Instability of bound states for abstract nonlinear Schrödinger equations
The author studies abstract Hamiltonian systems of the form \[ u'(t) = \tilde{J}E'(u(t)), \] where \(E\) is the energy functional on a real Hilbert space \(X\), \(J\) is a skew symmetric operator on \(X\), and \(\tilde{J}\) is a natural extension of \(J\) to the dual space \(X^*\). One assumes that the above equation is invariant under a one parameter group \(\{T(s); \, s\in \mathbb{R}\}\) of unitary operators on \(X\). The instability of the bound states \(T(\omega t)\varphi_{\omega}\) is studied, where \(\omega \in \mathbb{R}\) and \(\varphi_{\omega}\) is a solution of the corresponding stationary problem. Applications to nonlinear Schrödinger equations are given.
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