Equazione di Korteweg-deVries: teoria ed applicazioni

1 INTRODUZIONE Il concetto di onda solitaria fu introdotto nella nascente scienza dell'idrodinamica ben oltre cento anni fa da John Scott Russell; nel suo articolo "Report on waves" pubblicato nel 1844 egli descrive così la sua sensazionale scoperta: "I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward whit great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears." All'inizio, tale rivelazione di Russell fronteggiò grande ostilità e scetticismo da parte delle persone influenti della comunità scientifica dei suoi tempi. Furono due studiosi olandesi, Diederik Johannes Korteweg e Gustav de Vries, a risolvere tale questione definitivamente; nel 1895 pubblicarono l'articolo "On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves" [1], nel quale introdussero la seguente equazione differenziale: )(1 x3 1 3 2 2 1 x g 2 3 t 2 2 2       ∂ η∂ σ+αη+η ∂ ∂ = ∂ η∂ λ