Abstract

In 1986 Floer and Weinstein gave a rigorous proof that a solution of the stationary, one-dimensional, cubic, non-linear Schrödinger equation,

−ε² u′′ + V(x)u − u³ = 0, −∞ < x < ∞

exists that concentrates at a non-degenerate critical point of V as ε → 0. This means that the solution differs substantially from 0 only in a small neighbourhood of the critical point. Y-G. Oh then constructed solutions that concentrate simultaneously at a finite set of non-degenerate critical points, the so-called multibump solutions. This work led to much interest in the concentration properties of solutions of this, and more general equations, in R ⁿ and with more general non-linearities. These have the form

−ε² ∆u + F (V (x), u) = 0, x ∈ R ⁿ.

In this talk I shall discuss the construction of solutions that concentrate simultaneously at an infinite set. I shall try to explain the key ideas, derived from PDE theory and functional analysis, used in the proofs.