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AARMS Course: Homework 4
Problem 1:
Consider the coupled RD system on −1 < x < 1 for u(x, t) and w(x, t), given by
ut = uxx + f (u) − 2δb(x)w ,
τ wt = Dwxx − w + c(x)u, ,
with wx = ux = 0 at x ± 1. Assume that f (u) is smooth with f (0) = 0 and f ′ (0) = 1. In our RD system, δ > 0 is a
coupling constant measuring the coupling of u and w.
(1) For the equilibrium state (ue , we ) = (0, 0) introduce the perturbation u = ue + eλt Φ(x) and w = we + eλt η(x).
For the limiting problem D → ∞, referred to as the shadow limit, show that Φ(x) is an eigenfunction of the
nonlocal eigenvalue problem
Φ′′ −
δb(x)
1 + τλ
Z
1
c(x)Φ(x) dx = (λ − 1)Φ(x) ,
|x| ≤ 1 ;
Φ′ (±1) = 0
(0.1)
−1
(2) Let τ = 0 in (0.1). Prove that the eigenvalues λ of (0.1) must be real when b(x) = kc(x) for some constant k
independent of x. (Hint: just form the usual Lagrange identity from Sturm-Liouville (SL) theory and a simple
adaptation of the proof that the eigenvalues of a SL operator are real.)
(3) For τ = 0 and δ = 0 show that the eigenfunctions of the now Sturm-Liouville problem (0.1) can be written in
terms of an even and odd eigenfunctions as
φ̂2m = cos(πmx) ,
φ̂2m+1 = sin (π(2m + 1)πx/2) ,
(0.2)
for m = 0, 1, 2, ....
(4) Suppose that τ = 0 and
b(x) = b0 + b1 cos(πx) + b2 cos(2πx) ,
c(x) = c0 + c1 cos(πx) + c2 cos(2πx) ,
(0.3)
for some given coefficients bj , cj for j = 0, 1, 2. Look for an eigenfunction for (0.1) in the form
Φ(x) = s0 + s1 cos(πx) + s2 cos(2πx) .
(0.4)
Derive a 3 × 3 matrix eigenvalue problem of the form (Λ − δD)s = σs, where s = (s0 , s1 , s2 )T and σ ≡ λ − 1.
Here Λ is a diagonal matrix, while D is a 3 × 3 rank-one matrix determined in terms of bj , cj , for j = 0, 1, 2.
For b(x) and c(x) of the form (0.3), what happens to all the other eigenfunctions φ̂2m and φ̂2m+1 of the Sturm
Lioville problem as δ is increased from zero?
(5) Consider the eigenvalues σ of your matrix system in the previous part as a function of δ on the range 0 < δ < 5
for the specific choice
b0 = 2.5,
b1 = 1.0 ,
b2 = 1.0 ,
c0 = 1.1 ,
c1 = −0.5 ,
c2 = 0.5 .
(0.5)
I would like two plots: Plot 1: Re(σ) versus δ. Plot 2: Im(σ) versus Re(σ) (with δ parameterizing the curve in
this plane). What can you say about the stability of (ue , we ) = (0, 0) for τ = 0 as δ is varied?
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(6) Finally, take b(x) = c(x) = 1 for all x and let τ > 0. Consider for δ ≥ 0,
Z 1
δ
Φ′′ −
Φ(x) dx = (λ − 1)Φ(x) , |x| ≤ 1 ;
1 + τ λ −1
Φ′ (±1) = 0
(0.6)
Find the range of τ and δ for which Re(λ) < 0 for any eigenvalue of (0.6). Can we get a Hopf bifurcation?
Problem 2: let q > 1, and suppose that w(y) is the homoclinic of
w′′ − w + w2q−1 = 0 .
(0.7)
L0 Φ = Φ′′ − Φ + (2q − 1)w2q−2 Φ .
(0.8)
Consider the linearized operator
(1) Verify that L0 (wq ) = (q 2 − 1)wq .
(2) Now let w be the homoclinic of w′′ − w + w5 = 0. Show that any unstable eigenvalue of the NLEP
R∞ 3
w Φ dy
′′
4
3 −∞
Φ − Φ + 5w Φ − χ(λ)w R ∞ 2
= λΦ
w dy
−∞
(0.9)
with Φ → 0 as |y| → ∞, muust satisfy the simple transcendental equation
χ(λ) =
2
(8 − λ) .
3
(0.10)
(Hint: use the identity in the first part with an appropriate q, combined with Lagrange’s identity for w3 L0 Φ −
ΦL0 w3 , and then finally combined with some integral identity given earlier in the notes.)