Consequences of scaling in nonlinear partial differential equations

Abstract

Scaling of the form x=ax′, t=a^ht′, q=a^mQ of a nonlinear partial differential equation for q is connected with the form of the auxiliary functions in an inverse scattering method (AKNS scheme). Solvability of an equation by this scheme is treated. It is shown that only equations with m=2 are solvable by using the method of inverse scattering in conjunction with the Schrödinger eigenvalue equation. The criterion m=2 restricts the form of the terms in these equations. The terms, powers of q and its derivatives, can be found by inspection. A separate problem, the decay of a single soliton in the Korteweg–de Vries equation with damping, is solved using only scaling properties.