Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrodinger Equation

In the first part of these notes, we deal with first order Hamiltonian systems in the form Ju'(t) = del H(u(t)) where the phase space X may be in infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian H is an element of C-1( X, R) is required to be invariant with respect to the action of a group {e(tA) : t is an element of R} of isometries where A is an element of B(X, X) is skew-symmetric and JA = AJ. A standing wave is a solution having the form u(t) = e(t lambda A)phi for some lambda is an element of R and phi is an element of X such that lambda JA phi = Delta H(phi). Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves.