Two verifications of the determinantal solutions of the korteweg-de vries and kadmotsev-petviashvili equations

Abstract

It is well known that the Korteweg-de Vries equation

is satisfied by a function of the form

and that the Kadomtsev-Petviashvili equation

is satisfied by a function of the form

where H is a certain determinant of arbitrary order whose elements are functions of x, y, t, and H₁ is a special case of H.

These equations have been solved in the context of the theory of solitons but no satisfactory verifications of the solutions by differentiation and substitution have been published. It is the object of this paper to fill these gaps in the literature. Each solution has been verified by two distinct methods each of which apply recently developed techniques in the theory of determinants.

AMS(MOS)::

• 15A15
• 35Q20