On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type
It is shown that the solutions of the Sine Gordon equation with a source of the integral type can be found by the method of the inverse scattering problem for the Dirac type operator on the real line.
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| Опубліковано в: : | Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type / A.B. Khasanov, G.U. Urazboev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 287-298. — Бібліогр.: 11 назв. — англ. |
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Khasanov, A.B. Urazboev, G.U. 2016-10-01T13:31:16Z 2016-10-01T13:31:16Z 2006 On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type / A.B. Khasanov, G.U. Urazboev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 287-298. — Бібліогр.: 11 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106620 It is shown that the solutions of the Sine Gordon equation with a source of the integral type can be found by the method of the inverse scattering problem for the Dirac type operator on the real line. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type Article published earlier |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type Khasanov, A.B. Urazboev, G.U. |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type |
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On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type |
| title_sort |
on the sine-gordon equation with a self-consistent source of the integral type |
| author |
Khasanov, A.B. Urazboev, G.U. |
| author_facet |
Khasanov, A.B. Urazboev, G.U. |
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2006 |
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English |
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Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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It is shown that the solutions of the Sine Gordon equation with a source of the integral type can be found by the method of the inverse scattering problem for the Dirac type operator on the real line.
|
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106620 |
| citation_txt |
On the Sine-Gordon Equation with a Self-Consistent Source of the Integral Type / A.B. Khasanov, G.U. Urazboev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 287-298. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
AT khasanovab onthesinegordonequationwithaselfconsistentsourceoftheintegraltype AT urazboevgu onthesinegordonequationwithaselfconsistentsourceoftheintegraltype |
| first_indexed |
2025-11-25T12:08:14Z |
| last_indexed |
2025-11-25T12:08:14Z |
| _version_ |
1850511975054835712 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 3, pp. 287�298
On the Sine�Gordon Equation with a Self-Consistent
Source of the Integral Type
A.B. Khasanov, G.U. Urazboev
Urgench State University
14 H. Olimjan Str., Urgench, 740000, Uzbekistan
E-mail:ahasanov2002@mail.ru
Received May 27, 2004
It is shown that the solutions of the Sine�Gordon equation with a source
of the integral type can be found by the method of the inverse scattering
problem for the Dirac type operator on the real line.
Key words: Sine�Gordon equation, inverse scattering method, Jost solu-
tions, scattering data.
Mathematics Subject Classi�cation 2000: 37K40, 37K15, 35Q53, 35Q55.
1. Introduction
In this paper we consider the problem of integration of the following system
of equations 8<
: uxt = sinu+
1R
�1
�
�
2
1 � �
2
2
�
d� ;
L� = ��;
(1)
u(x; 0) = u0(x); x 2 R; (2)
where L(t) = i
�
d
dx
ux
2
ux
2
� d
dx
�
, ux =
@u(x;t)
@x
; uxt =
@
2
u(x;t)
@x@t
; and u0(x)
(�1 < x <1) is a function satisfying the conditions:
1) u0(x) � 0(mod 2�) as jxj ! 1;
1Z
�1
�
(1 + jxj)
��u00(x)��+ ��u000(x)��� dx <1;
(3)
2) the operator L(0) does not have the points of spectral singularity (see
[6]) and has only simple eigenvalues �1(0); �2(0); : : : ; �N (0).
c
A.B. Khasanov, G.U. Urazboev, 2006
A.B. Khasanov, G.U. Urazboev
We assume that the vector function � = (�1(x; �; t); �2(x; �; t))
T is a solution
of the equation L� = �� satisfying the condition
�! A (�; t)
�
exp(�i�x)
exp(i�x)
�
as x!1; (4)
where A(�; t) is a continuous function satisfying the condition
A(��; t) = A(�; t);
1Z
�1
jA(�; t)j2d� <1; (5)
for all nonnegative values of t.
We assume that the solution u(x; t) of the problem (1)�(5) exists, possesses
the required smoothness, and tends to its limits su�ciently rapidly as x! �1,
i.e., for all t � 0 it satis�es the condition
u(x; t) � 0(mod 2�) as jxj ! 1;
1R
�1
((1 + jxj) jux(x; t)j+ juxx(x; t)j) dx <1:
(6)
The main objective of this paper is to derive representations for the solutions
u(x; t), �(x; �; t) within the framework of the inverse scattering method for L(t)
operator.
The full description of the solutions of the Sine�Gordon equation without
sources was given in [1�2].
The scattering problem for L(t) operator was studied in the papers by
V.E. Zakharov, A.B. Shabat [3], L.P. Nizhnik, Fam Loy Woo [4], I.S. Frolov
[5], A.B. Khasanov [6] and in many others.
Note that the similar problem for the KdV equation was considered
in the paper [7]. In the V.K. Mel'nikov's paper [8] there was obtained
evolution of the scattering dates for the selfadjoint Dirac type operator with
the potential which is a solution of the NLS equation with the integral type
source. Notice however that in our case operator L(t) is not self-adjoint. As
it is well known, under the condition (6) the not self-adjoint operator L(t)
has a �nite number of complex eigenvalues (in general multiple). Moreover,
operator L(t) may have a �nite number of real points of spectral singularity.
The continuous spectrum of the operator L(t) �lls up the real line, i.e.,
�ess(L(t)) = (�1; 1). For simplicity we suppose that operator L(t) has
a �nite number of simple complex eigenvalues, and does not have points of
singular spectrum.
288 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Sine�Gordon Equation...
2. Scattering Problem for Zakharov�Shabat Eigenvalue Problem
In this section we present some facts from the theory of the direct and
inverse scattering problems for the operator L(t) (for example, see [9]). For
a while in this section we omit the dependence of functions on t.
We consider the eigenvalue problem�
v1x + i�v1 = u0(x)v2
v2x � i�v2 = �u0(x)v1;
(7)
on the interval �1 < x <1. The potential u0(x) is assumed to satisfy the
condition
u(x) � 0(mod 2�) as jxj ! 1;
1Z
�1
((1 + jxj) ju0(x)j) dx <1: (8)
We de�ne the Jost solution of the problem (7)�(8) with the following
asymptotic values
' �
�
1
0
�
e�i� x
�' �
�
0
�1
�
ei� x
9>>=
>>; as x! �1;
�
�
0
1
�
ei� x
� �
�
1
0
�
e�i� x
9>>=
>>; as x!1:
For real � the pairs of functions f'; �'g and
�
; �
are the pairs of
linearly independent solutions of (7), and therefore
' = a(�) � + b(�) ; �' = ��a(�) +�b(�) � ; (9)
where a(�) = W f'; g � '1 2 � '2 1, b(�) = W
�
� ; '
, a(�)a(��) +
b(�)b(��) = 1.
For real � the coe�cient b(�) has the following asymptotic b(�) = O
�
1
j�j
�
as j�j ! 1, Im� = 0. The coe�cient a(�) (�a(�)) can be analytically
extended into the upper (lower) half-plane Im � > 0 (Im� < 0). The
function a(�) has the asymptotic a(�) = 1 +O
�
1
j�j
�
as j�j ! 1, Im� � 0.
Besides, in the half-plane Im � > 0 (Im� < 0) the function a(�) (�a(�))
has a �nite number of zeros at the points �k
�
��k
�
, and these points are the
eigenvalues of the operator
L = i
d
dx
u
0(x)
2
u
0(x)
2
�
d
dx
!
;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 289
A.B. Khasanov, G.U. Urazboev
so that '(x; �k) = Ck (x; �k) ( �'(x; �k) = �Ck � (x; �k)), k = 1; 2; : : : ; N . It is
clear that the function 'k � '(x; �k) is an eigenfunction of the operator L
corresponding to the eigenvalue �k.
We assume that the operator L does not have multiple eigenvalues. The
requirement of absence of the points of spectral singularity of the operator
L(t) means the absence of real zeros of function a(�). The class of the
potentials satisfying a(�) 6= 0 as � 2 R1 is not empty. For example, this
class contains �unre�ected� potentials, i.e., potential for which b(�) = 0. In
this case the equation a(�)a(��) = 1, � 2 R1 is valid.
We have the following integral representation for the function ' [9]
=
�
0
1
�
ei� x +
1Z
x
K (x; s) ei� sds; (10)
where the kernel K (x; s) =
�
K1 (x; s)
K2 (x; s)
�
does not depend on � and is
related to the potential u(x) by the formulae
u0 (x) = 4K1 (x; x) ; (u0 (x))
2
= 8
dK2 (x; x)
dx
: (11)
Components K1(x; y), K2(x; y) of the kernel K (x; y) in the representa-
tion (10), for y > x are solutions of the integral Gelfand�Levitan�Marchenko
equations
K1(x; y)� F (x+ y) +
1R
x
1R
x
K1(x; z)F (z + s)F (s+ y)dsdz = 0;
K2(x; y) +
1R
x
F (x+ s)F (s+ y)ds+
1R
x
1R
x
K2(x; z)F (z + s)F (s+ y)dsdz = 0;
where F (x) = 1
2�
1R
�1
b(�)
a(�)
ei�xd� � i
NP
j=1
Cje
i�jx.
Now the potential can be expressed via K1 (x; y) by the formula (11).
The set of the quantities
n
r+ (�) = b(�)
a(�)
; �k; Ck; k = 1; 2; : : : ; N
o
is called
the scattering data for equations (7).
It is worthy to remark that the vector functions
hn (x) =
d
d�
('� Cn )
���� � = �n
_a (�n)
; n = 1; 2; : : : ; N; (12)
290 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Sine�Gordon Equation...
are solutions of the equations Lhn = �nhn and have the following asymp-
totics
hn � �Cn
�
0
1
�
ei�nx as x! �1;
hn �
�
1
0
�
e�i�nx as x!1:
(13)
According to (13) we obtain
Wf'n; hng � 'n1hn2 � 'n2hn1 = �Cn; n = 1; 2; : : : ; N: (14)
It is easy to see that the following statement is true.
Lemma 1. If Y (x; �) and Z (x; �) are solutions of the equations LY =
�Y and LZ = �Z, then
d
dx
(y1z2 � y2z1) = �i (� � �) (y1z2 + y2z1) ;
d
dx
(y1z1 + y2z2) = �i (� + �) (y1z1 � y2z2) :
3. Evolution of the Scattering Data
Let the potential u (x; t) of the problem (7) be a solution of the system
of equations 8<
: uxt = sinu+
1R
�1
(�21 � �22) d� ;
L� = ��:
(15)
We put G(x; t) =
1R
�1
(�21 � �22) d�. According to (4)
�(x; �; t) = A(�; t)
�
� (x; �; t) + (x; �; t)
�
;
and therefore, by using (9), as well as the asymptotic for the Jost solution
and a(�); b(�) and Riemann�Lebesgue lemma in each nonnegative t, we have
G(x; t) = o(1) as x ! �1. The �rst equation of (15) can be rewritten in
the form
uxt = sin u+G: (16)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 291
A.B. Khasanov, G.U. Urazboev
Lemma 2. If potential u (x; t) of the problem (7) is a solution of equation
(16), then the scattering data depend on t as
dr+
dt
= �
i
2�
r+ +
1
2a2
1Z
�1
�
G'2
2 +G'2
1
�
dx; (Im� = 0) ;
dCn
dt
=
0
@� i
2�n
+
1Z
�1
G
2
(hn2 n2 + hn1 n1) dx
1
ACn;
d�n
dt
=
i
1R
�1
(G'2
n2 +G'2
n1) dx
4
1R
�1
'n1'n2dx
; n = 1; 2; : : : ; N:
P r o o f. Here we use the method of [10] (see also [11]).
We set
A =
�
i cosu
4�
i sinu
4�
i sinu
4�
�
i cos u
4�
�
:
It is easy to see that
[L;A] � LA� AL = �i
�
0 sinu
2
sinu
2
0
�
: (17)
The operator L (t) depends on time t as a parameter and therefore
@L
@t
= i
�
0 uxt
2
uxt
2
0
�
: (18)
Comparing formulas (17) and (18) with the equation (16), we can see
that the equation (16) is identical to the operator relation
@L
@t
+ [L;A] = iR; (19)
where R =
�
0 G
2
G
2
0
�
.
Let ' (x; �; t) be the Jost solution of the equation
L' = �':
292 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Sine�Gordon Equation...
We di�erentiate this relation with respect to time
Lt'+ L't = �'t; (20)
and substitute Lt from (19) into (20). This results to
(L� �) ('t � A') = �iR': (21)
We seek the solutions of (21) in the form
't � A' = � (x) + � (x)': (22)
To �nd �(x) and �(x) we use the equation
M�x +M�x' = �R'; (23)
where
M =
�
1 0
0 �1
�
:
According to (9)
̂TM' = �'̂TM = a; ̂TM = '̂TM' = 0;
where '̂ =
�
'2
'1
�
.
Multiplying (23) by '̂T and ̂T we yield
�x =
'̂TR'
a
; �x = �
̂TR'
a
: (24)
On the basis of (6) and the asymptotic of the Jost solution we have
't � A'! �
i
4�
�
1
0
�
e�i�x as x! �1:
Therefore from (22) one gets
� (x)! �
i
4�
; � (x)! 0 as x! �1:
By solving (24) we obtain
� (x) =
1
a
xZ
�1
'̂TR'dx; � (x) = �
1
a
xZ
�1
̂TR'dx�
i
4�
:
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 293
A.B. Khasanov, G.U. Urazboev
Therefore the relation (22) can be rewritten in the form
't � A' =
1
a
xZ
�1
'̂TR'dx � +
0
@�1
a
xZ
�1
̂TR'dx�
i
4�
1
A': (25)
Using (9) we take the limit in (25) as x!1 and obtain
at = �
1Z
�1
̂TR'dx;
bt = �
i
2�
b +
1
a
1Z
�1
'̂TR'dx�
b
a
1Z
�1
̂TR'dx:
Consequently, for Im� = 0 we get
dr
+
dt
= �
i
2�
r
+ +
1
2a2
1Z
�1
�
G'
2
2 +G'
2
1
�
dx:
We di�erentiate the relation 'n = Cn n with respect to t
@'
@t
���� � = �n
+ @'
@�
���� � = �n
d�n
dt
=
dCn
dt
n + Cn
@
@t
���� � = �n
+ Cn
@
@�
���� � = �n
d�n
dt
; (26)
and substitute d
d�
('� Cn )
���� � = �n
from (12) into (26). This results in the
following formula:
@'n
@t
=
dCn
dt
n +Cn
@ n
@t
� _a (�n) hn
d�n
dt
; (27)
where @'n
@t
� @'
@t
���� � = �n
.
Similarly to the continuous spectrum case, by using (14) for the discrete spec-
trum, we have
@'n
@t
�A'n =
0
@� 1
Cn
xZ
�1
'̂
T
nR'ndx
1
Ahn +
0
@ 1
Cn
xZ
�1
ĥ
T
nR'n dx�
i
4�n
1
A'n:
294 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Sine�Gordon Equation...
Hence, according to (27), we have
dCn
dt
n + Cn
@ n
@t
� _a (�n)
d�n
dt
hn � CnA n
=
� 1
Cn
xR
�1
'̂
T
nR'ndx
!
hn +
1
Cn
xR
�1
ĥ
T
nR'n dx�
i
4�n
!
Cn n :
(28)
Using (13) we pass to the limit in (28), as x!1, and obtain
dCn
dt
=
0
@� i
2�n
+
1Z
�1
ĥ
T
nR ndx
1
ACn;
d�n
dt
=
1R
�1
'̂
T
n
R'n dx
Cn _a (�n)
:
Therefore
dCn
dt
=
0
@� i
2�n
+
1Z
�1
G
2
(hn2 n2 + hn1 n1) dx
1
ACn ;
d�n
dt
=
1R
�1
�
G'
2
n2 +G'
2
n1
�
dx
2Cn _a (�n)
:
Hence, according to the relation
_a (�n) = �
2i
Cn
1Z
�1
'n1'n2dx;
we have
d�n
dt
=
i
1R
�1
�
G'
2
n2 +G'
2
n1
�
dx
4
1R
�1
'n1'n2dx
:
Lemma 2 is proved.
Let in Lemma 2
G =
1Z
�1
�
�
2
1 � �
2
2
�
d�:
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 295
A.B. Khasanov, G.U. Urazboev
According to Lemma 1
1Z
�1
�
�
2
1(x; �) � �
2
2(x; �)
� �
'
2
1(x; �) + '
2
2(x; �)
�
dx
= i
2
lim
R!1
�
(�1(x;�)'1(x;�)+�2(x;�)'2(x;�))
2
�+�
+
(�1(x;�)'2(x;�)��2(x;�)'1(x;�))
2
���
����R
�R
:
By using (4), (5), (9) and the Riemann�Lebesgue lemma, we obtain
1Z
�1
�
G'
2
2 +G'
2
1
�
dx = 2ab
0
@�A2(�; t) + iV:p:
1Z
�1
A
2(�; t)
� + �
d�
1
A :
Similarly,
1Z
�1
�
G'
2
n2 +G'
2
n1
�
dx = 0;
1Z
�1
(Ghn2 n2 +Ghn1 n1) dx = 2i
1Z
�1
A
2(�; t)�a(�; t)a(�; t)
� + �n
d�:
By using Lemma 2 and the relation �a(�)a(�) = 1
1+r+(�)r+(��)
, we have the
following theorem
Theorem. If the functions u (x; t), �1(�; x; t), �2(�; x; t are solutions of the
problem (1)�(6), then the scattering data of the operator L (t) depend on t as
dr
+
dt
=
0
@� i
2�
+ �A
2(�; t) + iV:p:
1Z
�1
A
2(�; t)
� + �
d�
1
A r
+
; (Im� = 0) ;
dCn
dt
=
� i
2�n
+ i
1R
�1
A
2(�;t)
(1+r+(�;t)r+(��;t))(�+�n)
!
Cn;
d�n
dt
= 0; n = 1; 2; : : : ; N:
The above relations determine completely the evolution of the scattering data
for the operator L (t), which allows us to �nd the solutions of problem for (1)�(6)
by using the inverse scattering problem method.
In conclusion we consider the following example. Let
uj
t=0 = 4arctg
�
e
2x
�
; A(�; t) =
�
1 + �
2
�� 1
2 :
296 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Sine�Gordon Equation...
In this case r+(�; 0) = 0, �1(0) = i, C1(0) = �2i.
Therefore, by using the theorem
r
+(�; t) = 0; �1(t) = i; C1(t) = �2i exp
�
� � 1
2
t
�
:
According to the inverse scattering problem method
u(x; t) = 4arctg
�
exp
�
2x�
� � 1
2
t
��
;
�1(x; �) =
1p
�2+1
�
cos �x+
(1�e�2x+g)(cos �x�� sin �x
(1+�2)ch(2x�g)
�
+ ip
�2+1
�
� sin �x+
(1+e�2x+g)(sin �x+� cos �x
(1+�2)ch(2x�g)
�
;
�2(x; �) =
1p
�2+1
�
cos �x� (1+e�2x+g)(cos �x�� sin �x
(1+�2)ch(2x�g)
�
+ ip
�2+1
�
sin �x+
(1�e�2x+g)(sin �x+� cos �x
(1+�2)ch(2x�g)
�
;
where g(t) =
(��1) t
2
.
References
[1] V.E. Zakharov, L.A. Takhtajan L, and L.D. Faddeev, A Complete Description of
the Solutions of the Sine�Gordon Equation. � Dokl. Akad. Nauk USSR 219 (1974),
No. 6, 1334�1337. (Russian)
[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, Method for Solving the
Sine�Gordon Equation. � Phys. Rev. Lett. 30 (1973), No. 25, 1262�1264.
[3] V.E. Zakharov and A.B. Shabat, Exact Theory of Two-Dimensional Self-Focusing
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[7] V.K. Mel'nikov, Integration of the Korteweg-de Vries Equation with a Source. �
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[8] V.K. Mel'nikov, Integration of the Nonlinear Schrodinger Equation with a Source.
� Inverse Probl. 8 (1992), 133�147.
[9] Mark J. Ablowitz and Harley Segur, Solitons and the Inverse Scattering Transform.
� SIAM, Philadelphia, 1981.
[10] V.I. Karpman and E.M. Maslov, The Structure of Tails, Appearing under Soliton
Perturbations. � JETP 73 (1977), 2 (8), 537�559. (Russian)
[11] G.L. Lamb, Jr., Elements of Solution Theory. A Wiley Intersci. Publ. John Wiley
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298 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
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