Inverse scattering problem for a wave equation with absorption
We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator.
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| Date: | 2007 |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509499934113792 |
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| author | Tarasova, E. V. Тарасова, Е. В. Тарасова, Е. В. |
| author_facet | Tarasova, E. V. Тарасова, Е. В. Тарасова, Е. В. |
| author_sort | Tarasova, E. V. |
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| description | We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. |
| first_indexed | 2026-03-24T02:42:05Z |
| format | Article |
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UDK 517.9
E. V. Tarasova (Evrop. un-t, Ûytomyr. fyl.)
OBRATNAQ ZADAÇA RASSEQNYQ
DLQ VOLNOVOHO URAVNENYQ S POHLOWENYEM
We prove a theorem on the uniqueness in the inverse scattering problem for the wave equation with
absorption. We also develop an algorithm for the solution of this problem on the basis of the given
scattering operator.
Dovedeno teoremu [dynosti v obernenij zadaçi rozsiqnnq dlq xvyl\ovoho rivnqnnq z
pohlynannqm ta vkazano alhorytm rozv’qzku ci[] zadaçi za zadanym operatorom rozsiqnnq.
V nastoqwej rabote yzuçaetsq nestacyonarnaq obratnaq zadaça rasseqnyq dlq
volnovoho uravnenyq s pohlowenyem vyda
∂
∂
∂
∂
∂
∂
2
2
2
2 0u
t
u
x
g x t u
t
− + =( , ) , (1)
hde x, t ∈ −∞ +∞( ; ) , u ( x, t ) — yskomoe reßenye, koπffycyent g ( x, t ) opys¥vaet
pohlowagwyesq svojstva sred¥. Otmetym, çto prqm¥e y obratn¥e zadaçy ras-
seqnyq dlq volnov¥x uravnenyj yzuçen¥ dostatoçno polno. V odnomernom
sluçae obratnaq zadaça rasseqnyq dlq uravnenyq Íturma – Lyuvyllq polno-
st\g reßena V.8A.8Marçenko [1] y M.8H.8Krejnom [2]. Dlq trexmernoho stacyo-
narnoho uravnenyq Íredynhera rqd postanovok obratn¥x zadaç yssledovan
G.8M.8Berezanskym [3]. Polnoe yssledovanye obratnoj zadaçy rasseqnyq dlq
trexmernoho uravnenyq Íredynhera v¥polneno L.8D.8Fadeev¥m [4]. P.8D.8Lak-
som y R.8S.8Fyllypsom [5] rassmotrena zadaça ob opredelenyy rasseyvagweho
obæekta po operatoru rasseqnyq dlq volnovoho uravnenyq vo vneßnosty ohra-
nyçennoj oblasty. L.8P.8NyΩnykom y eho uçenykamy [6 – 11] vperv¥e yzuçen¥
nestacyonarn¥e prqm¥e y obratn¥e zadaçy rasseqnyq dlq volnovoho uravnenyq
s potencyalom, zavysqwym ot vremeny.
V xarakterystyçeskyx peremenn¥x uravnenye (1) ymeet vyd
∂
∂ ∂
2
0u
x y
Au x y+ =( )( , ) , (2)
hde operator A opredelqetsq ravenstvom
( )( , )Au x y = g x y u x y u x yx y( , ) ( , ) ( , )′ + ′( ).
Pust\ v uravnenyy (2) funkcyq g ( x, y ) qvlqetsq kompleksnoznaçnoj yzme-
rymoj funkcyej po peremenn¥m x, y, kotoraq udovletvorqet uslovyqm
sup ( , )
x
g x y dy
−∞
+∞
∫ < ∞ , sup ( , )
y
g x y dx
−∞
+∞
∫ < ∞ . (3)
V dal\nejßem budem rassmatryvat\ tol\ko te reßenyq u ( x , y ) uravne-
nyq8(2), kotor¥e qvlqgtsq neprer¥vno dyfferencyruem¥my funkcyqmy yz
prostranstva C E1 2( ) y udovletvorqgt uravnenyg (2) v sm¥sle teoryy obob-
wenn¥x funkcyj. Takye reßenyq budem naz¥vat\ dopustym¥my.
V rabote [12] yzuçena zadaça rasseqnyq dlq uravnenyq (2). Dokazano, çto
operator rasseqnyq S dlq uravnenyq (2) v prostranstve L C2
2( , ; )− ∞ + ∞ sovpa-
daet s operatorom rasseqnyq dlq system¥ uravnenyj Dyraka vyda
∂
∂
ϕ
x
x y1( , ) + g x y( , )( )ϕ ϕ1 2+ = 0,
(4)
∂
∂
ϕ
y
x y2( , ) + g x y( , )( )ϕ ϕ1 2+ = 0.
© E. V. TARASOVA, 2007
1580 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
OBRATNAQ ZADAÇA RASSEQNYQ DLQ VOLNOVOHO URAVNENYQ … 1581
Operator rasseqnyq opredelqetsq formuloj
Sϕ ϕ
− +
= , (5)
hde ϕ
±
— asymptotyky na beskoneçnosty dopustymoho reßenyq system¥ (4):
ϕ1( , )x y = ϕ
∓
1 1( ) ( )y o+ , x → ∞∓ ,
(6)
ϕ2( , )x y = ϕ
∓
2 1( ) ( )x o+ , y → ∞∓ ,
ymegwye sm¥sl profylej padagwej ϕ
−
= col( , )ϕ ϕ
− −
1 2 y rasseqnnoj ϕ
+
=
= col( , )ϕ ϕ
+ +
1 2 voln. Pry πtom dokazano, çto operator rasseqnyq S dlq uravne-
nyq (2) qvlqetsq ohranyçenn¥m, lynejn¥m, matryçn¥m operatorom v prost-
ranstve vektor-funkcyj L C2
2( , ; )−∞ +∞ , dlq kotoroho suwestvuet ohranyçen-
n¥j obratn¥j operator S−1
, y
S = P + F, S P G− −= +1 1
, (7)
hde F, G — matryçn¥e yntehral\n¥e operator¥, a P — matryçn¥j dyahonal\-
n¥j operator umnoΩenyq v prostranstve vektor-funkcyj L C2
2( , ; )−∞ +∞ vyda
P P P= { }diag 11 22, , (8)
πlementamy kotoroho qvlqgtsq operator¥ umnoΩenyq
( )( , )P f x y11 = p y f x y11( ) ( , ) ,
( )( , )P f x y22 = p y f x y22( ) ( , )
na funkcyy
p y g y d11( ) exp ( , )= −
−∞
+∞
∫ ξ ξ , p x g x d22( ) exp ( , )= −
−∞
+∞
∫ η η . (9)
Otmetym, çto slahaem¥e v formulax (7) opredelqgtsq odnoznaçno po zadan-
nomu operatoru rasseqnyq S, t.8e. ymeet mesto sledugwaq lemma.
Lemma 1. Esly yzvesten operator rasseqnyq S = P + F dlq volnovoho
uravnenyq (2), to operatorn¥e slahaem¥e P y F opredelqgtsq odnoznaçno.
Dokazatel\stvo. Rassmotrym v kaçestve profylej padagwyx voln vek-
tor-funkcyy
ϕ ε1( , )y = col ω ε( ),y −( )1 0 , ϕ ε2( , )x = col 0 1, ( )ω εx −( ),
hde ω ( s ) — funkcyq s kompaktn¥m nosytelem, prynadleΩawaq klassu C0
∞
,
ω (0) = 1, ω∫ ( )s ds = 1. Tohda
S yϕ τ ε1( , )− = col S y S y11
1
21
1ω τ ε ω τ ε( ) , ( )−( ) −( )( )− −
y
lim ( , )
ε τϕ τ ε
→ =−( )
0
1
1
S y y = lim ( )
ε τω τ ε
→
−
=−( )( )
0
11
1S y y .
Poskol\ku
S y y11
1ω τ ε τ( )−( )−
= = p y y y11
1( ) ( )ω τ ε τ−( )−
= + F y y d
y
y11
1+
−∞
−
=∫ −( )( , ) ( )η ω η ε η τ ,
uçyt¥vaq svojstva funkcyy ω ( s ), poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1582 E. V. TARASOVA
lim ( , )
ε τϕ τ ε
→ =−( )
0
11 1S y y = p y11( ) . (10)
Analohyçno dokaz¥vaetsq, çto
lim ( , )
ε τϕ τ ε
→ =−( )
0
22 2S x x = p x22( ) . (11)
Slahaemoe F opredelqetsq yz formul¥ F = S – P.
Obratnaq zadaça rasseqnyq dlq uravnenyq (2) sostoyt v opredelenyy koπf-
fycyenta g ( x, y ) po yzvestnomu operatoru rasseqnyq S.
Dlq reßenyq πtoj zadaçy yspol\zuem rezul\tat¥ L.8P.8NyΩnyka [10] po ob-
ratnoj zadaçe rasseqnyq dlq kanonyçeskoj system¥ uravnenyj Dyraka vyda
∂ψ
∂
ψ1
12 2 0
x
u x y x y+ =( , ) ( , ) ,
(12)
∂ψ
∂
ψ2
21 1 0
y
u x y x y+ =( , ) ( , ) ,
k kotoroj moΩno perejty ot system¥ (4) s pomow\g zamen¥
ψ1( , )x y = ϕ ξ ξ1( , )exp ( , )x y g y d
x
−∞
∫
,
(13)
ψ2( , )x y = ϕ η η2( , )exp ( , )x y g x d
y
−∞
∫
.
Pry πtom
u x y12( , ) = g x y x y( , )exp ( , )α{ } ,
u x y21( , ) = g x y x y( , )exp ( , )−{ }α , (14)
α( , )x y = g y d
x
( , )ξ ξ
−∞
∫ – g x d
y
( , )η η
−∞
∫ .
Otmetym, çto uslovye (3) na koπffycyent g x y( , ) obespeçyvaet prynad-
leΩnost\ u x y12( , ) y u x y21( , ) prostranstvu L E2
2( ).
Yz sootnoßenyj (14) poluçaem g x y2 ( , ) = u x y u x y12 21( , ) ( , ) y
g x y( , ) = signu x y u x y u x y12 12 21( , ) ( , ) ( , ) . (15)
Suwestvuet tesnaq svqz\ meΩdu operatorom rasseqnyq S dlq volnovoho
uravnenyq (2) y operatorom rasseqnyq SN dlq system¥ Dyraka (12).
Lemma 2. Operator rasseqnyq S dlq volnovoho uravnenyq s pohlowenyem
(2) y obratn¥j k nemu operator S−1
svqzan¥ s operatorom rasseqnyq SN dlq
system¥ Dyraka (12) formulamy
S P SN = −1
, S S PN
− −=1 1
, (16)
hde P — matryçn¥j operator umnoΩenyq, opredelenn¥j v (8), (9).
Dokazatel\stvo. Sohlasno rabote [10], operator rasseqnyq SN dlq syste-
m¥ Dyraka (12) y obratn¥j k nemu operator SN
−1
qvlqgtsq matryçn¥my opera-
toramy v prostranstve L C2
2( , ; )−∞ +∞ :
S I FN
N= + , S I GN
N− = +1
,
y svqz¥vagt asymptotyky dopustymoho reßenyq ψ i x y( , ) , i = 1, 2, πtoj syste-
m¥ na beskoneçnosty
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
OBRATNAQ ZADAÇA RASSEQNYQ DLQ VOLNOVOHO URAVNENYQ … 1583
ψ1( , )x y = a y o1 1( ) ( )+ , x → −∞,
ψ2( , )x y = a x o2 1( ) ( )+ , y → −∞ ,
(17)
ψ1( , )x y = b y o1 1( ) ( )+ , x → +∞,
ψ2( , )x y = b x o2 1( ) ( )+ , y → +∞ ,
t.8e.
S a bN = . (18)
Perexodq k predelu v formulax (13) y uçyt¥vaq (6), (17), poluçaem
a y y1 1( ) ( )=
−
ϕ , a x x2 2( ) ( )=
−
ϕ ,
b y y p y1 1 11
1( ) ( ) ( )=
+ −ϕ , b x x p x2 2 22
1( ) ( ) ( )=
+ −ϕ ,
otkuda sleduet formula (16).
Yz (16) poluçaem formul¥, svqz¥vagwye qdra yntehral\n¥x operatorov SN
y S:
F y sN
12 ( , ) = p y F y s11
1
12
− ( ) ( , ), F x sN
21 ( , ) = p x F x s22
1
21
− ( ) ( , ) ,
(19)
G x sN
21( , ) = p s G x s11 21( ) ( , ), G y sN
12( , ) = p s G y s22 12( ) ( , ).
Ymeet mesto teorema edynstvennosty v obratnoj zadaçe rasseqnyq dlq urav-
nenyq (2).
Teorema 1. Po zadannomu operatoru rasseqnyq S koπffycyent pohlowe-
nyq g x y( , ) v volnovom uravnenyy (2) opredelqetsq odnoznaçno. Reßenye
πtoj obratnoj zadaçy moΩet b¥t\ poluçeno s pomow\g sledugweho alhoryt-
ma:
1) po operatoru rasseqnyq S dlq volnovoho uravnenyq (2), sohlasno for-
mulam (10), (11), naxodym funkcyy p y11( ) , p x22( ) , t.1e. matryçn¥j operator
umnoΩenyq P;
2) yspol\zuq formul¥ (19), stroym operator rasseqnyq SN dlq kanony-
çeskoj system¥ uravnenyj Dyraka (12);
3) po operatoru SN reßaem obratnug zadaçu rasseqnyq dlq system¥ urav-
nenyj Dyraka (12), sohlasno alhorytmu rabot¥ [10], y naxodym funkcyy
u x y12( , ) , u x y21( , );
4) po funkcyqm u x y12( , ) , u x y21( , ) , sohlasno formule (15), naxodym koπf-
fycyent pohlowenyq g ( x, y ) v volnovom uravnenyy (2).
Dokazatel\stvo. Pust\ zadan operator rasseqnyq S dlq volnovoho urav-
nenyq (2). Tohda po nemu, sohlasno lemme 1, odnoznaçno opredelqetsq operator
umnoΩenyq P. Po formulam (19) stroym operator rasseqnyq SN dlq sootvet-
stvugwej kanonyçeskoj system¥ uravnenyj Dyraka (12). Dalee, sohlasno rabo-
te [10], po operatoru SN sostavlqem system¥ osnovn¥x uravnenyj vyda
B x y s11
+ ( , , ) + B x y G s d
x
N
12 21
−
∞
∫ ( , , ) ( , )ξ ξ ξ = 0, y s> ,
B x y s12
− ( , , ) + B x y F s d
y
N
11 12
+
−∞
∫ ( , , ) ( , )ξ ξ ξ + F y sN
12 ( , ) = 0, s y> ,
(20)
B x y s21
+ ( , , ) + B x y G s d
x
N
22 21
−
∞
∫ ( , , ) ( , )ξ ξ ξ + G x sN
21( , ) = 0, y s> ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
1584 E. V. TARASOVA
B x y s22
− ( , , ) + B x y F s d
y
N
21 12
+
−∞
∫ ( , , ) ( , )ξ ξ ξ = 0, s x> ,
kotor¥e svqzan¥ s koπffycyentamy system¥ Dyraka (12) formulamy
B x y x12 0− +( , ; ) = u x y12( , ), B x y y21 0+ −( , ; ) = u x y21( , ) . (21)
Dalee, koπffycyent volnovoho uravnenyq g ( x, y ) naxodytsq po formule (15).
Zameçanye 1. Yz dokazatel\stva teorem¥ 1 vydno, çto v opredelenyy koπf-
fycyenta g ( x, y ) uçastvugt tol\ko try funkcyy : F x y12( , ), G x y21( , ) (qdra
yntehral\n¥x operatorov F12, G21) y ln ( )p y11 = −
−∞
+∞
∫ g y d( , )ξ ξ . Poπtomu es-
testvenno πty funkcyy naz¥vat\ dann¥my rasseqnyq dlq volnovoho uravne-
nyq8(2).
Yz teorem¥ 1 sleduet, çto po yzvestn¥m dann¥m rasseqnyq koπffycyent
g ( x, y ) v volnovom uravnenyy (2) naxodytsq odnoznaçno.
Zameçanye 2. Esly koπffycyent g ( x, y ) v volnovom uravnenyy (2) qvlq-
etsq çysto mnymoj velyçynoj
g x y( , ) = −g x y( , ) , (22)
to yz (14) ymeem
u x y12( , ) = −u x y21( , ),
t.8e. potencyal v kanonyçeskoj systeme uravnenyj Dyraka (12) qvlqetsq koso-
symmetryçeskym.
Tohda, kak pokazano v rabote [10], operator rasseqnyq SN (18) qvlqetsq uny-
tarn¥m operatorom ( )S SN N
− ∗=1
v prostranstve L C2
2( , ; )−∞ +∞ y odnoznaçno
opredelqetsq po operatoru otraΩenyq FN
21 .
Takym obrazom, çysto mnym¥j koπffycyent pohlowenyq g ( x, y ) v volno-
vom uravnenyy (2) odnoznaçno opredelqetsq po yzvestn¥m funkcyqm F x y12( , )
y p y11( ) , lybo po funkcyqm F x y21( , ) y p x22( ) , hde funkcyy p11 y p22 op-
redelqgtsq formulamy (9), a F x y12( , ) y F x y21( , ) qvlqgtsq qdramy ynteh-
ral\n¥x operatorov — πlementov matryçnoho operatora rasseqnyq S.
1. Marçenko V. A. Operator¥ Íturma – Lyuvyllq y yx pryloΩenyq. – Kyev: Nauk. dumka,
1977. – 331 s.
2. Krejn M. H. Reßenye obratnoj zadaçy Íturma – Lyuvyllq // Dokl. AN SSSR. – 1951. – 76,
# 1. – S. 21 – 24.
3. Berezanskyj G. M. O teoreme edynstvennosty v obratnoj zadaçe spektral\noho analyza
dlq uravnenyq Íredynhera // Tr. Mosk. mat. o-va. – 1958. – 7. – S. 3 – 62.
4. Fadeev L. D. Obratnaq zadaça kvantovoj teoryy rasseqnyq // Ytohy nauky y texnyky. Ser.
Sovrem. probl. matematyky / VYNYTY. – 1974. – 3. – S. 93 – 180.
5. Laks P., Fylyps R. Teoryq rasseqnyq. – M.: Myr, 1971. – 312 s.
6. NyΩnyk L. P. Zadaça rasseqnyq pry nestacyonarnom vozmuwenyy // Dokl. AN SSSR. –
1960. – 132, # 1. – S. 40 – 43.
7. NyΩnyk L. P. Korrektnaq zadaça rasseqnyq bez naçal\n¥x dann¥x dlq volnovoho urav-
nenyq // Ukr. mat. Ωurn. – 1968. – 20, # 6. – S. 802 – 813.
8. NyΩnyk L. P. Obratnaq zadaça nestacyonarnoho rasseqnyq // Dokl. AN SSSR. – 1971. –
196, # 5. – S. 1016 – 1019.
9. NyΩnyk L. P. Obratnaq zadaça rasseqnyq na nestacyonarnom potencyale v trexmernom
prostranstve // Metod¥ funkcyonal\noho analyza v zadaçax matematyçeskoj fyzyky. –
Kyev: Yn-t matematyky AN USSR, 1978. – S. 38 –862.
10. NyΩnyk L. P. Obratn¥e zadaçy rasseqnyq dlq hyperbolyçeskyx uravnenyj.88– Kyev: Nauk.
dumka, 1991. – 232 s.
11. Fam Loj Vu. Obratnaq nestacyonarnaq zadaça rasseqnyq dlq vozmuwennoho uravnenyq
strun¥ na vsej osy // Ukr. mat. Ωurn. – 1980. – 32, # 5. – S.8630 – 637.
12. Tarasova E. V. Zadaça rasseqnyq dlq volnovoho uravnenyq s pohlowenyem // Tam Ωe. –
2004. – 56, # 2. – S. 221 – 227.
Poluçeno 06.02.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 11
|
| id | umjimathkievua-article-3413 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:05Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/77/317604d4f77caf7b5cb4b70283dd3b77.pdf |
| spelling | umjimathkievua-article-34132020-03-18T19:53:28Z Inverse scattering problem for a wave equation with absorption Обратная задача рассеяния для волнового уравнения с поглощением Tarasova, E. V. Тарасова, Е. В. Тарасова, Е. В. We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. Доведено теорему єдиності в оберненій задачі розсіяння для хвильового рівняння з поглинанням та вказано алгоритм розв'язку цієї задачі за заданим оператором розсіяння. Institute of Mathematics, NAS of Ukraine 2007-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3413 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 11 (2007); 1580–1584 Український математичний журнал; Том 59 № 11 (2007); 1580–1584 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3413/3569 https://umj.imath.kiev.ua/index.php/umj/article/view/3413/3570 Copyright (c) 2007 Tarasova E. V. |
| spellingShingle | Tarasova, E. V. Тарасова, Е. В. Тарасова, Е. В. Inverse scattering problem for a wave equation with absorption |
| title | Inverse scattering problem for a wave equation with absorption |
| title_alt | Обратная задача рассеяния для волнового уравнения с поглощением |
| title_full | Inverse scattering problem for a wave equation with absorption |
| title_fullStr | Inverse scattering problem for a wave equation with absorption |
| title_full_unstemmed | Inverse scattering problem for a wave equation with absorption |
| title_short | Inverse scattering problem for a wave equation with absorption |
| title_sort | inverse scattering problem for a wave equation with absorption |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3413 |
| work_keys_str_mv | AT tarasovaev inversescatteringproblemforawaveequationwithabsorption AT tarasovaev inversescatteringproblemforawaveequationwithabsorption AT tarasovaev inversescatteringproblemforawaveequationwithabsorption AT tarasovaev obratnaâzadačarasseâniâdlâvolnovogouravneniâspogloŝeniem AT tarasovaev obratnaâzadačarasseâniâdlâvolnovogouravneniâspogloŝeniem AT tarasovaev obratnaâzadačarasseâniâdlâvolnovogouravneniâspogloŝeniem |