Construction of infinite-bump solutions for non-linear-Schrödinger-like equations

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• Prof. Robert Magnus (Reykjavik/DPMMS)
• Monday 22 October 2012, 15:00-16:00
• CMS, MR11.

If you have a question about this talk, please contact Prof. Mihalis Dafermos.

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.

This talk is part of the Geometric Analysis and Partial Differential Equations seminar series.

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