On the existence of full dimensional KAM torus for nonlinear Schrödinger equation (Q2272761)

Consider the one-dimensional nonlinear Schrödinger equation with periodic boundary conditions, \[ \sqrt{-1} u_t - u_{xx} + V * u + \varepsilon f(x) |u|^4u=0, \quad x \in \mathbb R / 2 \pi \mathbb Z \] where \(V *\) is the Fourier multiplier defined by \(\widehat{(V * u)}_n = V_n \widehat u_n\), \(V_n \in [-1,1]\), and \(f(x)\) is Gevrey smooth. The authors show that for small enough \(\varepsilon\geq 0\), there exists a sequence \((V_n)_{n\in \mathbb Z}\) such that this equation admits a full-dimensional invariant KAM torus. Their main difficulty is that the zero momentum of the equation is absent.