Resonant solutions of the non-linear Schrodinger equation with periodic potential

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrodinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k(2). It is shown that the isoenergetic surface corresponding to k(2) is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.