Solitary wave ($u(x,t)=U(x-ct),U(\xi )\to 0\text{ as }|\xi |\to \infty$)

A solitary wave is a spatially localized travelling wave that retains its profile as it propagates. The displayed form uses a zero background; other constant backgrounds are possible. A soliton additionally has the characteristic elastic interaction property with other solitons. The existence of a single localized travelling profile does not alone establish that stronger property.