An elementary construction of complex patterns in nonlinear schrödinger equations

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε²u″ + V(x)u = |u|^p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x₁ < X₂ < ⋯ < x_m constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n _i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each x_i as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.