Prolongation structures of a higher-order nonlinear Schrodinger equation

Abstract

A higher-order Schrodinger equation containing parameters, which is used to describe pulse propagation in optical fibres, is shown to admit an infinite-dimensional prolongation structure for exactly four combinations of the parameters, besides the classical NLS equation. For each of these cases, the structure of the resulting prolongation algebra is determined explicitly. For the first three cases the prolongation algebra is essentially a sub-algebra of A1(1), the fourth case turns out to be sub-algebra of the twisted Kac-Moody algebra A2(2). Using vector-field representations, related systems of differential equations for the (pseudo-) potential functions are given for each of the cases. The cases found here correspond exactly to the derived NLS equations I and II, the Hirota equation and the equation recently considered by Sasa and Satsuma (1991). The result of this paper strongly indicates that the considered higher-order NLS equation is completely integrable for precisely these four cases.