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Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Asymptotic behaviour of the heat equation intwisted waveguides
Gabriela Malenova
Faculty of Nuclear Sciences and Physical Engineering, CTU, PragueNuclear Physics Institute, AS CR, Rez
Graphs and Spectra, Chemnitz,July 19, 2011
Jointly with: David Krejcirık and Milos Tater, NPI, AS CR, Rez.
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Introduction
• Straight waveguide Ω0 := R× ω, ω ∈ R2 is non-circularcross-section
• Twisted waveguide Ωθ, ω is rotating with respect tonon-constant angle θ(x1).
The Hamiltonian of a particle moving inside is descibed by theDirichlet Laplacian −∆Ωθ
D : L2(Ωθ)→ L2(Ωθ).It is associated with the quadratic form ψ 7→ ‖∇ψ‖2 with thedomain D(Ωθ) := H1
0 (Ωθ).
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Spectral stability
−∆ΩθD and −∆Ω0
D have the same spectrum as a set:
σ(−∆ΩθD ) = σess(−∆Ωθ
D ) = [E1,∞).
• E1 is the threshold energy of −∆ωD
• Difference: existence of the Hardy inequality [Ekholm,Kovarık, Krejcirık 2008]
Hardy-type inequality
−∆ΩθD − E1 ≥ ρ, ρ is a positive function.
• Operator −∆ΩθD − E1 is subcritical, −∆Ω0
D − E1 is critical
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Spectral stability
−∆ΩθD and −∆Ω0
D have the same spectrum as a set:
σ(−∆ΩθD ) = σess(−∆Ωθ
D ) = [E1,∞).
• E1 is the threshold energy of −∆ωD
• Difference: existence of the Hardy inequality [Ekholm,Kovarık, Krejcirık 2008]
Hardy-type inequality
−∆ΩθD − E1 ≥ ρ, ρ is a positive function.
• Operator −∆ΩθD − E1 is subcritical, −∆Ω0
D − E1 is critical
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Heat equation on Ωθ
ut(x , t)−∆u(x , t) = 0
subject to Dirichlet boundary conditions on ∂Ωθ.
From the semigroup theory: ∀u0 ∈ L2(Ωθ) exists uniquelydetermined generalized solution of the heat equation in the form
u(x , t) = e∆ΩθD tu0(x),
where e∆ΩθD t : L2(Ωθ)→ L2(Ωθ) is the semigroup operator
associated with the Laplacian −∆ΩθD .
It follows from the spectral mapping theorem that
‖e∆ΩθD t‖L2(Ωθ)→L2(Ωθ) = e−E1t .
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Decay rate
We are interested in an additional (polynomial) decay of thesemigroup, which will follow if we restrict the class of initial datato the weighted space L2(Ωθ,K ), where K (x) := ex2
1/4. Let usdefine the decay rate:
γ(Ωθ) := supγ∣∣ ∃Cγ > 0, ∀t ≥ 0,
∥∥e(∆θ+E1)t∥∥
K≤ Cγ(1 + t)−γ
,
where ‖.‖K : L2(Ωθ,K )→ L2(Ωθ).
Theorem
We have γ(Ωθ) = 1/4 if Ωθ is untwisted, while γ(Ωθ) = 3/4 if Ωθ
is twisted.
[D. Krejcirık and E. Zuazua, 2011]
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Self-similarity transformation
Motivation: The Dirichlet Laplacian converges in the normresolvent sense to one dimensional Schrodinger operator whosepotential holds the information about twisting [Sedivakova,Krejcirık, 2011].One-dimensional heat equation:
ut − uxx + V (x)u = 0, with V = Cω θ2.
Self-similarity transformation
u(x , t) = (t + 1)−1/4w(y , s),
y := (t + 1)−1/2x , s := ln (t + 1).
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Using self-similarity transformation we arrive at:
ws −1
2ywy −
1
4w − wyy + esV (es/2y)w = 0.
Finally, obeyingz(y , s) := ey2/8w(y , s),
we get expression in (y , s) ∈ Ω0 × (0,∞):
zs − zyy + y2
16z + esV (es/2y)z = 0
This is a parabolic equation with time-dependent coefficients.However, this form is advantageous because of the compactness ofthe resolvent. Self-similarity transformation is unitary:‖u(t)‖ = ‖w(s)‖. According to theorem, we expect
V = 0, ‖u(t)‖ ∼ t−1/4, V ≥ 0, ‖u(t)‖ ∼ t−3/4.
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Exact solution for V = 0
Semigroup operator of the heat equation in an integral operator
u(x , t) =
∫R
G (x , y , t)u0(y)dy ,
where the heat kernel is defined as
G (x , y , t) =e−|x−y|2
4t
√4πt
.
We are able to find numerical solution (Wolfram Mathematica 7.0).
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Expansion to oscillator basis
The first method uses the expansion to the eigenbasis of theharmonic oscillator:
u(x , t) =∞∑
n=1
an(t)ψn(x).
ψ satisfies the Helmoltz equality −ψ′′ + x2
16ψ = λψ, we knowexplicitly the eigenvalues and -vectors from quantum mechanics:
λn =1
2
(n +
1
2
), ψn(x) = NnHn
(x
2
)e−
x2
8 .
Plugging into the heat equation:
a(t) = e−Mta(0), where Mmn = λnδmn −⟨ψm,
x2
16ψn
⟩.
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Self-similarity solution
The solution of self-similarity-transformed equation
zs − zyy + y2
16z = 0 may be found again as expansion to harmonicoscillator eigenbasis. Then
a(t) = e−Mta(0), where Mmn = λnδmn.
Finally, we apply backward self-similarity transformation.The decay rate function is defined
q(t) := − ln ‖u(t)‖ln (1 + t)
.
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Comparison
Time evolution of the norm ‖u‖:
Decay rate q(t):
20 40 60 80 100 120
-0.2
0.2
0.4
0.6
0.8
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Layout
Heat equation in twisted waveguideQuantum waveguidesHardy inequalitiesHeat equation on Ωθ
Asymptotic behaviourDecay rateSelf-similarity transformation
Numerical solutionNumerical methodsComparisonThe twisted case
Conclusion
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
The twisted waveguide
The potential increases the decay rate about 1/2. Analytically, themathematical background lies in scalling which is more singularthan the Dirac’s delta interaction and thus leads to the Dirichletcondition at the origin.This problem is already explicitely solvable: the first eigenvalue is3/4, which coincides with the second eigenvalue of the harmonicoscillator without Dirichlet condition. Decay rate q(t):
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
Conclusion
• Aim: support the data given by Krejcirık and Zuazua; showthat the decay rate posseses increase about 1/2 in the twistedwaveguide in comparison to the untwisted case.
• Possible extensions: Computation for non-approximated 3Dwaveguides.
Heat equation in twisted waveguide Asymptotic behaviour Numerical solution Conclusion
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