Global attraction to solitary waves for Klein-Gordon equation with mean field interaction

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension one or larger, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges (as time goes to infinity) to the two-dimensional set of all ``nonlinear eigenfunctions'' of the form \phi(x)e\sp{-i\omega t}. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation

On linear stability of crystals in the Schroedinger-Poisson model

We consider the Schr\"odinger--Poisson--Newton equations for crystals with a cubic lattice and one ion per cell. We linearize this dynamics at the ground state and introduce a novel class of the ion charge densities which provide the stability of the linearized dynamics. This is the first result on linear stability for crystals. Our key result is the {\it energy positivity} for the Bloch generators of the linearized dynamics under a Wiener-type condition on the ion charge density. We also assume an additional condition which cancels the negative contribution caused by electrostatic instability. The proof of the energy positivity relies on a special factorization of the corresponding Hamilton functional. We show that the energy positivity can fail if the additional condition breaks down while the Wiener condition holds. The Bloch generators are nonselfadjoint (and even nonsymmetric) Hamilton operators. We diagonalize these generators using our theory of spectral resolution of the Hamilton ope\-rators {\it with positive definite energy} \ci{KK2014a,KK2014b}. Using this spectral resolution, we establish the stability of the linearized crystal dynamics.Comment: 23 page