to describe approximately the slow evolution of long water waves of moderate amplitude as they propagate under the influence of gravity in one direction in shallow water of uniform depth. We now know that the KdV equation describes approximately the evolution of long, one-dimensional waves in many physical settings, including long internal waves in a density-stratified ocean, ion-acoustic waves in a plasma, acoustic waves on a crystal lattice, and more.
If one relaxes the restriction that the waves be strictly one-dimensional, then one often derives instead a natural generalization of KdV that was first discovered by Kadomtsev & Petviashvili,
Depending on the physical problem, one can derive one of two KP equations, which differ in the sign of their uxxxx-terms. The equation given above is sometimes called KP2. In particular, this KP equation describes approximately the slow evolution of gravity-induced waves of moderate amplitude on shallow water of uniform depth when the waves are nearly one-dimensional.
The KP equation admits a large family of exact quasiperiodic solutions. Each such solution has N independent phases. Recent comparisons with experiments show that the family of two-phase solutions of the KP equation describes waves in shallow water with surprising accuracy. This success suggests that more complicated KP solutions might provide accurate physical models of more complex wave phenomena.