Exact theory of solitary waves in a stratified fluid with surface tension. I: Nonoscillatory case (Q687489)

The subject is to give a rigorous proof of existence of an elevation solitary wave solution to a full system of partial differential equations describing two-dimensional motion of an inviscid incompressible stratified liquid over a solid bottom (the liquid has a free surface). The existence proof is given for parametric regions in which the full system of equations can be reduced asymptotically to the Korteweg-de Vries equation for long small-amplitude waves, which has the explicit soliton solution. The proof is based on analysis of two auxiliary Sturm- Liouville problems. An explicit example, with a particular distribution of the density, is considered, to which the general results apply to guarantee existence of the solitary wave.