Gauge symmetry and the generalization of hirota’s bilinear method

Abstract

One of the most powerful methods for finding and solving integrable nonlinear partial differential equations is Hirota’s bilinear method. The idea behind it is to make first a nonlinear change in the dependent variables after which multisoliton solutions of integrable systems can be expressed as polynomials of exponentials eηi where the (Formula presented.) are linear in the independent variables. Among all quadratic expressions homogeneous in the derivatives, Hirota’s bilinear form can be isolated by a gauge symmetry: it is the only one that is invariant under f α → e θ f α where θ is linear in the variables. This suggest a generalization to multilinear equations using the same gauge symmetry. The set of gauge invariant multilinear differential equations can then be studied and integrable equations identified e.g. by the Painlevé method. Some interesting new equations have been found in this way. © 1996 Taylor & Francis Group, LLC.