Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids

Abstract

Abstract In this article, the prolongation structure technique is applied to a generalised inhomogeneous Gardner equation, which can be used to describe certain physical situations, such as the stratified shear flows in ocean and atmosphere, ion acoustic waves in plasmas with a negative ion, interfacial solitary waves over slowly varying topographies, and wave motion in a non-linear elastic structural element with large deflection. The Lax pairs, which are derived via the prolongation structure, are more general than the Lax pairs published before. Under the Painlevé conditions, the linear-damping coefficient equals to zero, the quadratic non-linear coefficient is proportional to the dispersive coefficient c(t), the cubic non-linear coefficient is proportional to c(t), leaving no constraints on c(t) and the dissipative coefficient d(t). We establish the prolongation structure through constructing the exterior differential system. We introduce two methods to obtain the Lax pairs: (a) based on the prolongation structure, the Lax pairs are obtained, and (b) via the Lie algebra, we can derive the Pfaffian forms and Lax pairs when certain parameters are chosen. We set d(t) as a constant to discuss the influence of c(t) on the Pfaffian forms and Lax pairs, and to discuss the influence of d(t) on the Pfaffian forms and Lax pairs, we set c(t) as another constant. Then, we get different prolongation structure, Pfaffian forms and Lax pairs.