Some physical applications of solitons

§1.1 I n t r o d u c t i o n Two equations t ha t have been studied intensively in recent years are the Korteweg-de Vries (KdV; 1805) equation, uT + Buu x + Uxx x = 0 , (1) and a generalization of it due to Kadomtsev and Petviashvili (KP; 1970}, (u~ + 6u~ x + ~xxx)x + 3 ~ = 0. (3) Most of this interest is due to the remarkable fact t ha t each equation can be solved exactly as an initialvalue problem by a method now known as the Inverse Scattering Transform. This method was first discovered for the KdV equation on c ~ < X < vo, in the famous papers of Gardner, Greene, Kruskal and Miura (1967, lg74). The corresponding work for (1) on a periodic interval was published by several people during lg74-1978 [Novikov (lg74), Dubrovin and Novikov (1974), Dubrovin (1975), Lax (1975), Its and Matveev (1975), McKean and van Moerbeke (lg75), McKean and Trubowitz (1976), Dubrovin, Matveev and Novikov (1976)]. For the KP equation on ~ < X, rl < c~, a method of solution was given very recently by Ablowitz, Bar Yaaeov, and Fokas (1982); e/. Ablowitz and Fokas, in these Proceedings). As we shall see below, it happens tha t both (1) and (2) also model the evolution of water waves of moderate amplitude as they propagate in one direction in relatively shallow water. In physical By Harvey Segur, Allan Finkel (Institute for Advanced Study, Princeton), and Hilda Philander (A.R.A.P., Princeton). 1.2. Derivat ion of the equations 213 z :t] (x, y , t ) g ~ z =0 ~ : \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z : h Figure 1.1. Physical configuration showing notat ion for Equations (3)-(6). terms, the KdV equation arises if the waves are strictly one-dimensional (i.e., one spatial dimension plus time), while the KP equation arises if they are only nearly one-dimensional. Because both equations are completely integrable, they provide very precise predictions about the evolution of water waves under appropiate conditions. This paper has two objetives. The first is to examine the validity of (1) and (2) as models of water waves, by comparing their solutions with some of the available experimental data. The comparison given here will be brief, but much more detailed verifications of (1) have been made elsewhere (e.g., Hammack and Segur, 1974, 1978). The second objetive is to describe in detail a special family of solutions of (2), which are periodic in each of two independent variables. It appears tha t these doubly-periodic solutions may have great practical importance as models of long water waves. Among other things, they describe: L the nonlinear interaction of two trains of finite amplitude waves in shallow water; iL the reflection of a t rain of finite-amplitude shallow-water waves from a vertical wall (by replacing an appropriate line of symmetry with the wall); IlL the reflection of such a wave t rain by a change in bot tom topography; or iv. typical finite-amplitude, short-crested waves in shallow water. In this last sense, they are the natural generalizations to two dimensions of cnoidal waves in one dimension. §1.2 D e r i v a t i o n of t h e e q u a t i o n s The classical problem of water waves is to find the irrotational motion of an inviseid, incompressible, homogeneous fluid, subject to a constant gravitational force (g). The fluid rests on a horizontal and impermeable bed of infinite extent at z --~ h and has a free surface at z = f(x, y, t); see Figure 1.1. In this derivation we neglect the effects of surface tension at the free surface, al though it can be included without difficulty (e.g., see Ch. 4 of Ablowitz and Segur, 1981). The fluid has a velocity potencial, ~b, which satisfies VZ¢---0 , h < z < f ( z , y , t ) ; (3) (irrotational motion of an incompressible fluid). It is subject to boundary conditions on the bottom, Z ~ h : Cz = o, (4) (impermeable bed); and along the free surface, z ~f: D~ Dt s't + ¢~s'~ + ~b~'y = ~ (5)