude interactions. This updates the prior numerical observations of inelastic interactions in Fenton and Rienecker [J. Fluid Mech. 118, 411 (1982)]. In the case of two nonidentical solitary waves, our precision wavetank experiments are compared with numerical simulations, again observing the run-up, phase lag, and generation of a residual from the interaction. Considering overtaking solitary wave interactions, we compare our experimental observations, numerical simulations, and the asymptotic predictions of Zou and Su [Phys. Fluids 29, 2113 (1986)], and again we quantify the inelastic residual after collisions in the simulations. Geometrically, our numerical simulations of overtaking interactions fit into the three categories of Korteweg-deVries two-soliton solutions defined in Lax [Commun. Pure Appl. Math. 21, 467 (1968)], with, however, a modification in the parameter regime. In all cases we have considered, collisions are seen to be inelastic, although the degree to which interactions depart from elastic is very small. Finally, we give several theoretical results: (i) a relationship between the change in amplitude of solitary waves due to a pairwise collision and the energy carried away from the interaction by the residual component, and (ii) a rigorous estimate of the size of the residual component of pairwise solitary wave collisions. This estimate is consistent with the analytic results of Schneider and Wayne [Commun. Pure Appl. Math. 53, 1475 (2000)], Wright [SIAM J. Math. Anal. 37, 1161 (2005)], and Bona, Colin, and Lannes [Arch. Rat. Mech. Anal. 178, 373 (2005)]. However, in light of our numerical data, both (i) and (ii) indicate a need to reevaluate the asymptotic results in Su and Mirie [J. Fluid Mech. 98, 509 (1980)] and Zou and Su [Phys. Fluids 29, 2113 (1986)].