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AMS/IP Studies in Advanced MathematicsVolume 13, 1999

NONLINEARITY AND SELF-SIMILARITY: WAVELETS ANDCOMPACTONS ON A PHYSICAL BACKGROUND

A. LUDU, J. P. DRAAYERDepartment of Physics and Astronomy, Louisiana State University,

Baton Rouge, LA 70803-4001U.S.A.

E-mail: ludu@epscor.phys.lsu.edu

AbstractWe propose a generalized Lagrangian capable of describing in an uni­fying mode, different nonlinear dynamical systems and wavelets; forexample Korteweg-de Vries (KdV) solitons, K(2,2) compactons andMorlet continuous wavelets. We give a procedure to associate anydiscrete wavelet, that is any multi-scale finite-difference equation acorresponding Lagrangian and algebraic structure. We focus uponintroduction of a nonlinear wavelet-like basis made by localized ana­lytical nonlinear solutions. We introduce a method, based on Morletwavelet, to provide a relation between the amplitude, width and ve­locity of the traveling solutions of certain nonlinear equations.

1 Introduction

Nonlinear science, though relatively new and far less understood, is an importantfrontier for probing fundamentals of Nature. Intense research has developed world­wide in the mathematics and physics of nonlinear dynamical systems, occuring indifferent fields, from fluid dynamics through string theory. Among other, nonlinearphysics presents a variety of patterns and particle-like traveling waves 1. Notableexamples include dynamics of solitons or breathers 2, nonlinear molecular and solidstate physics, nonlinear optics, nonlinear atomic physics and laser beams and non­linear nuclear and particle physics 3.

The construction of analytical, localized and finite support solutions for anygiven NPDE is still an open question, especially because of this superposition. Re­cently, contra-examples were found, where the traditional nonlinear tools (inversescattering, group symmetry, functional transform) are inapplicable, 4,5,6. Also, ob­served patterns in Nature, either stationary, growing or propagating, are generaly offinite space-time extension. But all conventional soliton solutions though localizedare of infinite extent. These theoretical gaps may be filled by finding an adequatenonlinear basis. Such a nonlinear basis, made of compact supported functions, ul-

© 1999 American Mathematical Societyand International Press

387

A. LUDU and J. P. DRAAYER

timately orthonormalized, could diagonalize nonlinear operators, or could be usefulin perturbation and quantization procedures, or in any physical nonlinear modelinvolving patterns. One good support for this challenge is given by the multires­olution analysis 7. Fourier or other harmonic analysis are inefficient for nonlinearsystems. Wavelets are self-similar bases analysing signals at different scales,7. Founding their first applications in signal processing, pattern recognition andfractal theory, 8, wavelets have now a strong impact in turbulence and propagationof singularities, singular potentials in quantum mechanics, anyons and in statisti­cal physics/coherent states, 9 or in the theory of q-algebras and nuclear physics, 10.

Wavelets provide new numerical possibilities for analysing of NPDE involving manyscales or self-similarity.

In this contribution we have constructed an unifying Lagrangian which yieldsa general NPDE having both compact supported solution, (and in the limit KdV,MKdV solitons) and continuous wavelet solutions. This basis can be used to investi­gate other NPDE, as we provide some examples. We have proved that the dilationequations for discrete wavelets are related with infinite-dimensional Hamiltoniansystems, too.

2 Structure of wavelets

There are two different decomposition principles into series of standard functionsand hence, two categories, the Taylor and Fourier series. Following from the twobasic commutative real operations, addition and multiplication. The Taylor (orany orthogonal polynomial) series is based on polynomials and the decompositioncriterion is transiation. Translation spreads the local properties of a signal outalong the whole axis. The dilation does not affecting this structure. The smootherthe function the farther the translation. A Fourier series is tuned to dilation, withfunctions decomposed according to different scales. Basis elements are dilated copiesof the same unique mother function ( for example sine) and its symmetrized image(cosine, respectively) at any scale.

Each of these two categories has its own algebraic structure, through the repre­sentation theory of the corresponding involved operators. While the Taylor polyno­mials provide a harmonic oscillator or a central potential basis, the Fourier basis isconstructed by exponentials In >= einx, which are the eigenfunctions of the deriva­tive operator, and the corresponding representation space of the 8l(2, R) angularmomentum algebra. The Fourier spectrum generating algebra, denoted here F, isgenerated by Jo = -iox, J± = -ie±ixox. This realization is an algebraic extensionof the affine Lie algebra generated by ax, xox, which is exactely the translation­dilation algebra. The ladder operators act on In >, changing the scale in steps andIn > are eigenfunctions for Jo. A diagram displaying the algebraic structure of Fis presented bellow

J±1-1 >

J±

10 >J±

11>J± J±+----t

+----t+----t+----t+----tIn>

IJo

IJoIJo IJo_ie±ixa

1-1 >_ie±ixa

10 :'>

_ie±ixa

11>_ie±ix8

_ie±ixa+----t

+----t+----t+-->+--> In>

In this case translation is ineffective, since the basis function are infinite ex­tended and periodic.

The new wavelet bases use a windowed Fourier transform and involves bothtranslation of compactly supported (or at least rapidly decreasing) functions and di­lation. They form a bi-dimensional system describing localization and scale changes.Though it would be expected that a system which unifies both multiscale and lo­calization properties fails to have a coherent algebraic structure, we proved that anonlinear algebraic structure exists and that it can be determined through a defor­

mation of the Fourier spectrum generating algebra, [10]. A wavelet basis for L[R]

consists in a set of dilated and translated copies Wn,k(X) = w(2nx - k), k, nEZ,of an initial wavelet w(x) defined itself as a dilation of a linear combination oftranslations of a scaling function <I>(x)

LUDU and J. P. DRAAYER

1

w(x) = L (-1)kC1_k<I>(2x - k).k=-N+l

The scaling function <I>(x) fulfills the two-scale equation (or dilation equation)

N N

<I>(x) = L Ck <I>(2x - k) = L Ck<I>l,k,k=O k=O

389

(1)

(2)

with coefficients Ck restricted by the conditions 2:~=oCk = 2, 2:~=oCkCk+21 =2801. The operations involved in Eq. (1,2) are handled by the translation operatorTk f(x) = f(x + k) and the dilation operator Dnf(x) = f(2nx). Eq. 2 can also bewritten as <I> = Dj(T)<I> where j(T) is a N-th order polynomial in T-1. The twooperators can be expressed in terms of infinite series of derivatives, if we introducethe differential realizations Tk = ekox, Dn = enx In 2ox, valid only when acting on

CCR) functions. Eq. (1,2) describe the selfsimilarity of the basis {Wn,k(X)}n,kEZ and

L2(R) = EBnEZ Vn where each Vn is generated by all k-translations of Dnw for any

positive integer n. Recently, a complete algebraic description was introduced, [10],proving that for any discrete wavelet system, a nonlinear algebra can be constructedin terms T and D.

A wavelet system is defined by the dilation equation Eq. 2, that can be writtenin the form of an fixed point problem j_<I> = <I>, where we introduce the operatorsj±(T) as the product between a dilation and a polynomial j in T, j± = D~OIj(T±).

The defining equation for the wavelet can be written as W = DOl j ( - T-1 ) <I> , orsimply as W = j-jo<I>, where we introduced a new operator jo(T). The waveletsystem is entirely described by the operators j(T), jo(T) and the parameter a. Theresulting the wavelet algebra is described by the relations

(3)

where the functional operators F(jo), CUo) fulfils a set of nonlinear closure condi­tions. The commutators in eq.(3) define a nonlinear algebra which we denote by

A. LUDU and J. P. DRAAYER

Ajaj. The construction of this algebra starts with a given wavelet system, Eqs.(1,2)which introduce two operators ), )0' Solving the closure conditions with respectto F, G as functions of the given ),)0 provides the wavelet algebra defined by thecommutation relations, Eqs.(3).

Conversely, for a given algebra AjaJ> (that is, given functions G, F) we solvethe closure conditions with respect to the functions )o,)± and we obtain the cor­responding dilation equation in the form ) _ cp = CP. The last step is to find thecombination of generators which produces the difference equation for III, eq.(l).

One application of this algebraic structure is the construction of wavelets assolutions of the two-scale equation. We use linear combinations of q-periodic func­tions II(x) (defined as II(qx) = II(x), q = 213). Examples are the logarithmic chirpsof period 1 and argument log x / {1log2. By using a test solution

CP&~8t= LAkII(x - k),k

the dilation equation can always be reduced to a finite-difference expression. Inthe Haar case ({1= 1) the step solution becomes CP(x) = H(x) - H(x - 1), whereH(x) is the Heaviside distribution. With the choice for the initial test function.600 = tanhx ~n 2H(x) - 1 the scaling function is expressible in terms of thegenerators of Ajaj only, and is given by

As an example, for the Haar scaling function, we associate the algebra Aiaj ~ AHwith s = 1 and

G = - 2)5 + )0 + 1,

and

Evidently, in the case when CPj,k is non-differentiable function the action of dif­ferential operators should be understood in the sense of distributions. A diagramdisplaying the algebraic structure of Ajaj, and the action of its generators on thescaling functions frame, is presented below

A. LUDU and J. P. DRAAYER 391

Wspace ~o(localized)Cl)oo

1j-jo

~limooOn1j-

<PO,-l

T±l

Cl)o,o

T±l

<PO,l

T±l T±l+----+

+----++----++----+<P0, n

1D±T±!

1D±

T±!

1D±

T±!

1D±

<Pl,-l

<Pl,O<Pl,lT±!

+----++----++----++----+<P1,n

1D±

1D±1D±1D±

~

TUo)===}

1D±T±2-j

1D±

T±2-j

1D±

T±2-j

ID±

<Pj,-l

<Pj,O<Pj,lT±2-J

+----++----++----+

+----+ <Pj, n

3

The generalized Lagrangian

The generalization which we will study in this section is described by the Lagrangian[5]

L[p, m, n, l] =1dx [~<Px<Pt + (p + l~P + 2) (<Px)p+2

-~( <px)m( <Pxxt + ~(<Px)l (<Pxx)q (<pxxxr] ,

(4)

which includes extra terms with higher order derivatives compared to the tradi­tional non-relativistic field theory Lagrangians. Here p, m, n, l, p, q and a,~, 'Yarefree parameters. In any case, after writing the Hamilton equation, we make thefunctional substitution TJ = <Px(x, t). The first term in eq.(4) gives the dynamicalstructure of the resulting equation, that is the time dependence. The second termis the typical term, most involved in nonlinear field equations. This term of termgenerates tne nonlinear termas in the correspondin NPDE. The next two terms areresponsible for the dispersion. The balance between the second term and the third(and further) terms controls the existence of localized solutions.

We begin our investigation by studying the simpler version of eq.(4), that is aLagrangian with only three terms, 'Y= O. The parameter n is the most importantone and controls the different cases. For n = 0,1, the third term in eq.(4) reducesto the second one or vanishes, respectively. The corresponding equation does notcontain dispersion so it is not interesting for our purposes. It's solutions are instablein time, decaying or blowing up. The next situation, n = 2, leads to a generalizedsequence of KdV-like equation of the form

TJt + aTJPTJx + ~[2TJmTJxxx + 4mTJm-lTJxTJxx

A. LUDU and J. P. DRAAYER

+m(m - 1)1]m-2(1]x)3],

called the K(p+2,m) nonlinear equation. These equations admit compact supportedtraveling solutions known as compactons. The compactons are trigonometric func­tions defined only on a half-period, and zero in the rest. In general they have theform Acosad(x - ct). In the special case when 0 < m < 2 and p = m, the solutionsare compactons for which the width is independent of the amplitude. This is thefact which provides the connection with wavelet bases. For m = p = 1, the resultingequation

a 2 2

1]t + 2(1] )x + 2{3(1] )xxx - 8{31]x1]xx = 0,(6)

is a generalized KdV equation with nonlinear dispersion (the third term in eq.(6))and a suplimentar nonlinear term. A compact on solution of eq.(6) is

(7)

if Ix - Vtl < rr.J3{3/a2, where 8 and 1]0 are free parameters and the velocity is

a function of the amplitude. We notice that the half-width L = .J24{3 / a of thewave is independent of the amplitude. This equation has the same terms and thesame compacton solution as the K(2,2) considered in [4]' and deriving from a denseanharmonic chain with many neighbors interaction. The quadratic dispersion termis characteristic for the nonlinear coupling in the chain.

For m 0 eq.(5) reduces to the class of modified KdV equations, namelyMKdV(p).

1]t + a1]P1]x + 2{31]xxx = 0

which are integrable for p = 1,2 with soliton traveling solutions, respectively

{ 1]osech2 [~(x - yt)), p = 11](x - Vt) = [( )]1]osech ~ x - ~t , p = 2.

(8)

(9)

For n > 2 the equations become too complex and their investigation beyond theaim of the present paper. Moreover, we have no knowledge of physical systemsdescribed by third power of the derivative terms.

In the case when the last term in eq.(4) is not any more neglected b =1= 0) itwas shown that compact on solutions occur only if p = m = l + q and r = 2, [5].Another interesting case occurs if l = q = 2, r = 1. In this case, if m = 1 and r = 4,the last term in eq.(4) reduces to the third one

(10)

The corresponding NPDE, after the traveling wave u(x, t) = 'Px(x-ct) substitution,has the form

A. LUDU and J. P. DRAAYER 393

(11)

where 'Px(x, t) is a travelling wave u(x - vt). One specific solution of eq(11) is amodulated (or complex shifted) Gaussian

1 ·k( ) (x-vt)2

'Px(x, t) = u(x - vt) = vrrre-' x-vt e---2-. (12)

This is a traveling soliton with fixed velocity and amplitude and arbitrary half­width. The function is known from the squeezed state formalism. Also, for t = 0,this function becomes the continuous Morlet wavelet [7], and can be extended toa basis of localized solutions, [8]. Another interpretation for solution in eq.(12) isgiven in the frame of the covariant phase-space representation for light, since it givesa localized probability for photons as localized waves. The Hamiltonian associatedwith eq.(12) has the form:

J fJL ] J 1 4 1 2 2Hwavelet = dX['Pt fJ'Pt - L = dX[-16 u + 2U Ux

(13)

We notice that there are lower order NPDE which have the Morlet wavelet, eq.(13),for solution

220uUxx - Ux + u = . (14)

Lwaveletl'P] leads to the three laws of conservation: mass, energy and momentum.Also, Hwavelet is conserved by Noether's theorem under the transformations: 'P--+

'P + cl, X --+ X + C2 and t--+ t + C3, where CI,C2,C3 are constants. By introducingthe Morlet solution in the Hamiltonian, eq.(13), we can analyse Hwavelet(k) as afunction of k. The dependence of the real part of Hwavelet on k shows that thespectrum is bounded. For small k the system has one local minimum, since in thisapproximation the system is a harmonic oscillator. For large k's the Hamiltonianshows a self-similar behavior. It provides a countable sequence of states degeneratedin energy with the ground state, (k = 0) for kj = 2VJ7i for integer j.

4 Discrete wavelets. Hamiltonian densities for the dilation equation

In this section we show that any scaling function is the stationary solution of acertain type of Lagrange or Hamiltonian system of equations. The procedure worksfor functions having a certain degree of smoothness such that both the translationT and dilation D operators can be expanded in formal operator Taylor series. Un­der such circumstances any q-difference (two or many scales) or finite-difference

A. LUDU and J. P. DRAAYER

equation, containing T and D operators, is equivalent with an infinite-order PDEequation.

We introduce a dynamical system described by a pair of functions u(±x - Vt),representing two traveling profiles of shapes U(6,2) (6,2 = ±x - Vt), running withfixed velocity V in the same direction on the real axis x. We introduce an infinite­dimensional Hamiltonian for U(6,2), associated with the two-scale equation

(15)

depending on the arbitrary constants tn, dn. By taking the functional derivative,the Hamilton equations associated with eq.(15)

u(±x - Vt) = ()x8HDT/8u(~x - Vt),

for u(±x - Vt) become

N

2: 2:( -1)ndnVnkx2n-ku(2n-k) (6,2) + VU(6,2)n;:::Ok=O

± 2:( -1)ntnu(n)(6,2) = O.n;:::O

(16)

Eq.(16) can be related with any dilation equation for an appropriate choice of thecoefficients tn and dn. By using the differential realization for T and D operatorsand by identifying in eq.(2) and eq.(16) the same order of derivative for the function(Le. <I>in eq.(2) and u in eq.(16)) one obtains an algebraic system oflinear equations.For the coefficients related with translation we have

For the coefficients related to the dilation part we have

(17)

(213 _ l)n

n!

n

2: (-I)kdkVnk'k=[(n+l)/2]

(18)

The system eqs. (17-18) can be solved in both directions. That is, for any Hamilto­nian in the form eq.(15) with given coefficients tk and dk, we can solve eqs.(17-18)

with respect to the coefficients Ck and /3and obtain the two-scale equation as Hamil­ton equation, associated to the Hamiltonian system, eq.(16). Conversely, for a giventwo-scale eq.(2), starting from its coefficients Ck and dilation factor /3, we can solveeqs.(17-18) and obtain the coefficients in the Hamiltonian in eq.(16). The series in

A. LUDU and J. P. DRAAYER 395

Nn+1eqs.(29-30) are convergent for a large class of functions since Itnl ~ ---nr- -t 0 whenn -t 00. If u(±x - Vt) = ±<I>(x, t) in HDT, eq.(16) is recovered but contains onlyeven (odd) powers of T±l, respectively. This duality suggests, as a further study,the introduction of a pair (<1>, Ill) for u(±x - Vt).

In the case of the dilation equation of the Haar wavelet, the solution is a dis­continuous distribution, <1>0,0 = H(x) - H(x - 1). In order to calculate the formalcoefficients for the Hamiltonian in eq.(15) (with /3 = -1) we can use the dilationexpansion. In the second order, the corresponding Haar Hamiltonian density is

h = _<1>_(x_)<I>_(_l_-_x_)_+_<1>_( x_-_1 )_<1>_( -_x_)2

(19)

One can neither calculate the derivatives in the above expression, nor in eqs.(15-16).To avoid this problem we have to use a functional sequence of differentiable functions~n, generated by a test function ~o, weakly converging towards <1>0,0. Hence, foreach n we can use the corresponding for the differential terms, the correspondingterm in the functional sequence.

5 Qualitative wavelet analysis of nonlinear equations

We present a simple qualitative analysis of any NPDE having localized soliton-liketraveling solutions, based on wavelet theory. This localized solutions are character­ized by three parameters: the amplitude A, the width L and the velocity V. Fara given NPDE we make the substitution of all its terms as follows Ut -t ±Vux,u -t ±A, UX -t ±A/ L, Uxx -t ±A/ L2 and so on. Consequently, the NPDE trans­forms into an algebraic equation providing a relation between the A, L, V. In Table1 there are example of this method on seven important NPDE in physics and inother applications.In all these equations their terms have the same parity.

The basic idea of this substitution is to write the NPDE on a very narrowinterval where the behavior of the localized function is very close to a modulatedGaussian, and then to apply a Morlet wavelet transform to the equation. In Table 1we have choosen some specific NPDE given in the first column. In the second columnit is presented one of their tipicallocalized solution and in the thrid column there aregiven the amplitude, width and velocity or general relations among them. Finaly,in the last column is presented the result of our method, that is the relation betwenthe three parameters obtain by the above substiution, without actualy solving theequation. For example in the case of the KdV equation we have a general expressionfor L. If we want L to depend only on A, we have to ask V to be proportional with

Table 1: Nonlinear equations, exact solutions and wavelet analysis.

Nonlinear Eq.

SolutionRelationsWaveletLiniar wave L:Ckeik(x±t/ oX)

V=AV=JXUxx - AUtt = 0

A L arbitra

Ut + 6uux +

Asech2x-VtL=f{L - 1L - vlV-6AIuxxx = 0

V=2A

L = l/A

L

Ut + 6u2ux +Asechx-Vt 1L A=VVVIV-6A21Uxxx = 0

sme-

or onAtan-l'"Ye~

A=4-14 = sinAUxt - sin U = 0

V=L2

K n,mUt + (un)x +

Not known n(n2+1)An-l(um)xxx = 0

±V±mAm-l

K(n,n) [Acos2 ( x-it) ] n~lUt + (un)x + (un)xxx = 0

K(2,2)

L=4~Ut + (u2)x + A 2x-Vtcas -L-X[- L; ,L2"J A- 4V1V-2AI

---.

(u2)xxx = 0

- 32V2

A. LUDU and J. P. DRAAYER

A. In this case we obtain the real expression among the parameters provided bythe exact solution.

The power of prediction of the method is exemplified in cases when there is noknown localized (soliton or compacton) solution of the NPDE, as for the K(n,m)equation. We provide here a relation among the parametrs which approaches thecorrect relations in the exact solutions, in same particular cases (n = m, n = 2m =2, etc), and also approaches our relations for those cases (lower columns). Like forexample in the last two columns of the Table 1. When n = m, there is an exactcompact on solution given here in the Table (K(n, n) row, second column). Theoriginal form of this solution was written with a small error in eq.(3) of the firstarticle in [6]. Many other examples can be taken for different values of n,m withthe correct prediciton for the behavior of the solutions (just put the values of nandm in fifth row, last column formula in Table 1. See for example the first article in[4]' page 567 the series of four equations. However, using this method, we notice amistake in the third equation where we thing it should be K(2, 3) instead of K(3, 2).In the last line of the Table 1 we have the welknown compacton solutior.. In ourapproch, if we assume that V is proportional with A we obtain for L a number,which may be not the exact solution, but is close to it.

The reason for this simple estimation works so good is provided by the possi­bility of wavelet analysis of localized solutions. We stress that this method, though

apparently similar, has little to do with the dimensional or similairty analysis. Inthat case one obtains relations only among powers of A, L, V and not relations withnumeric coefficients; like in our case.

A. LUDU and J. P. DRAAYER 397

Conclusions

In this contribution we introduce the problem of the physical interpretation ofwavelets, in the frame of localized physical nonlinear solutions. Self-similarity, clus­ter expansion, bifurcations and universality are all We proved that starting fromany unique soliton-like solution of a nonlinear partial differential equation, we canconstruct a whole basis which generates a special Hilbert space of solutions. Thisbasis is a wavelet system, hence providing similarity properties. We found a gen­eral Lagrangian out of which classical NLPDE and wavelets occur. This formalismprovide the possibility of constructing nonlinear basis and nonlinear superpositionprinciples for NLPDE. We show that frames of self-similar functions are related withnonlinear problems: nonlinear algebraic structures, nonlinear Hamiltonian systems,and especially with solitons with compact support. To any scaling function wecan associate a nonlinear finite generated algebra. Also, for any two-scale equationwe can construct a special infinite-dimensional Hamiltonian system such that thecorresponding scaling function is one of its extrems.

This unifying direction between nonlinearity and selfsimilarity, could bring newapplications of wavelets in cluster formation at any scale from supernovae throughfluid dynamics to atomic and nuclear systems, droplet bubbles and shell physics,stable traveling patterns, fragmentation, cold fission, the dynamics of the pelletsurface in inertial fusion, stellar models, etc.

Acknowledgments

Supported by the U.S. National Science Foundation through a regular grant, No.9603006, and a Cooperative Agreement, No. EPS-9550481, that includes matchingfrom the Louisiana Board of Regents Support Fund.

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