Wavelet-based Integral Representation of Solutions of Wave and … · 2010-04-28 · The Integral Representation We consider here the initial value problem for wave equation and for

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.


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


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.


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.


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.


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 <∞.


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)


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.


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.


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

).


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.


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


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.


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.


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).


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

.


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.


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


Thank you for the attention !


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Wavelet-based Integral Representation of Solutions of Wave and … · 2010-04-28 · The Integral Representation We consider here the initial value problem for wave equation and for