Geometry of Solitons, Volume 47, Number 1

parameterized by θ ∈ R . But we do not expect that the “sum” of two such solutions will again be a solution. However, the special class of soliton equations, the subject of this article, does have a form of nonlinear superposition. An n-soliton solution is a solution that is asymptotic to a nontrivial sum of n solitary waves ∑n i=1 fi(x− cit) as t → −∞ and to the sum of the same waves ∑n i=1 fi(x− cit + ri) with some nonzero phase shifts ri as t →∞. In other words, after nonlinear interaction the individual solitary waves pass through each other, keeping their velocities and shapes but with phase shifts. Equations with multisoliton solutions are very rare (they occur nearly always in one space dimension); these equations are called soliton equations. The Korteweg-de Vries equation