Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation
Stuart, C.
In: Milan Journal of Mathematics, 2008, vol. 76, no. 1, p. 329-399
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- Abstract.: In the first part of these notes, we deal with first order Hamiltonian systems in the form $$Ju\prime(t) = \bigtriangledown H(u(t))$$ where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian $$H \,\epsilon\, C^{1}(X,{\mathbb{R}})$$ is required to be invariant with respect to the action of a group $$\{e^{tA} : t \,\epsilon\, {\mathbb{R}}\}$$ of isometries where $$A \in B (X,X)$$ is skew-symmetric and JA =AJ. A standing wave is a solution having the form $$u(t) = e^{t\lambda A}\varphi$$ for some $$\lambda \,\epsilon\, {\mathbb{R}}$$ and $$\varphi \in X$$ such that $$\lambda JA_{\varphi} = \bigtriangledown H(\varphi)$$ . 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. In the second part we consider the nonlinear Schrödinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability