Bifurcation of Relative Equilibria in Infinite-Dimensional Hamiltonian Systems

Bifurcations of relative equilibria in perturbed infinite-dimensional Hamiltonian systems are studied. We assume that the unperturbed system has several symmetries and a family of relative equilibria and that the perturbation breaks some of the symmetries. Using the Lyapunov--Schmidt method, we detect saddle-node and pitchfork bifurcations of relative equilibria along with their linear stability. Our theory is illustrated for solitary waves of the nonlinear Schrodinger equation with a small potential and the theoretical results are demonstrated with the numerical computations.