In this paper we consider relativistic models that contain in their spectra of solutions extended topological defects. We find the geometrical constrains that describe deformed vortices and domain walls of constant width. Analytical form of these solutions in co-moving coordinates is identical with analytical form of the appropriate static solutions in the laboratory Cartesian coordinates. The geometrical constrains presented here describe fully the shape and the evolution of the vortices and domain walls of constant width.

The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The momentum map from the action-angle scattering variables $T^*({\mathbb{T}^N})$ to the flow momentum provides the Eulerian representation of the $N$-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. The peakon solutions of the dispersionless CH equation in one dimension are shown to generalize in higher dimensions to peakon wave-front solutions of the EPDiff equation whose associated momentum is supported on smoothly embedded subspaces. The Eulerian representations of the singular solutions of both CH and EPDiff are given by the (cotangent-lift) momentum maps arising from the left action of the diffeomorphisms on smoothly embedded subspaces.

The stability properties of the Einstein Static solution of General Relativity is altered when corrective terms arising from modifications of the underlying gravitational theory appear in the cosmological equations. Employing dynamical system techniques and numerical integrations, we discuss the stability of static cosmological solutions in the framework of two recently proposed quantum gravity models, namely Loop Quantum Cosmology and Horava-Lifshitz gravity.

In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very “ascetic” in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.