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Joe Keller and the Courant Institute 1970-71:Beginning a Mathematical Journey

Brian Sleeman

School of Mathematics, University of Leeds,/Division of Mathematics, Universityof Dundee

Mathematics in the Spirit of Joseph Keller, Isaac NewtonInstitute, Cambridge, 2-3 March, 2017

Brian Sleeman Joe Keller and the Courant Institute 1970-71: Beginning a Mathematical Journey

Outline

The Courant Institute 1970-71

The Watson Transformation in Obstacle Scattering

Validity of the Geometrical Theory of Diffraction

Inverse Problems

Nagumo Equation

Nerve Impulse Propagation

Outline

Inverse Problems

Nagumo Equation

Outline

Inverse Problems

Nagumo Equation

Outline

Inverse Problems

Nagumo Equation

Outline

Inverse Problems

Nagumo Equation

Outline

Inverse Problems

Nagumo Equation

Courant Institute

Douglas Samuel Jones photo.jpg

Figure : Douglas Jones

The story starts here!

Courant Institute

Figure : Courant Institute

Courant Institute

Courant.jpg

Figure : Richard Courant

Some Notable Faculty Jerome Berkowitz; Alexander Chorin;Richard Courant; Kurt Friedrichs; Paul Garabedian; James Glimm;Harold Grad;

Courant Institute

Figure : Joe

Warren Hirsch; Lars Hormander; Eugene Isaacson; Fritz John;Joseph Keller:Morris Kline; Heinz-Otto Kreiss; Peter Lax: DonaldLudwig; Wilhelm Magnus; Henry McKean; Cathleen Morawetz;Jurgen Moser; Louis Nirenberg; Jacob Schwartz; James Stoker;Olof Widland; Shmuel Winograd.

Courant Institute

Jurgen Moser; Louis Nirenberg; Jacob Schwartz; James Stoker;Olof Widland; Shmuel Winograd.Joe’s Graduate CourseSpecial Topics in Applied Mathematics: Perturbation Theory andAsymptotics.

G.N.Watson 1918, A.S.Fokas and E. A. Spence 2012, E.A. Spence2014.The acoustic scattering of an analytic incident field ui (r , θ) by asound soft circular cylinder of radius a is solved using separation ofvariables to give the scattered field u(r , θ) representation

u(r , θ) =1

n=∞∑n=−∞

H(1)n (kr)

H(1)n (ka)

e inθU(a,−in), (1)

U(a,−in) = −∫ 2π

0e−inωui (r , ω)dω. (2)

This series can be represented as the combined contour integrals

u(r , θ) =1

(∫ ∞+ic

−∞+ic−∫ ∞−ic−∞−ic

)e iνθU(a,−iν)

1− e−2πiν

H(1)ν (kr)

H(1)ν (ka)

Deform contour to surround simple the zeros of H(1)ν (ka). Rapid

convergence.

Contour.jpg

Figure : Poles and contours of the Watson Transformation, Spence 2014

Obstacle scattering of a simple source by a prolate spheroid, BDS1968, 1969.Here we use Spheroidal coordinates

x = c(ξ2 − 1)1/2(1− η2)1/2 cosφ,

y = c(ξ2 − 1)1/2(1− η2)1/2 sinφ,

z = cξη.

ξ ∈ (1, ∞), η ∈ (−1, 1), φ ∈ (0, 2π).

Using separation of variables we encounter two separationconstants µ, ν in the spheroidal ODEs. Using the Watson idea wearrive at the double contour integral.

G (ξ, η, φ; ξ′, η

′, φ

′) =

∫Γµ

∫Γν

G1(φ, φ′; µ)G2(η, η

′; µ, ν)G3(ξ, ξ

′; µ, ν)dµdν. (4)

G1(φ, φ′; µ) = −cos(µ(π− | φ− φ′ |)

sin(µπ)

and G2, G3 are complicated angular and radial Green’s functionsinvolving spheroidal wave functions.

Figure : Double contours for the spheroid

Keller (1962), R. M. Lewis, N. Bleistein and D. Ludwig (1967), C.A. Bloom and B. J. Matkowsky (1969) BDS (1976).

figure.jpg

Figure : Perturbed ellipse

Theorem (BDS 1976)

As k →∞

U(r; rs; k) = Uellipse(r; rs; k)[1 +O(exp−k

1/3µ)]

uniformly in r, rεS<2 (rs)− R, where µ is positive and independentof k and r, where S<2 (rs) is the ”deep shadow” of Ω2 and R is the”region of influence” of B2.

A 3D analogue of this result is needed and may depend on full highfrequency analysis of scattering by a prolate spheroid.

Inverse problem of Obstacle scattering

Keller (1976): What is the question to which the answer is NineW?D a bounded, simply connected domain of class C 2 with boundary∂D.ν the inward unit normal to ∂D.ui (x , k) is the incident field and u∞(x) is the far field pattern. LetΦ(x , y) be the fundamental solution of the Helmholtz equation.Given knowledge of the far field pattern determine the unknownscattering obstacle D.An early attempt at a solution is due to Imbriale and Mittra(1970),BDS(1982). Here they fix a coordinate origin, calculate thescattered field from u∞(x) and then use least squares to findpoints where the total field vanishes. Choose another origin ofcoordinates and repeat the process until a sufficient number ofsurface points have been obtained to ascertain the shape of D.

At high frequency Keller (1959) used a physical opticsapproximation which, except in special cases, rests on the solutionof Minkowski’s problem. Is an obstacle uniquely determined bysurface Gaussian Curvature?Huge progress since (1970) c.f.Colton Kress (2013).We have the field representations

ui (x , k) = −∫∂D

Φ(x , y)∂u

∂νds(y), x ∈ ∂D, (5)

u∞(x , k) = γ

∫∂D

e−ikx ·y∂u

∂νds(y). (6)

This is the basis of a Newton type method BDS (1982), Johanssonand BDS (2007).Need for a rigorous convergence theory.

Theorem (Schiffer 1967,Lax and Philips (1967))

Assume D1 and D2 are two sound soft scatterers such that the farfield patterns coincide for an infinite number of incident planewaves with distinct directions and one fixed wave number. ThenD1 = D2.

Theorem (Colton and BDS 1983)

Let D1 and D2 be two sound soft scatterers which are contained ina ball of radius R such that kR < π and assume that the far fieldpatterns coincide for one incident plane wave with wave number k .Then D1 = D2.

Problems Does the far field pattern for scattering of one incidentplane wave at one single wave number determine the scatterer?What can be said about sound hard or impedance scatterers?

Nagumo Equation

H. P. McKean (1970)Nagumo’s equation is a starting point for modelling and studyingnerve impulse propagation in unmyelinated nerve axons.

∂t=∂2u

∂x2+ u(1− u)(u − a).

Travelling wavesu(x , t) = φ(x + ct),

Unique stable heteroclinic orbit in which , with ξ = x + ct,

φ(ξ) =1

1 + exp(−ξ/√

2), c =

2− a)

McKean gives full phase plane analysis. model does not accountfor recovery to a resting state.

Recovery is achieved with the FitzHugh-Nagumo Model

∂x2+ u(1− u)(u − a)− w ,

∂t= bu − γw .

Look for travelling waves u(x , t) = φ(x + ct) andw(x , t) = ψ(x + ct)Need 3D phase space analysis

Rinzel and Keller (1973) simplified matters by modelling the cubicin u by

F (u) = −u + H(u − a), H -Heaviside step function

Later BDS (see Jones, Plank and BDS (2010)) treated the moregeneral form

F (u) = −u, u ≤ a/2,

= u − a, a/2 < u ≤ (1 + a)/2,

= 1− u, (1 + a)/2 ≤ u.

Open Problems: evolutionary structure of the F-N PDE; Thresholdbehaviour; more on stability of travelling waves. What aboutmylinated fibres (essential to mammals)?

Thank you Joe for so much joy and inspiration.Brian Sleeman Joe Keller and the Courant Institute 1970-71: Beginning a Mathematical Journey

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