Slowly varying solitary waves. I. Korteweg-de Vries equation

Abstract
The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.