Elementary but accurate analytical approximation for a one-dimensional soliton, conservative or not

Abstract

Well-known rigorous methods have been elaborated long ago to exactly solve conservative soliton equations. Mainly, there are the inverse scattering transform, Hirota's direct method, and Sato's formalism. These methods are fully satisfactory: analytical expressions for solitons are obtained in any spatial dimension as well as multisoliton solutions. Therefore, there is no need for an additional approach, especially if it is an approximate one, restricted to a one-dimensional situation and to a single soliton solution. Except, our approach is really elementary, straightforward, and unexpectedly accurate. It provides a physical background to the newfound exp-function method and, most importantly, it furnishes an analytical description of front solution in a nonconservative equation for which no other rigorous methods exist.