H.T. Banks
Center for Research in Scientic Computation
North Carolina State University
Raleigh, NC 27965-8205
Gabriella A. Pinter
Department of Mathematical Sciences
University of Wisconsin Milwaukee
Milwaukee, WI 53201-0413
March 15, 2002
ABSTRACT
We establish well-posedness results for a model describing the propagation of
high-intensity electromagneticwaves ina nonlinear medium. The nonlinear
ma-terial properties are represented by a nonlinear polarization in the form of a
convolution. We alsoinclude some remarks onpotentialapplications.
1 Introduction
Inthis paperwe consideratime-domainmodelfor thepropagationofhigh-intensity
electro-magnetic waves (e.g.,laserbeams)indielectricmaterials. Thisworkis acontinuationof the
eorts in the monograph [1], where the authors investigated a similar model using the full
Maxwell'sequations together with linearconstitutive relationsina variationalapproachfor
thepropagationofmicrowavesthroughdielectriclayers. Thevariationalapproachfacilitates
theincorporationofantennasourcesinthemodelandprovidesanapproximatecomputation
of the electromagnetic transients like the Sommerfeld and Brillouin precursors. The
oreticalandcomputationalbasis forpotentialnew electromagneticinterrogationtechniques.
Our goal in this paper is to broaden the applicability of the results in [1] in the following
sense: the modelinthemonographcontains alinearconstitutiverelationshipdescribing
ma-terial polarizationin a convolution representation. Such a formulation includes Debye and
Lorentzpolarizationmodels and is adequateto describe the interaction of microwaves with
dielectric materials. However, the polarization mechanism cannot be assumed to be linear
in the case of very high intensity electromagnetic waves, like lasers. Thus inthis paper we
develop theoreticalresultsusing arepresentationofthe polarizationby anonlinear
convolu-tion. In particular, we take aphysical modelsimilar to that in[1], introduce a polarization
law
P(t;z)= Z
t
0
g(t s;z)(E(s;z)+f(E(s;z)))ds;
andestablishthewell-posednessoftheresultingsystem. Thisformulationcanbeinterpreted
asa generalizationof the Debye or Lorentz polarizationmodels inthe sense that the
polar-ization dynamics is driven by a nonlinear function of the electriceld. We show the global
existence-uniqueness and continuous dependence on data of weak solutions under general
assumptions on the nonlinearity f: These conditions are satised by a number of locally
polynomial nonlinearities proposed in the nonlinear optics literature (e.g., see the \second
order" optical models involving cubic nonlinearitiesin [6, 7].) The analysis uses techniques
similar to those in [2] with the added diÆculty of treating the terms that result from the
boundary conditions. Basedonthe theoretical results,rigorouscomputationalmethodscan
be developed forthe forward problemand subsequently for the associatedinverse problems.
The literature on nonlinear optics and Maxwell's equations is, of course, vast. Good
introductions tononlinear optics canbefound in[6,11,13,15]to namejusta few. Related
results and some background information on similar electromagnetic models are given in
[1, 5, 8, 10, 14]. For example in [10] the authors develop theoretical results in an operator
theoretic context for systems with nonlinear polarization and magnetization under global
Lipschitz conditions.
We consider this paper to be a rst, but nontrivial, step in analyzing the propagation
of highintensity electromagneticwavesthrough dielectric materialsusingthe fullMaxwell's
equationsin the time-domain. Numerousinteresting questionsremain: forexample, can we
theoreticallyand/orcomputationallydemonstrateself-focusingofthebeamorwavecollapse
using this approach when an array of sources is used instead of a point source? These
phenomena are usually discussed in the context of the nonlinear Schrodinger equation in
weaklynonlinear (Kerrmedium)dielectricse.g., in[13,15,16]. Inthat approachthecrucial
assumption is that the relevant electric eld can be represented by a nite basis of weakly
coupledwavepackets andthe wavepacketenvelopesevolvevianonlinearequations. However,
when light intensities become suÆciently high so that the material behaves like a plasma,
this weak coupling theory breaks down [13]. This is exactly what happens when a laser
pulse collapses intomultiple self-focused laments. The initialphases of the event are
well-understood but as the collapseevolves anelectron plasmaand very large thermalgradients
The structureof thepaperisasfollows: InSection2we presentashortderivationof the
model and the existence result for weak solutions. Section 3 and 4 contain the uniqueness
and continuous dependence arguments, respectively. We close with a brief discussion of
applications tononlinear materials.
2 Weak formulation and existence result
As described in the Introduction, we consider Maxwell's equations applied to a specic
physical problem as depicted in Figure 1. An innite slab of material is interrogated by
a normally incident polarized plane wave windowed pulse originatingat an antenna source
z = 0 in free space
0
= [0;z
1
]: The slab of material in = [z
1 ;z
2
] is assumed to be
homogeneous in the directions orthogonal to the direction z of propagation of the plane
wave. Under these assumptions it is possible to represent the strength of the electric and
magneticeldsinand
0
bythe scalarfunctionsE(t;z)andH(t;z),respectively. Onecan
readily eliminatethe magnetic eld fromthe full Maxwell equationsto arrive at the strong
formulation of the problem(for adetailed derivation see [1])
0
E+
0 I
(z)
P +
0
_
E E
00
=
0 _
J
s
0<z <z
2
; t >0; (2.1)
1
c @E
@t
@E
@z
z=0
=0 t>0; (2.2)
E(t;z
2
)=0 t>0; (2.3)
E(0;z)=(z); _
E(0;z)= (z) 0<z<z
2
; (2.4)
where P denotes the polarizationand I
isthe indicatorfunction
I
(z) =
(
0 if 0<z <z
1
1 if z
1
z z
2 ;
)
;
where z
1
< 1 is the front boundary of the slab and z
2
is the (sometimes unknown in
in-terrogation problems) back boundary (see Figure 1). We note that anabsorbing boundary
condition is placed at z = 0 to prevent the reection of waves. We assume that there
is a supraconductive backing on the slab at z = z
2
and specify the boundary conditions
accordingly. Equation (2.1) can begiven in the form
0
E+ 1
0 I
(z)
P + 1
0
_
E c
2
E 00
= 1
0 _
J
s ;
where =
0
(1+(
r 1)I
);and c 2
= 1
0
0
:We denote
0 by "^
r :
We assume that the frequency of the interrogatingwave is sohigh that the dependence
of the polarization on the electric eld can no longer be adequately described by a linear
constitutive law. Specically, we consider a polarization mechanism that depends on the
strength of the electriceld in the convolution
P(t;z)= Z
t
0
z
H(t;z)
E(t;z)
z
1
z
2
y
Figure1: Geometryof the physical problem
with nonlinearity f. Thus
P(t;z) = Z
t
0
g(t s;z)f(E(s;z))ds+g(0;z) @
@t
f(E(t;z))+g(0;_ z)f(E(t;z))
+ Z
t
0
g(t s;z)E(s;z)ds+g(0;z) _
E(t;z)+g(0;_ z)E(t;z);
wheref :IR!IRisanonlinear(notnecessarilysmall)perturbationofthelinearmechanism.
This yieldsthe strong formof the equation
^ "
r
E(t;z) + 1
0 I
(z)((z)+g(0;z)) _
E(t;z)
+ 1
0 I
(z)g(0;_ z)E(t;z)+ Z
t
0 1
0 I
(z)g(t s;z)E(s;z)ds
+ 1
0 I
(z)g(0;_ z)f(E(t;z))+ Z
t
0 1
0 I
(z)g(t s;z)f(E(s;z))ds
+ 1
0 I
(z)g(0;z) d
dt
f(E(t;z)) c 2
E 00
(t;z)= 1
0 _
J
s
(t;z); 0<z <z
2 :(2.6)
Inthe physicalproblemz
2
isassumed tobeunknown, andit isdesirabletoestimateitfrom
given data. Since theexistenceproof isconstructiveinthe sense thatthe numericalmethod
we use tosolve this problem(forboth forward and inverse problems) follows the theoretical
functions for dierent z
2
; but the same numerical framework can be used throughout the
optimizationprocedure inthe inverse problems (again,this isexplained indetail in [1]).
We rst multiply (2.6) by , where 2 H 1
(0;z
2
) and integrate from 0 to z
2 using
integration by parts in the lastterm in the left. Then we introduce achange of variableby
letting
~
z =h(z) = (
z
1
if 0<z <z
1
z
1
+(z z
1 ) 1 z1 z2 z1 if z 1
z z
2 ;
)
i.e., h(z)= z+( 1)(z z
1 )I
[z1;z2]
(z) with = 1 z 1 z 2 z 1
: Then h 0
(z) =1+( 1)I
(z) and
~
h 0
(~z)=1+( 1)I
~
(~z);where ~
=[z
1
;1]andweadoptthenotationthat ~
E; ~
h; ~
;g;~ etc., are
the mapsE;h;;g;etc., afterthey havebeen mapped fromthedomain[0;z
2
]tothedomain
[0;1]:Usingthis transformation we obtain
h 1 ~ h 0 ~ ^ " r ~
E(t;);'i~ +h 1 ~ h 0 1 0 I ~
(~+g(0;~ )) _
~
E(t;);'i~
+ h 1 ~ h 0 1 0 I ~ _ ~ g(0;)
~
E(t;);'i~ +h 1 ~ h 0 Z t 0 1 0 I ~ ~
g(t s;) ~
E(s;)ds;'i~
+ h 1 ~ h 0 1 0 I ~ _ ~
g(0;)f( ~
E(t;));'i~ +h 1 ~ h 0 Z t 0 1 0 I ~ ~
g(t s;)f( ~
E(s;))ds;'i~
+ h 1 ~ h 0 1 0 I ~ ~ g(0;)
d
dt f(
~
E(t;));'i~ +hc 2 ~ h 0 ~ E 0
(t;);'~ 0
i+c _
~
E(t;0)'(0)~
= h 1 ~ h 0 1 0 _ ~ J s
(t;);'i;~ (2.7)
whereh;iistheL 2
(0;1)innerproduct. Thiseectivelymapsoursystemtoaxedreference
domain [0;1] and we use this form to dene the weak solution of the problem. We let
H =L 2
(0;1); V =H 1
R
(0;1)=f2H 1
(0;1)j(1)=0gleadingtotheGelfandtriple([9,17])
V ,! H ,!V
: We say that E 2L 1
(0;T;V) with _
E 2L 2
(0;T;H);
E 2L 2
(0;T;V
);is a
weak solutionif it satises forevery '2V
h" r E;'i V ;V +h _
E;'i+hE;'i+h Z
t
0
(t s;)E(s;)ds;'i
+hf(E);'i+h Z
t
0
(t s;)f(E(s;))ds;'i+h^ d
dt
f(E);'i
+hc 2 h 0 E 0 ;' 0
i+c _
E(t;0)'(0)=hJ(t;);'i
V
;V
(2.8)
and
E(0;z)=(z); _
E(0;z)= (z); (2.9)
wherein(2.8) wedropthe overtildaonvariablesand functionsforsimplicationof notation
and for z~2[0;1]; we dene
(~z) =
~
h 0
0 I
~
_
~ g(0;z);~
(~z) = 1
~
h 0
1
0 I~
(~(~z)+g(0;~ ~z));
^
(~z) = 1
~
h 0
1
0 I
~
~ g(0;z);~
J(t;z)~ = 1
~
h 0
1
0 _
~
J
s (t;z);~
"
r =
1
~
h 0
~
^ "
r :
We note that since 1
~
h 0
(z)
> 0 is a piecewise constant function, the inclusion of this term in
;;;;^ "
r
;J is essentially equivalentto using amodied
0
: In developing the theoretical
existence result we can also make the simplifying assumption that "
r
= 1 without loss of
generality. (This, of course, is not true for computationaleorts.) Our goalis to establish
the existence and uniqueness of weaksolutions.
We makethe following assumptions:
A1) The functions ;;^ 2L 1
(0;1) with
j(z)jL
1
; j^(z)jL
2 :
A2) The function is bounded on [0;T](0;1) with
j (t;z)jL
3 :
A3) g(0;z) 0;(z) 0: (We note that this assumption implies that (z) 0;(z)^ 0
for every z 2[0;1]:)
A4) The nonlinear function f :IR!IR is C 1
, with f(0)=0; and f 0
(z)>0for all z 2IR :
f isalsoassumed tobeaÆneat innity,i.e., there existconstantsR ;L
R
andK
R such
that for every jxj>R ; jf(x)jL
R
jxj+K
R :
Remark 2.1 We note that A4) implies that f is locally Lipschitz, i.e., for any r >0 there
exists L
r
>0 such that
jf(x) f(y)jL
r
jx yj for every jxj;jyj<r:
Alsothere is a constant M with jf(x)jMjxj for all x2IR :
Werst prove the followingtheorem:
Theorem 2.1 Under assumptions A1)-A4) the system (2.8)-(2.9) has a weak solution for
any 2V; 2H and J 2H 1
(0;T;V
the nonlinear terms. Thus we rst add hkE;'i to both sides of (2.8) where k is chosen in
suchawaythat ^
k+ satises ^
"
1
>0on[0;1]forsomeconstant "
1
:Suchak exists
since by assumption 2L 1
(0;1):Next we denethe sesquilinear form
1
:V V !Cl by
1
(; )=hc 2
h 0
0
; 0
i
H +h
^
; i
H
for ; 2V:
It is readily seen that
1
is V-continuous and V-elliptic, i.e., there are positive constants
c
1 ;c
2
such that
j
1
(; )j c
2 kk
V k k
V
for all ; 2V; (2.10)
1
(;) c
1 kk
2
V
for all 2V: (2.11)
Let 2 V and 2 H and x T > 0: We choose a subset fw
i g
1
i=1
spanning V. Without
loss of generality we assume that the elements w
i
are linearly independent. Let V m
=
span fw
1
;:::;w
m
g;and denethe Galerkin approximations
E
m
(t;z)= m
X
i=1 e
m
i (t)w
i (z);
where the fe m
i (t)g
m
i=1
are obtained by solving the m-dimensional system of nonlinear delay
dierentialequations:
h
E
m (t);w
i i
H +h
_
E
m (t);w
i i
H +h
Z
t
0
(t s;)E
m
(s;)ds;w
i i
+
1 (E
m (t);w
i )+c
_
E
m (t;0)w
i
(0)+h^ d
dt f(E
m (t));w
i i+h
Z
t
0
(t s;)f(E
m
(s;))ds;w
i i
+hf(E
m (t));w
i
i=hJ(t);w
i i
V
;V
+hkE
m (t);w
i
i; (2.12)
for i=1:::m; with
E
m
(0) = m
(2.13)
_
E
m
(0) = m
; (2.14)
where m
! in V; and m
! in H. Multiplying each equation by e_ m
i
(t) and adding
them we obtain
1
2 d
dt k
_
E
m (t)k
2
H +h
_
E
m (t);
_
E
m (t)i
H +h
Z
t
0
(t s;)E
m
(s;)ds; _
E
m (t)i
H
+
1 (E
m (t);
_
E
m
(t))+cj _
E
m (t;0)j
2
+h^ d
dt f(E
m
(t)); _
E
m (t)i
H
+h Z
t
0
(t s;)f(E
m
(s;))ds; _
E
m (t)i
H
+hf(E
m (t));
_
E
m (t)i
H
=hJ(t); _
E
m (t)i
V
;V
+hkE
m (t);
_
E
m (t)i
H
E
m
(0)=
m ;
_
E
m (0)=
1 2 d dt n k _ E m (t)k 2 H + 1 (E m (t);E m (t)) o +k q
+^f 0 (E m (t)) _ E m (t)k 2 H +cj _ E m (t;0)j 2 =h Z t 0
(t s;)E
m (s;)ds; _ E m (t)i H +h Z t 0
(t s;)f(E
m (s;))ds; _ E m (t)i H
+h f(E
m (t)); _ E m (t)i H
+hJ(t); _ E m (t)i V ;V +hkE m (t); _ E m (t)i H ;
where we used that +f^ 0
(E) 0 by assumptions A3) and A4). Integration from 0 to t,
where t2[0;T] yields
k _ E m (t)k 2 H +c 1 kE m (t)k 2 V +2 Z t 0 k q
+f^ 0 (E m (s)) _ E m (s)k 2 H
ds+2cj _ E m (;0)j L 2 (0;t) k _ E m (0)k 2 H +c 2 kE m (0)k 2 V +2j Z t 0 F m
()dj (2.15)
with
F
m
() = h Z
0
( s;)E
m (s;)ds; _ E m ()i H +hkE m (); _ E m ()i H
+ hJ(); _ E m ()i V ;V +h Z 0
( s;)f(E
m (s;))ds; _ E m ()i H
+ h f(E
m ()); _ E m ()i H T 1
()+T
2
()+T
3
()+T
4
()+T
5 ():
Let 0< <t bearbitrary. Wenext estimatethe termsT
i
() one by one. Thus,
jT 1 ()j Z 0 L 3 kE m (s)k H dsk _ E m ()k H 1 2 Z 0 L 3 kE m (s)k H ds 2 + 1 2 k _ E m ()k 2 H 1 2 L 2 3 t Z t 0 kE m (s)k 2 H ds+ 1 2 k _ E m ()k 2 :
Usingthis estimatewe obtain
Z t 0 jT 1 ()jd 1 2 L 2 3 t 2 Z t 0 kE m (s)k 2 H ds+ 1 2 Z t 0 k _ E m (s)k 2 H ds: (2.16)
Next, we have
Z t 0 jT 2 ()jd Z t 0 jhkE m (); _ E m ()ijd Z t 0 1 2 k 2 kE m ()k 2 H + 1 2 k _ E m ()k 2 H d: (2.17) Since _
J 2L 2
(0;T;V
)we have
d
dt
hJ(t);E
m (t)i V ;V =h _
J(t);E
m (t)i
V
;V
j Z t 0 T 3
()dj Z t 0 ( d d
hJ();E
m ()i V ;V h _
J();E
m ()i V ;V ) d
= jhJ(t);E
m (t)i
V
;V
hJ(0);E
m (0)i V ;V Z t 0 h _
J();E
m ()i V ;V dj 1 2Æ kJ(t)k 2 V +ÆkE m (t)k 2 V + 1 2 kJ(0)k 2 V + 1 2 kE m (0)k 2 V + Z t 0 1 2 k _
J()k 2 V + 1 2 kE m ()k 2 V d; (2.18)
whereÆ >0isarbitrary. Usingthe assumptionsA2)andA4)wecan estimatethenext term
as aboveto yield
jT
4
()j = jh Z
0
( s;)f(E
m (s;))ds; _ E m ()i H j Z 0 L 3 kf(E m (s))k H dsk _ E m ()k 1 2 L 2 3 M 2 t Z t 0 kE m (s)k 2 H ds+ 1 2 k _ E m ()k 2 H :
Thuswend
Z t 0 jT 4 ()jd 1 2 L 2 3 M 2 t 2 Z t 0 kE m ()k 2 H d + 1 2 Z t 0 k _ E m ()k 2 H d: (2.19)
Similarly the lastterm can be estimated
Z t 0 jT 5 ()jd Z t 0 jhf(E m ()); _ E m ()ijd Z t 0 L 1 kf(E m ())k H k _ E m ()k H d Z t 0 1 2 L 2 1 M 2 kE m ()k 2 H + 1 2 k _ E m ()k 2 H d: (2.20)
Combiningthe estimates (2.16)-(2.20)we obtain from(2.15)
k _ E m (t)k 2 H +(c 1 2Æ)kE m (t)k 2 V +2ck _ E m (;0)k 2 L 2 (0;t) k _ E m (0)k 2 H +(c 2 +1)kE m (0)k 2 V
+kJ(0)k 2 V + Z t 0 k _
J()k 2 V d +(L 2 3 t 2 +k 2 +L 2 3 M 2 t 2 +L 2 1 M 2 ) Z t 0 kE m ()k 2 H d + Z t 0 kE m ()k 2 V d + 2 Æ kJ(t)k 2 V +4 Z t 0 k _ E m ()k 2 H d:
We choose Æ to be such that =c
1
2Æ > 0: Since V ,! H there exists a constant >0
with kuk
H
kuk
V
; for every u2V:Also, E
m
(0) is bounded in V; _
E
m
and J 2H (0;T;V ),so we can conclude that
k _
E
m (t)k
2
H
+kE
m (t)k
2
V
+2ck _
E
m (;0)k
2
L 2
(0;t) B
1 +4
Z
t
0 k
_
E
m ()k
2
H d
+B
2 Z
t
0 kE
m ()k
2
V
d; for all t2[0;T]; (2.21)
where B
1 ;B
2
are independent of m. Nowby the Gronwallinequality we arriveat
k _
E
m (t)k
2
H
+kE
m (t)k
2
V
+2ck _
E
m (;0)k
2
L 2
(0;t)
C(T;f;g;; ;J); (2.22)
where C is a positive constant independent of m: Hence
fE
m
g is bounded in C(0;T;V)L 2
(0;T;V) (2.23)
f _
E
m
g is bounded in C(0;T;H)L 2
(0;T;H) and (2.24)
f _
E
m
(;0)g is bounded in L 2
(0;T): (2.25)
Consequently there existsubsequences (againdenoted by the subscript m) such that
E
m
! E weakly in L 2
(0;T;V) (2.26)
_
E
m !
~
E weakly in L 2
(0;T;H) (2.27)
_
E
m
(;0) ! E
L
weakly in L 2
(0;T): (2.28)
First we show that ~
E = _
E; E
L =
_
E(;0)insome sense. For each m wehave that
E
m
(t) = E
m (0)+
Z
t
0 _
E
m
(s)ds (2.29)
E
m
(t;0) = E
m
(0;0)+ Z
t
0 _
E
m
(s;0)ds: (2.30)
Passing to the limitin(2.29) (in the weak H sense)and in(2.30), weobtain
E(t) = + Z
t
0 ~
E(s)ds; (2.31)
E(t;0) = (0)+ Z
t
0 E
L
(s;0)ds; (2.32)
where (2.31) holds in the sense of H for each t 2 [0;T]: Thus (2.31) implies that ~
E = _
E,
whilefrom(2.32)wecanconclude thatE(t;0)exists andiscontinuousint:Actually,E(t;0)
is absolutelycontinuous with _
E(t;0)=E
L
(t) foralmost every t.
We proceed to show that the convergence (2.26) can be strengthened to provide the
strongconvergence E
m
!E inC(0;T;H):We notethat this strongconvergence enablesus
to pass to the limit as m ! 1 in expressions involvingthe nonlinear function of E
m ; and
thus it is the key to establishing the existence of weak solutions for systems with nonlinear
polarizationwith the techniques employed here. Toachieve this result we use the following
relativelycompactifandonlyifF isequicontinuousandff(t):f 2Fgisrelativelycompact
in Y foreach t2[0;T]:
Here we letY =H and F =fE
m g
1
m=1
C(0;T;H):First weshow thatthe subset fE
m gis
equicontinuous inC(0;T;H):Using the estimate(2.21) we have
kE
m
(t+t) E
m (t)k
2
H
= k Z
t+t
t _
E
m
()dk 2
H
Z
t+t
t k
_
E
m ()k
H d
!
2
t
Z
t+t
0 k
_
E
m ()k
2
H d
(t)
2
C
Sinceforeacht 2[0;T]fE
m (t)g
1
m=1
isuniformlyboundedinV bytheaprioriestimate(2.22)
and V iscompactlyembedded inH wecan conclude that fE
m (t)g
1
m=1
isrelatively compact
in H for each t 2 [0;T]: Thus the Arzela-Ascoli theorem guarantees that fE
m (t)g
1
m=1 is
relatively compact in C(0;T;H) and hence there exists a subsequence (again denoted by
E
m
) that converges strongly to E in C(0;T;H): Now we claim that E (which must be
equivalent to the limit in(2.26)) is a weak solution of (2.8)-(2.9). To establishthis fact we
rst take 2 C 1
(0;T) with (T) = 0 and choose
j
(t) (t)w
j
where fw
j g
1
j=1
are the
basis elements as before. For a xed j we have that E
m
satises the following relation for
allm >j :
Z
T
0 n
h
E
m (t);
j
(t)i+h(+^f 0
(E
m (t)))
_
E
m (t);
j
(t)i+
1 (E
m (t);
j (t))
+h Z
t
0
(t s;)E
m
(s;)ds;
j
(t)i+c _
E
m (t;0)
j
(t;0)+h Z
t
0
(t s;)f(E
m
(s;))ds;
j (t)i
+hf(E
m (t));
j
(t)igdt = Z
T
0
fhJ(t);
j (t)i
V;V
+hkE
m (t);
j
(t)igdt: (2.33)
Weintegrateby partsinthersttermintheleftusingthe propertythat (T)=0toobtain
Z
T
0 n
h _
E
m (t);
_
j
(t)i+h(+^f 0
(E
m (t)))
_
E
m (t);
j
(t)i+
1 (E
m (t);
j (t))
+h Z
t
0
(t s;)E
m
(s;)ds;
j
(t)i+c _
E
m (t;0)
j
(t;0)+h Z
t
0
(t s;)f(E
m
(s;))ds;
j (t)i
+hf(E
m (t));
j
(t)igdt = Z
T
0
fhJ(t);
j (t)i
V
;V
+hkE
m (t);
j
(t)igdt+h ;
j (0)i:
Now we can take the limitas m ! 1 using the convergences (2.26)-(2.28) and the strong
convergence of E
m
!E inC(0;T;H) toobtain
Z
T
0 n
h _
E(t); _
j
(t)i+h(+f^ 0
(E(t))) _
E(t);
j
(t)i+
1 (E(t);
+h t
0
(t s;)E(s;)ds;
j
(t)i+c _
E(t;0)
j
(t;0)+h t
0
(t s;)f(E(s;))ds;
j (t)i
+hf(E(t));
j
(t)igdt= Z
T
0
fhJ(t);
j (t)i
V
;V
+hkE(t);
j
(t)igdt+h ;
j (0)i:
Wenotethatpassingtothelimitinthe secondtermundertheintegralinthe leftispossible
by the convergences (2.26),the strongconvergence E
m
!E in C(0;T;H)and the factthat
f 0
is assumed to be continuous. It follows that for every
j
we have in the L 2
(0;T) sense
(except inthe rst term which is inthe distributional sense int)
d
dt h
_
E(t);
j
(t)i+h+^f 0
(E(t))) _
E(t);
j
(t)i+
1 (E(t);
j (t))
+h Z
t
0
(t s;)E(s)ds;
j
(t)i+c _
E(t;0)
j
(t;0)+h Z
t
0
(t s;)f(E(s))ds;
j (t)i
+hf(E(t));
j
(t)i=hJ(t);
j (t)i
V
;V
+hkE(t);
j
(t)i: (2.34)
Since fw
j g
1
j=1
is total in V we can conclude that
E 2 L 2
(0;T;V
) and E satises (2.8).
From (2.31) it follows that E(0) = and the fact that _
E(0) = can be established
exactlyas in [4]. Hence wend that E isa solution of (2.8)-(2.9). It remainsto argue that
E 2L 1
(0;T;V):WeshowusingtheArzela-AscolitheoremthattheGalerkinapproximations
satisfyanadditionalconvergence property, namely,that E
m
!E inC
W
(0;T;V):Since this
implies E
m
(t) !E(t) weakly in V, we can conclude from the a priori estimate (2.22) that
for allt2 [0;T]
kE(t)k 2
V
1
C;
and hence E 2L 1
(0;T;V):
To establish the equicontinuity of fE
m
g in C
W
(0;T;V) we use similar arguments as
those in [2]. Namely, we note that the mapping A
1
: V ! V
: hA
1 ; i
V
;V =
1 (; )
is a topological isomorphism. We dene domA
1
= fv 2 V : A
1
v 2 Hg = A 1
1
(H): Since
H is dense in V
, domA
1
is dense in V. We assume that V is equipped with the inner
product
1
(;); which is in fact equivalent to the original inner product in V. Let us rst
take v 2domA
1
: Then we have
j
1 (v;E
m
(t+t) E
m
(t))j=jhA
1 v;E
m
(t+t) E
m
(t)ijkA
1 vk
Z
t+t
t k
_
E
m
()kd
kA
1 vk
p
Cjtj; (2.35)
where C is the constant from (2.22). Nowlet 2V be arbitrary, v 2domA
1
; and >0:
j
1 (;E
m
(t+t) E
m
(t))jj
1 (v;E
m
(t+t) E
m (t))j
+j
1
( v;E
m
(t+t) E
m
(t))jkA
1 vk
p
Cjtj+2Ck vk
V
: (2.36)
Since domA
1
is dense in V, we can choose v 2 domA
1
such that 2Ck vk
V <
2
: Thus if
jtj<Æ=
2kA
1 vk
p
C ; then
j
1 (;E
m
(t+t) E
m
m
t, we can choose Y in the Arzela-Ascoli theorem to be a closed and bounded subset of
V containing fE
m
(t)g: We equip Y with the weak topology. Thus Y is a compact metric
space, and fE
m
(t)g is relatively compact in Y for each t. Since we already established the
equicontinuity of fE
m g in C
W
(0;T;V); we obtain by the Arzela-Ascoli theorem that fE
m g
is relatively compact in C(0;T;Y); i.e., we have E
m
(t) ! E(t) weakly in V uniformly in t
for asuitable subsequence. ThusE 2L 1
(0;T;V) and Theorem 2.1. is proved.
3 Uniqueness
WenextestablishtheuniquenessofweaksolutionsinL 1
(0;T;V)\C(0;T;H):Thussuppose
that E
1 ;E
2
both solve (2.8)-(2.9) with the same data ; ;J: Then E E
1 E
2
satises
E(0;z)=0; _
E(0;z)=0and
h
E;'i
V
;V +h
_
E;'i+
1
(E;')+h Z
t
0
(t s;)E(s;)ds;'i+h(f(E
1
) f(E
2 ));'i
+h Z
t
0
(t s;)(f(E
1
(s;)) f(E
2
(s;)))ds;'i+h^ d
dt (f(E
1
) f(E
2 ));'i
+c _
E(t;0)'(0) khE;'i=0 for every '2V: (3.37)
For arbitrarybut xed in (0;T) dene
(t) =
(
R
t
E()d fort < ;
0 fort ;
so
(T)=0 and
(t)2V for eacht 2[0;T]: We alsohave that
Z
0 h
E(t);
(t)i
V
;V +h
_
E(t);E(t)idt = Z
0 d
dt h
_
E(t);
(t)idt=0: (3.38)
Let '=
(t) in(3.37) and integrate overt from0 to :
Z
0
h _
E(t);E(t)i h _
E(t);
(t)i
1 (E(t);
(t)) h Z
t
0
(t s;)E(s;)ds;
(t)i
h(f(E
1
(t)) f(E
2 (t)));
(t)i h Z
t
0
(t s;)(f(E
1
(s;)) f(E
2
(s;)))ds;
(t)i
h^ d
dt (f(E
1
(t)) f(E
2 (t)));
(t)i c _
E(t;0)
(t)(0)+khE(t);
(t)i
)
dt=0:(3.39)
By usingthe relations
Z
0 h
_
E;
i+hE;Eidt= Z
0 d
dt hE;
idt =0 (3.40)
d
dt
1 (
(t);
(t))=2
1 (E(t);
0 n
c _
E(t;0)
(t)(0)+cE(t;0) 2
o
dt=
0 d
dt
cE(t;0)
(t)(0)dt=0 (3.42)
d
dt hk
(t);
(t)i=2hkE(t);
(t)i (3.43)
Z
0 (
h^ d
dt (f(E
1
) f(E
2 ));
(t)i+h^(f(E
1
) f(E
2
));E(t)i )
dt
= Z
0 d
dt
h^(f(E
1
) f(E
2 ));
(t)i
= h^(f(E
1
(0;)) f(E
2 (0;));
(0))i=0; (3.44)
we obtainfrom (3.39) that
1
2
kE()k 2
+ 1
2
1 (
(0);
(0))+
1
2 kh
(0);
(0)i+
Z
0 n
hE(t);E(t)i+cE(t;0) 2
Reh Z
t
0
(t s;)E(s;)ds;
(t)i
o
dt Re Z
0 n
h(f(E
1
) f(E
2 ));
(t)i
h Z
t
0
(t s;)(f(E
1
(s;)) f(E
2
(s;)))ds;
(t)i+h^(f(E
1
) f(E
2 ));Ei
o
dt=0:
Now
1
2
kE()k 2
+ 1
2
1 (
(0);
(0))
1
2 jhk
(0);
(0)ij+ Z
0 n
jhE(t);E(t)ij
+jh Z
t
0
(t s;)E(s;)ds;
(t)ij+jh(f(E
1
) f(E
2 ));
(t)ij
+jh Z
t
0
(t s;)(f(E
1
(s;)) f(E
2
(s;)))ds;
(t)ij+jh^(f(E
1
) f(E
2 ));Eij
o
dt:
(3.45)
From the denition of
we have that for eacht 2[0;T]
k
(t)k
2
Z
0
kE()kd
2
T Z
0
kE()k 2
d; (3.46)
so that
jhk
(0);
(0)ijkk
(0)k
2
=kT Z
0
kE()k 2
d: (3.47)
Weuse (3.47)and arguments similar tothose behind the estimate (2.16)to obtain
jh Z
t
0
(t s;)E(s;)ds;
(t)ij Z
t
0 L
3
kE()kd
k
(t)k
1
2 Z
t
0 L
3
kE()kd
2
+ 1
2 k
(t)k
2
K
1 Z
0
kE()k 2
for all t ; where K
1 =
2 (L
2
3
T +T): Next we estimate the terms in (3.45) containing the
nonlinear function f using the local Lipschitz property of f and the fact that E
1 ;E
2 are
uniformlybounded, i.e., inL 1
(0;T;V):Thus for t
jh(f(E
1
(t)) f(E
2 (t)));
(t)ijL
1
KkE(t)kk
(t)k
1
2 L
2
1 K
2
kE(t)k 2
+ 1
2 k
(t)k
2
1
2 L
2
1 K
2
kE(t)k 2
+ T
2 Z
0
kE()k 2
d (3.49)
and
jh Z
t
0
(t s;)(f(E
1
(s;)) f(E
2
(s;)))ds;
(t)ijL
3 K
Z
t
0
kE(s)kdsk
(t)k
K
2 Z
0
kE(k 2
d; (3.50)
where K
2 =
1
2 (L
2
3 K
2
T +T):Wealso have that
Z
0
jh^(f(E
1
(t)) f(E
2
(t)));E(t)ijdt Z
0 L
2
KkE(t)k 2
dt: (3.51)
Usingthe estimates (3.47)-(3.51)in (3.45) wend that forT t
1
2
kE()k 2
C Z
0
kE(t)k 2
dt; (3.52)
where C is a constant. By the Gronwall inequality this yields that E() 0 on (0,T), and
thus the solutionof (2.8)-(2.9) is unique.
4 Continuous Dependence
In this section we show that the weak solution of the system (2.8)-(2.9) depends
continu-ously oninitialconditions and the source term. More precisely, we prove that the mapping
(; ;J)!(E; _
E)iscontinuousfromV HH 1
(0;T;V
)toC(0;T;H)L 2
weak
(0;T;H):
To this end letus x (; ;J) 2V HH 1
(0;T;V
)and consider sequences n
! in
V, n
! in H and J n
!J in H 1
(0;T;V
): We denote the unique weak solution
corre-sponding todata ( n
; n
;J n
)by (E n
; _
E n
) for eachn. Our goal isto showthat E n
!E in
C(0;T;H)and _
E n
! _
E weaklyinL 2
(0;T;H)and(E; _
E)satises(2.8)withdata(; ;J):
We rst note that from the a priori estimate (2.22) we can conclude that the Galerkin
approximations fE
m
g alsosatisfy
k _
E
m k
2
L 2
(0;T;H)
+kE
m k
2
L 2
(0;T;V)
+2cTk _
E
m (;0)k
2
L 2
(0;T)
CT;
(4.53)
for each m, where the constant C is given as before and depends continuously on ; ;J:
Since we established the convergences E
m
! E; _
E
m !
_
E; _
E
m
L (0;T;V); L (0;T;H) and L (0;T);respectively, by the weak lowersemicontinuity of the
norms we obtain that E; the unique weak solution obtained in Section 1 also satises the
inequality above,i.e.,
k _
Ek 2
L 2
(0;T;H)
+kEk 2
L 2
(0;T;V)
+2cTk _
E(;0)k 2
L 2
(0;T)
CT: (4.54)
Now if we consider the weak solution E n
corresponding to data ( n
; n
;J n
) it satises
the same inequality (4.54) with C n
= C n
(T;f;g; n
; n
;J n
) instead of C: However, since
( n
; n
;J n
) ! (; ;J); there exists ~
C = ~
C(T;f;g) independent of n such that C n
~
C
for eachn. Thus
fE n
g is bounded in L 2
(0;T;V) (4.55)
f _
E n
g is bounded in L 2
(0;T;H) and (4.56)
f _
E n
(;0)g is bounded in L 2
(0;T): (4.57)
We alsoknow that E n
2C(0;T;H) for all n. As in the proof of existence we can conclude
thatthereexists E2L 2
(0;T;V)suchthatforasuitablesubsequence(againdenotedby E n
)
E n
! E weakly in L 2
(0;T;V) (4.58)
_
E n
! _
E weakly in L 2
(0;T;H) (4.59)
_
E n
(;0) ! E(;0) weakly in L 2
(0;T): (4.60)
We can now proceed and nish the proof of the continuous dependence of weak solutions
exactlyas we did in the existence proof if wecan show the stronger convergence
E n
!E in C(0;T;H):
Notethat no otherproperty of the Galerkin approximations was used inthe previous proof
exceptthesefourconvergences andthefactthatE
m
satises(2.33). Weestablishthe
conver-genceE n
!E inC(0;T;H)usingtheArzela-Ascolitheoremagain. Followingthearguments
atthe end ofSection2 andusing the weaklower semicontinuityof normswe canargue that
for the weak solutionscorresponding to data( n
; n
;J n
); we have
kE n
(t)k 2
V
1
C
n
1
~
C:
Hence, we can conclude that fE n
(t)g isbounded inV.
We can alsosee that fE n
g isequicontinuous inC(0;T;H):
kE n
(t+t) E n
(t)k 2
=k Z
t+t
t _
E n
()dk 2
Z
t+t
t k
_
E n
()kd !
2
t
Z
t+t
t k
_
E n
()k 2
d t
~
C: (4.61)
By an application of the Arzela-Ascoli theorem we can show that fE n
g is relatively
com-pact in C(0;T;H), so it has a subsequence (again denoted by E n
) that converges to E
in C(0;T;H): Indeed, this argument reveals that any subsequence has in turn a
subse-quence converging tothe unique limitE and hence the originalsequence converges toE in
C(0;T;H):Now the proof of continuous dependence can be nishedin the same fashion as
The theoretical results in this paper are an initial eort in developing a foundation for
nonlinear electromagnetics and the needed computationalmethodsfor such. The potential
applications of nonlinear optics with light as an information carrier are widespread and
are the basis of optical information technology and the use of optical bers. They are of
increasing importance with the development of materials (e.g., GaAs, InP, KNbO
3
) which
possess outstanding nonlinear optical and electro-optical properties [7]. Our interests arise
fromthe potentialof usinginterrogatinginputpulsesinthe IR range(terahertz) inimaging
and more specically indetection algorithms.
Nonlinearopticaleectsareusuallydescribedthroughnonlinearpolarizationlaws[6]and
many of these laws can beformulated inthe framework of this paperwhere wetreat rather
general nonlinearities. These include locally polynomial nonlinearities (e.g., so-called third
order and higher eects [6, 7]) that are bounded as the magnitude of the E eld saturates
polarizationmechanisms(e.g.,the material"freezes"dielectricallyathigh eldvalues). The
assumption that the nonlinearity is increasing with the magnitude of the E eld is again a
realistic one.
Though our treatment is for full wave propagation (as opposed to the popular paraxial
approximate models [6] found widely in the literature), even here there are some tacit
as-sumptions that may result inlimiting approximations to phenomena that actually occur in
nonlinearmaterials. Specically,theone-dimensionalmodelformulatedinSection2depends
onthetacit assumptionthat thepolarizationeld
P inthe dielectricremainsparalleltothe
electriceld
E:Even then,theusualMaxwellequationr
D=0alongwiththeconstitutive
law
D=
0
r
E+f
1 (P)
P neednotresultinr
E =0:Thisisimportantinderivingthesecond
orderformofMaxwell'sequationwheretheidentityrr
E =r(r
E) r 2
E resultsin
the simpleLaplacianonlyifr
E =0oroneassumesthis termisnegligibleasoftendonein
nonlinearoptics([6]p.54-60). Evenwithsomeobviousinadequacies,themodelconsideredin
this paperdoes facilitatetreatmentof a numberof mechanismsof interest, includingDebye
and Lorentzmaterialsthat are driven by nonlinear E eld inputs, e.g., _
P + 1
P =f(E):
Acknowledgment
The authors are grateful to Dr. Richard Albanese for numerous technical discussions,
sug-gestions and encouragement to pursue eorts in nonlinear electromagnetics. This research
wassupportedinpartbythe USAirForceOÆceofScientic Researchundergrant
AFOSR-F49620-1-00-0026.
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