Lie symmetries of a generalised non-linear Schrodinger equation. II. Exact solutions

Abstract

For pt.I see ibid., vol.21, p.1493 (1988). The authors obtain group-invariant solutions of the non-linear equation i psi t+ Delta psi =a0 psi +a1 psi mod psi mod 2+a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended Galilei group for a1a2 not=0 and the extended Galilei-similitude group for a1=0 or a2=0. They use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved, whenever possible, with the help of a singularity analysis. Solutions are obtained in terms of elementary functions, Jacobi elliptic functions and Painleve transcendents.