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VOLTERRA SERIES AND OPERATORS
Dan Censor
Ben-Gurion University of the Negev
Department of Electrical and Computer Engineering
Beer Sheva, Israel 84105
http://www.ee.bgu.ac.il/~censor, [email protected]
INTRODUCTION AND LINEAR SYSTEMS
In addition to playing an important role in the development of theoretical biology, the
Italian mathematician Vito Volterra (1860-1940) also strongly influenced the
development of modern calculus. We deal here with the Volterra Functional Series
(VFS) and associated Volterra Differential Operators (VDO) which provide a
consistent mathematical framework for stating material properties in nonlinear wave
propagation systems, for example in nonlinear optics. See for example (Censor &
Melamed, 2002, Censor, 2000), and for some application to nonlinear optics and as a
further gateway to literature see (Sonnenschein & Censor, 1998). For simplicity the
present introduction is restricted to the spatial domain, adequate for time signals.
Waves require a spatiotemporal domain.
Modeling of physical systems requires material (or constitutive) relations. In
electromagnetics we need relations like D   E , B   H ; in acoustics the
compressibility (relation of pressure to volume) is needed, in elastodynamics we
include Hooke’s law (relation of stress to strain). Here D   E and its nonlinear
extensions are treated as prototypes.
Materials are dispersive, depending (in the restricted sense of temporal
dispersion discussed here) on frequency f . Consider the linear case first:
D( )   (i ) E ( ),  =2 f
(1)
where  is the angular frequency, and defining  (i ) instead of  ( ) is
convenient for subsequent applications.
The Fourier transform pair
F (t ) 
1
2




 i t
 d F ( )e , F ( ) 
 dtF (t )e
i t
(2)
relates the spectral and temporal domains. We use the same symbol F , although
F (t ) and F ( ) are different functions. Accordingly (1) becomes a convolution
integral

D(t ) 
 dt  (t ) E (t  t )
1

1
1
(3)
>2<
where D(t ),  (t ), E (t ) , are related to D( ),  (i ), E ( ) , respectively, according
to (2). Note that (3) can be viewed as an integral operation, acting on E (t ) . Also (3)
is the simplest form of a VFS.
Formally we can start with (1), transform according to (2), and note that in the
integral  (i ) (if it can be represented or approximated by a polynomial in i ),
can be considered as a polynomial operator  (t ) acting on the exponential, in which
every time derivative  t replaces a term i in  (i ) . Note that  (t ),  (t ) are
different functions, but  (i ),  (t ) posses the same functional structure. Thus
instead of (3) we now have the VDO representation
D(t )   (t ) E (t )   ( ) E ( ) | t
(4)
The last expression in (4) with the instruction   t is superfluous here but will be
important for the nonlinear case below. The possibility of using this technique for
 (t ) a rational function (ratio of polynomials) is discussed elsewhere (Censor,
2001).
As a trivial example for (3, 4), consider an harmonic signal
E (t )  E0eit
D(t )  E0eit

it
 it
 it
 dt1 (t1 )e 1   (i ) E0e   (t ) E0e
(5)

clarifying the role of the VDO in (4).
2. NONLINEAR SYSTEMS AND THE VOLTERRA SERIES AND OPERATORS
In nonlinear systems the material relations involve powers and products of fields. Can
we simply replace (1) by a series involving powers of E ( ) ? A cursory analysis
reveals that this (already appearing in the literature on the subject) leads to
inconsistencies. Instead we ask if (3) can be replaced by a "super convolution" and
what form should that take. Indeed, the Volterra series provides a consistent
mathematical answer to these questions. It is given by
D(t )   D ( m ) (t )
m

D
( m)
(t ) 
 dt
1

(6)

...
 dt 
m
( m)
(t1 , ... ,tm ) E (t  t1 )    E (t  tm )

Typically the VFS (6) contains the products of fields expected for nonlinear systems,
combined with the convolution structure (3). Various orders of nonlinear interaction
are indicated by m .
Theoretically all the orders co-exist (in practice the series will have to be
truncated within some approximation), and therefore we cannot inject a time
>3<
harmonic signal as in (5). If instead we start with a periodic signal, E (t )   Ene int
n
and substitute in (6), we find
D ( m ) (t ) 

 ( m ) (in1 , ... ,  inm ) En    En eiNt   DN eiNt
1
m
n1 , ... ,nm
N
(7)
N  n1  ... +nm
with (7) displaying the essential features of a nonlinear system, namely, the
dependence on a product of amplitudes, and the creation of new frequencies as sums
(including differences and harmonic multiples) of the interacting signals frequencies.
In addition, (7) contains the weighting function  ( m ) (in1 , ... ,  inm ) for each
interaction mode.
The extension of (4) to the nonlinear VDO is given by
D ( m ) (t )  ε ( m ) ( t1 , ... ,  tm ) E (t1 )    E (tm )
t1 , ... , tm  t
(8)
In (8) the instruction t1, ... , tm  t guarantees the separation of the differential
operators, and finally renders both sides of the equation to become functions of t .
The VFS (6), including the convolution integral (3), is a global expression
describing D(t ) as affected by integration times extending from  to  . Physically
this raises questions about causality, i.e., how can future times affect past events. In
the full-fledged four-dimensional generalization causality is associated with the so
called "light cone" (Bohm, 1965). It is noted that the VDO representation (4, 8) is
local, with the various time variables just serving for book keeping of the operators,
and where this representation is justified, causality problems are not invoked.
In general, the frequency constraint of (7) is obtained from the Fourier
transform of (6), having the form
D
( m)
( ) 

1
(2 )m1
 d
1


...
 d
 ( m) (i1, ... ,  im ) E (1 )    E (m )
m 1

(9)
  1  ... +m
It is noted that in (9) we have m  1 integrations, one less than in (6). This tallies with
the linear case where (1, 3) involve zero, one, integrations, respectively.
Consequently the left, right sides of (9) are functions of  , m , respectively. The
additional constraint   1  ... +m completes the equation and renders (9) selfconsistent.
3. SUMMARY
The modeling of nonlinear media using the VFS and VDO provides a mathematically
consistent framework which includes linear media as a special limiting case. The
model displays the typical ingredients of nonlinear circuits and wave systems, where
the nonlinear terms are proportional to the product of the amplitudes of the interacting
>4<
fields, and the newly created frequencies are sums (or differences, or harmonic
multiples) of the interaction frequencies, given by   1  ... +m . In the quantummechanical context this is an expression of the conservation of energy. Not shown
here is the associated wave propagation vector constraint k  k1  ... +k m , which in
the quantum-mechanical context expresses conservation of momentum. For further
reading on VFS in nonlinear systems see (Schetzen, 1980).
4. REFERENCES AND FURTHER READING
Bohm, D. The Special Theory of Relativity, Benjamin, 1965.
Sonnenschein, M & Censor, D., 1998, Simulation of Hamiltonian light beam
propagation in nonlinear media, JOSA-Journal of the Optical Society of America B,
(15): 1335—1345
Censor,D., 2000, A quest for systematic constitutive formulations for general
field and wave systems based on the Volterra differential operators, PIER-- Progress
In Electromagnetics Research, (25): 261--284
Censor, D., 2001, Constitutive relations in inhomogeneous systems and the
particle-field conundrum, PIER--Progress In Electromagnetics Research, (30): 305-335
Censor, D., & Melamed, T, 2002, Volterra differential constitutive operators
and locality considerations in electromagnetic theory, PIER- Progress in
Electromagnetic Research, 36: 121—137
For further reading and a plethora of early references, see
Schetzen, M., 1980, The Volterra and Wiener Theorems of Nonlinear Systems, New
York, Chichester, Brisbane and Toronto: John Wiley and Sons