The application is to the von Karaman Donnell equations, a coupled system of elliptic equations describing the equilibrium of an axially compressed cylindrical shell.
Earlier work suggests that axially-localised solutions are the physically prefered buckling modes.
Hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the axial variable and periodic circumferentially. Methods for location and parameter continuation of homoclinic solutions of ODEs are then adapted, making special use of the symmetry and reversibility properties of solutions observed experimentally. The method for location of solutions makes use of asymptotic approximations where available. The techniques illustrated here for the von Karaman Donnell equations are applicable to a wider class of PDEs.