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Wavelet-based Integral Representation of Solutionsof Wave and Klein-Gordon Equations
Maria V. Perel, Mikhail S. Sidorenko
St.Petersburg University, Russia,”Radioavionika” Corp., St Petersburg, Russia
New Trends and Directions in Harmonic Analysis,Approximation Theory, and Image Analysis
Inzell, Germany, 2007
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Wave Equation
The aim of this work is to obtain integral representations ofsolutions of wave equation [W]:
∂2u
∂t2−c2
(∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)= 0, u = u(r, t), r = (x, y, z).
and Klein-Gordon equation [KG]
∂2u
∂t2− c2
(∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)+m2 u = 0, u = u(r, t;m).
with constant coefficient c and m as the superposition of theirlocalized solutions from the wide class. The results are obtained bymethods of continuous wavelet analysis.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Most Widely Known Integral Representations
The Fourier analysis gives an exact representation as thesuperposition of plane waves.Disadvantage: Plane waves are not localized
There exists an approximate representation for wave equationas the sum of the Gaussian beams [Babich, Popov 1989][Heyman, Felsen 2001]Disadvantage: The representation is not exact. Gaussianbeams are localized only along the axis
Exact representation for the wave equation based on theAnalytic Signal Transform proposed by G. Kaiser [Kaiser 1994]Disadvantage: The very special fixed form of the sphericallysymmetric elementary solution
Our aim - exact representation of solution as thesuperposition of arbitrary localized solutions
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Integral Representation
We consider here the initial value problem for wave equation andfor Klein-Gordon equation. We seek the solution of each problemin the form
u(r, t) =∫dµ(ν)U(ν)ϕν(r, t)
where
ν denotes the set of the parameters
ϕν(r, t) denotes the family of elementary solutions
U(ν) are the coefficients
Obtained integral representations express the solutions of eachproblem as the superposition of localized solutions.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The motivation. Gaussian Wave Packet [Kiselev,Perel,2000]
The work was originally motivated by the invention of particle-likeexact solutions of the wave equation given by the explicit formula
ψ(r, t) =exp (−p
√1− iθ/γ)√
x+ ct− iε, θ = x− ct+
y2
x+ ct− iε,
where p,ε, γ are positive parameters.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Wavelet properties of the new solution [Perel, Sidorenko,2007]
It was noticed that this solution can be taken as a mother waveletfor continuous wavelet analysis if time is a parameter.The Fourier transform of the solution has been calculated exactly
ψ(k, t) =C exp
[−(k + kx)γ
2 − (k − kx) ε2 −
p2
2γ(k+kx) − ikct]
k(k + kx),
where k = |k|, C = 2π eiπ/4 pγ1/2 . This formula shows that
ψ(k)|k=0 = 0 and the wavelet has all zero moments also.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Space of Solutions
[W] We fix the space H of the solutions of the wave equation with
the property
∫R3
d3r |u(r, t)|2 <∞.
[KG] We fix the space Km of the solutions of Klein-Gordon
equation with the property
∫R3
d3r |u(r, t;m)|2 <∞.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Space of the Solutions
Each solution has the Fourier transform of the following form:
[W] u(k, t) = u+(k, 0) exp(i|k|ct) + u−(k, 0) exp(−i|k|ct)
[KG] u(k, t;m) = u+(k, 0) exp(it
√c2|k|2 +m2
)+u−(k, 0) exp
(−it
√c2|k|2 +m2
)We split the whole space of solutions into the direct sum ofpositive-frequency and negative-frequency subspaces
[W] H = H+ ⊕H−, u(r, t) = u+(r, t) + u−(r, t)
[KG] Km = Km+ ⊕Km
− , u(r, t;m) = u+(r, t;m) + u−(r, t;m)
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Family of Wavelets: Admissibility
[W] We choose an arbitrary solution ϕ+ ∈ H+ and ϕ− ∈ H− sothat ϕ+(r, 0) = ϕ−(r, 0) ≡ ϕ(r, 0) The admissibility conditionmust be satisfied
Cϕ ≡∫R3
d3k|ϕ(k, 0)|2
|k|3< ∞.
[KG] We choose an arbitrary solution ϕ+ ∈ Km+ and ϕ− ∈ Km
− sothat ϕ+(r, 0;m) = ϕ−(r, 0;m) ≡ ϕ(r, 0;m). Its Fouriertransform ϕ(k, 0) does not depend on m.The same admissibility condition must be satisfied.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Family of Wavelets: Initial guess
We define the family of wavelets ϕν by applying translations,rotations and dilations to the argument r in the moment of timet = 0:
ϕν+(r, 0) ≡ 1
a3/2ϕ+
(M−1
ϑ1,ϑ2,ϑ3
r − b
a, 0
),
ν = (a, b, ϑ1, ϑ2, ϑ3), a ∈ (0,∞), b ∈ R3,
and M−1ϑ1,ϑ2,ϑ3
is a rotation matrix parametrized by three Eulerangles ϑ1, ϑ2, ϑ3.
If a 6= 1, the function ϕ+
(M−1
ϑ1,ϑ2,ϑ3
r−ba , t
)is no longer a
solution of the wave or the Klein-Gordon equations.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Family of Wavelets
To overcome this problem we look at the Fourier transform:
[W] ϕν+(k, t) = ϕν(k, 0) exp(ia|k|ct),
[KG] ϕν+(k, t;m) = ϕν(k, 0) exp
(it
√c2a2|k|2 +m2
).
We modify the exponents in the following way:
[W] exp(ia|k|c t
a
),
[KG] exp(it
a
√c2a2|k|2 + a2m2
).
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Family of Wavelets: Final formula
The final formula for the family of wavelets reads[W]:
ϕν+(r, t) ≡ 1
a3/2ϕ+
(M−1
ϑ1,ϑ2,ϑ3
r − b
a,t
a
).
[KG]:
ϕν+(r, t;m) ≡ 1
a3/2ϕ+
(M−1
ϑ1,ϑ2,ϑ3
r − b
a,t
a; ma
).
For each set ν wavelets belong to corresponding spaces ofsolutions H+ and Km
+ respectively.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Decomposition Coefficients
For both cases we propose to put the coefficients U+(ν) equal tothe wavelet transform of u+(r, 0) with the mother waveletϕ+(r, 0):
U+(ν) ≡∫R3
d3r u+(r, 0)ϕν+(r, 0) =
∫R3
d3r u+(r, t)ϕν+(r, t).
The coefficients U+(ν) do not depend on time t.Isometry property is valid for both equations:∫R3
d3r|u±(r, t)|2 =1Cϕ
∫dµ(ν) |U±(ν)|2, ∀u± ∈ H±,Km
± ∀t
∫dµ(ν) ≡
2π∫0
dϑ1
π∫0
dϑ2 sinϑ2
2π∫0
dϑ3
∞∫0
da
a4
∫R3
d3b
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
The Integral Representation Formula for Both Equations
The integral representation formula:
u(r, t) =1C+
∫dµ(ν)
(U+(ν)ϕν
+(r, t) + U−(ν)ϕν−(r, t)
),
∫dµ(ν) ≡
2π∫0
dϑ1
π∫0
dϑ2 sinϑ2
2π∫0
dϑ3
∞∫0
da
a4
∫R3
d3b,
Cϕ ≡∫R3
d3k|ϕ(k, 0)|2
|k|3.
Here U+(ν) and U−(ν) are wavelet transforms of u+ and u−respectively.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Coefficients U+(ν), U−(ν)
We express the coefficients U+(ν), U−(ν) in terms of the initialdata
u|t=0 = w(r),∂u
∂t
∣∣∣∣t=0
= v(r), w, v ∈ L2(R3)
of the initial-value problem[W] for wave equation:
∂2u
∂t2− c2
(∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)= 0,
[KG] for Klein-Gordon equation:
∂2u
∂t2− c2
(∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)+m2 u = 0.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Coefficients U+(ν), U−(ν)
u(r, t) =1Cϕ
∫dµ(ν)
(U+(ν)ϕν
+(r, t) + U−(ν)ϕν−(r, t)
),
The coefficients U± are obtained from the formula
U+(ν) =12(W (ν) + aV (ν)), U−(ν) =
12(W (ν)− aV (ν)),
where W (ν) is the wavelet transform of w(r) with the motherwavelet ϕ(r, 0), and V (ν) - of v(r) with ψ(r, 0), where
ϕ(r, 0) =∂
∂tψ(r, t)
∣∣∣∣t=0
, ϕ(k, 0) = i|k| ψ(k, 0).
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Solution of initial value problem
Let us show that the expression for u is valid. We fix t = 0:
12Cϕ
∫dµ(ν) ([W (ν) + aV (ν)]ϕν(r, 0)
+[W (ν)− aV (ν)]ϕν(r, 0)) = w(r) = u(r, 0).
Now we consider the time derivative at the moment t = 0:
12Cϕ
∫dµ(ν)
([W (ν) + aV (ν)]
∂
∂tϕν
+(r, t)∣∣∣∣t=0
+[W (ν)− aV (ν)]∂
∂tϕν−(r, t)
∣∣∣∣t=0
)= v(r) =
∂u
∂t
∣∣∣∣t=0
.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
References
Kaiser G 1994 A Friendly Guide to Wavelets, Boston,Birkhauser
Babich, Popov 1989 Gaussian beams summation method(review),translated in Radiophysics and Quantum Electronics 32,1063-1081.
Popov 1982, A new method of computation of wave fields usingGaussian beams, Wave Motion, 4, 85-97.
Steinberg, Heyman and Felsen, 1981, Phase space beamsummation for time dependent radiation from large apertures:Continuous parametrization, J. Opt. Soc. Am., 8, 943-958.
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Kiselev A P and Perel M V 2000, Highly localized solutions of thewave equation J. Math. Phys. 41(4) 1934–55
Perel M.V. and Sidorenko M.S., 2007, New physical wavelet’Gaussian wave packet’, J. Phys. A: Math.and Theor. 40, pp3441-3461, http://stacks.iop.org/1751-8121/40/3441
Perel M.V. and Sidorenko M.S., 2003, Wavelet Analysis in Solvingthe Cauchy Problem for the Wave Equation in Three-DimensionalSpace In:Waves 2003, Ed G C Cohen, E Heikkola, P Jolly and PNeittaanmaki (Springer-Verlag) pp 794-798
Perel M.V. and Sidorenko M.S., 2006, Wavelet analysis for thesolution of the wave equation, In: Proc. of the Int. Conf. DAYSon DIFFRACTION 2006, Ed. I V Andronov (SPbU), pp 208-217
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation
Thank you for the attention !
Maria V. Perel, Mikhail S. Sidorenko Wavelet-based Integral Representation