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The inverse scattering problems for the hyperbolic equations and their application to non-linear integrable systems
Vol. 26 (1988)
THE
REPORTS
ON
MATHEMATICAL
No.
PHYSlCS
2
INVERSE SCATI’ERING PROBLEMS FOR THE HYPERBOLIC EQUATIONS AND THEIR APPLICATION TO NON-LINEAR INTEGRABLE SYSTEMS
L. P. NIZHNIK Institute
of Mathematics,
Academy
of Sciences of the Ukrainian
SSR, Kiev, USSR
(Recez’oed September 9, 1987) The results on two-dimensional inverse scattering problem for the two-component hyperbolic Dirac system of equations are formulated. The complete description of the scattering data is given and the algorithm of reconstructing the coefficients of equations from scattering data is formulated. The results have been applied to integration of the non-linear Schrodinger equation in two spatial dimensions by the inverse scattering method: the Cauchy problem is investigated, the exact soliton-like solutions are represented.
1. Introduction
In this paper the two-dimensional two-component the form
Dirac system of equations of
(1.1)
is considered under the assumption that the coefficients u1 (x, y), uz(x, y) are complex-valued measurable in x and y functions which are square integrable with respect to both variables, that is Ul, u2 E&(E2).
(1.2)
The direct and inverse scattering problems both for the Dirac system (1.1) in characteristic variables and for the two-dimensional nonstationary system
(1.3) *2 -=
w2 -7&+uzk
at
12611
0$1
L. P. NIZHNIK
262
have been studied in detail in [l-3]. In the papers [4-91 the results on the inverse problem for the system (1.1) have been applied to study the non-linear evolution equations in two spatial dimensions. The results of the papers quoted are used essentially in the following text. The inverse scattering problems for hyperbolic systems of n equations have been considered in [lO-151
2. Tbe scattering problem We shall consider the solutions of the system (1.1) such that for ‘any x the function tji (x, Y) is square integrable with respect to Y and for any y the function tiz(x, y) is square integrable with respect to x, that is Il/l(X, .), $2(-y Y)EJ%. Such solutions
(11/i, $J
(2.1)
of the system (1.1) will be called the admissible
solutions.
Let the coefficients of the system (1.1) satisfy the conditions (1.2). the admissible solution (til, ti2) the limits
THEOREM 2.1.
Then
for
$I(--co7
$I(+
$2 (x3 - a) = a2 (x),.
Y) = al(Y),
$zb,
00, Y) = bl (y),
(2.2)
+ ~0) = b,(x)
exist in L,. For one of the pairs (a,, 4, (bIy U, (aI, bJ, (k, e) of arbitrary functions belonging to the class L2 the admissible solution exists and is unique and this pair is the corresponding asymptotics at infinity for this solution. Proof:
If ($1, $2) is the admissible
$1(x, Y) = $l(O, Y)-
solution
of the system (1.1) then
j~~Cs,;M,(s, y)ds, b
(2.3) Ic/z(x, Y) = tizk Applying equations
adx,
s)IcIik
s)ds.
to the system (2.3) the Theorem on the solvability of the Volterra (see Appendix I), we obtain the boundedness of ll~$i(x, *)llL, and
lIti ( ‘9 YJIIL, uniformly $i(+a,
Q-
in x and y. Then (2.3) implies the existence
Y) = tbI(O, Y)-
‘~uI(s,
y)ti~h
Y)~S and $z(x,
of
+I;o)
0
=$,(x5
We now prove the second
part of the theorem.
0) - ‘Jmu,(z
s)$~(x,
s)ds.
Let .&, a2 ELM be given for
THE INVERSE SCATTERING
definiteness.
PROBLEMS
We shall consider
FOR THE HYPERBOLIC
the system
Ii/l@, Y) =%(Y)-
of integral
5 4(s7 -m
Y)$z(s,
EQUATIONS
263
equations y)ds,
(2.4)
According to Theorem A. I the solution of the system (2.4) exists, is unique’and II$1 (x7 *)I1Lz < C, jl$2 ( ., y)llL, d C. It follows from (2.4) that (11/I, 11/J is the admissible solution and 11/i(- CO,y) = a, (y), t,bz(x, - co) = a2 (x). When the functions bi, bz ELM are given the proof of this theorem to the analysis of the system
is reduced
$1 (x, Y) = bl (Y) + 7~1 (s, Y) $2 (sv Y) ds, x
$2(x,
Y) = bz(x)+
7~2(x,
(2.5) s)$I(x,
s)ds.
Y
Similarly we consider the cases when a,, b, EL, and bI, a, ELM are known. Theorem 2.1 gives the solution of the scattering problem for the system (1.1). The vector-function a = (a,, u2) defines the incident waves and b = (b,, b,) defines the scattered waves. The scattering problem consists in finding the solution of the system (1.1) when the incident wave is given. The operator S transforming the incident waves a into the scattered waves b is called the scattering operator b =Su.
We shall study the scattering 3. The transmutation
operator
(2.6)
in the space L, (- 00, + CC; E2).
operators
According to Theorem 2.1 the admissible solution is defined uniquely when the pair of asymptotics $r (+ cc, y), t,b2(x, + co) is given. The transmutation operators allow us to express explicitly the admissible solutions in terms of the asymptotics. LEMMA3.1. fying
The admissible solution of the system (1.1) with coefficients the c on d it’ions (1.2) can be represented in the form GI(x,Y)=~~(Y)+
j A,,(x,y;s)ui(s)ds+ -U3
sutis-
5 A,,(x.y;s)u,(s)ds, -CC
(3.1) 1(/2(x, Y) =
a,(x)+
j
A,,@,
-CO
Y; s)u,(s)ds+
-j -00
A,,@,
Y; s)u,(s)dk
’
264
L. P. NIZHNIK
Il/I(% Y) = b,(y)+
~~‘%I(X, Y; s)bt(s)ds+
~‘Mx,
Y
x
y; s)b2(s)ds, (3.2)
ti2k
Y) = J%(x)+ :A&
Y; s)br(s)ds+
:‘Mx,
$1(&Y)=
at(Y)+
$2(x9 Y) = b,(x)+
OD &(x9 ;
Y; s)a,(s)ds+
~&(x9
Y; s)a,(s)ds+
; &2(x, -52
Y) =b,(x)+
Y; s)&(s)& (3.3)
i &2(x,
y; s)b,(s)k
-03
Y
$I&
y; s)b,(s)&
x
Y
‘s 4,(x, -El
y; s)b,(s)ds+
T&,(x, x
Y; ~)a,(@& (3.4)
$2(x, Y) = az(y)+
j &,(x, -co
Y; s)br(s)ds+
where a,, u2, bl, b2 are the usymptotics (2.2) of Bij(X, y; s), i, j = 1, 2, are determined uniquely system (1.1) and umit the estimates JJ suPlA,j(x, x
u
~B,,(x, x
Y: s)+(s)&
solution. The kernels Aij(X, y; s), by the coeficients ul, u2 of the
JJ sup lA2j (X9y ; <)I2dxd5 < + 00 * (3.5)
Y; 012dyd5 < +m,
Y
The coefficients ul, u2 of the system (1.1) are expressed in terms of the transmutation operators kernels by means of formulas u~(x, Y) =
+A,,(x,
y; x310)
Y) =
+A,,(x,
Y; Y&O) =
=
fB,,(x,y;
xl@,
(3.6) u2(x,
+B,,(x,
Y; Y,O).
Proof: We shall prove the representation (3.3). The admissible solution of the system (1.1) with the given asymptotics u,(y) = tjr( - co, y), b,(x) = I(12(x, +co) satisfies the system of integral equations @1(x, Y) = al(~)-
j ur(s,
y)ti2h
y)ds,
-a,
(3.7)
cc $2(x>
Y) =b,(x)+
.fuz(x, Y
s)@l(x,
s)ds.
Conversely, the solution of the equations (3.7) with the uniformly bounded norms Ik (x9 911L2and lhk2( .9 Y)II~, is the admissible solution of the system (1.1) with the needed asymptotics.
THE INVERSE SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
265
If the solution of the equations (3.7) can be represented in the form of (3.3) for any al, b, ELM, then substituting (3.3) in (3.7) we obtain the system of equations for the kernels
(3.8)
B,l(X,
Y;
s)---2(x, s)-- Lu2(x,O&,(x,
Bl2k
YGs)+u,(s, y)+
r; d&f
jm
y)B,,(t,
= 0,
s 2 y,
y; S)d5 = 0,
s
(3.9)
m
&2(x, y; s)- ,$u2(x, OB,,(x, g; S)d5= 0,
s 6 x.
Conversely, if the kernels B,(x, y; s) satisfy the system (3.8H3.9) and the estimates (3.5) then the representation (3.3) gives the admissible solution for any a,, b, ELM. Thus to prove the representation (3.3) it is sufficient to prove that the systems (3.8), (3.9) have the solution admitting the estimate (3.5). This follows from the Volterra property of the systems (3.8), (3.9) on the basis of Theorem A. I. The equalities (3.6) follow directly from the systems (3.8), (3.9) with s = y, s = x. The proof of the representations (3.1), (3.2), (3.4) is similar to the proof of the representation (3.3). 4. The properties of the scattering operator The scattering operator is defined by the equality (2.6) as the operator connecting the asymptotics t+Gr (- co, a), t+Gz ( a, --co) and tii(+a, *), ti2(*, +a) of the admissible solutions of the system (1.1). If u1 = u2 = 0 then the scattering operator S is an identical operator I. THEOREM4.1. Let S be the scattering operator for the system (1.1) with the coefficients satisfying the conditions (1.2). Then in the space L, ( - co, + 03 ; E2) the operator S-l exists and s-‘=z+r,
S=I+F,
(4.1)
where F = IIFijll?j= 1) Y = Ilxjllilljfj= 1 are the matrix Hilbert-Schmidt operators S.). The diagonal elements of F and Y are the integral Volterra operators F,,
= W+,
F22
=
W2+,
51
=
w-9
Y22 = w,_
with the variable upper W, + , W2+ and lower W, _ , W, _ limits of integration. scattering operator S admits the two-sided factorization s = (I+AJ’(Z+A+)
= (z+B+)(I+B_)-‘,
(H.-
(4.2) The
(4.3)
L. P. NIZHNIK
266 where A-, A+, B-, B, corresponding polarity. Proof:
are the matrix
Comparing
Volterra Hilbert-Schmidt (3.1),
the representations
(3.2)
and
operators
assuming
of the s
y = x we
obtain (Z+A+)a
=(Z+A_)b,
(4.4)
where the kernels of the integral Volterra operators A+, A- are expressed in terms of the transmutation operators’ kernels by means of the equality A* (x, s) = llAij(X, x; s)ll~j=1. Due to the estimates (3.5), the operators A+, A_ are HilbertSchmidt operators. Taking into account the definition (2.6) of the scattering operator, we obtain from (4.4) the left factorization of the scattering operator. The right factorization (4.3) is obtained by comparing the representations (3.3), (3.4) with the operators B+, B_ which are connected with the transmutation operators by means of the equalities
B+(x, 4 =
f-41(x, x; 4 -&I (x, x; 4
--4,(x, x; 4 B,,(x, x; $ 1 ’
x 2s.
the first equalities in (3.1) and (3.4) and assuming y = x, a, = 0, b, = Sllal, we obtain Sii = (ZfB,l+)-‘(Z+A,l+) =Z+WI+. From the second equalities in (3.1), (3.3) it follows that Sz2 = Z+ W,, . The Volterra properties of Y,, and Yz2 follow correspondingly from the first equalities in (3.2), (3.3) and from the second equalities in (3.2), (3.4). The Volterra properties (4.22) of the diagonal elements of the scattering operator S and operator S-l lead to the important relations between their elements. LEMMA 4.1. The scattering operator is uniquely determined by one of the collecComparing
tions of the operators
F12,
Y,,
or F21,
Proof:
From the equality SS-’ + Y+ YF = 0. Hence we have
Y12.
= S-i S = Z we obtain
Z-J’12
L
=
U+J’,JU+
&A
I--
L
F12
Z-F,,
Y12
=
U+J’,,)U+
Y22L
I-
Y12F21
F+
Y+ FY = 0 and F
=
U+
Y,,)U+F22)7
(4.5)
=
(I+
Y,lIU+f’,,).
(4.6)
As the diagonal elements of S and S-l are the Volterra operators, these equalities are the right factorization of the operators I- F12 Yzl and Z-F,, Y12 and the left factorization of the operators I- Yzl F12 and I- Y,, Fzl. If the pair F12, Y21 or F 213 Yl2 is given, then all diagonal elements F, 1, F22, Y, 1, Y,; are found uniquely from the factorizations (4.5), (4.6). The remaining elements are found from the equalities f’12
=
-(I+
YlJ’
YlzU+F,,),
which follow from the identities
F21
=
-V+F22)
r,,U+
F + Y+ FY = F + Y+ YF = 0.
&,I-’
THE INVERSE
SCATTERING
DEFINITION 4.1.
PROBLEMS
The pair of operators
data for the system (1.1) connected
called the scattering Remark
4.1.
FOR THE HYPERBOLIC
267
Fi 2, Y,, will be called the scattering wave; the pair Fzl, YIz will be
with the jrst
data connected
If the
EQUATIONS
coefficients
with the second
wave.
of the
(1.1) are connected
system
by the
relation
then the elements
of S and S-’
Let the relation (1.1) the identity
a
this identity
(4.8)
of the system
a
(x, YV = -W2k
ay
with respect
vlcIlII/I(+~> ~~llt,-ll11/1(-~~9ll2,l The equality
Yz2 = F;,.
(4.7) be fulfilled. Then for the solution
‘l&h holds. Integrating
by the relations r,, = FT,,
Y,, = -YIFT,,
r,, = -?FZ,, Proof:
are connected
(4.7)
rl = ul,
%(X, Y) = vu1 (x9 Y),
Y)12
to x and y, we obtain
= lb/92( .? + 4112,
- w2 ( .Y -
4112,.
(4.9)
(4.9) means that r) [l/bill2 - lla1112] = llb2112- lla2112, i.e.
It leads to the equalities (4.8). Note that in the case u2 = vii,, 7 = if, the scattering operator determined according to (4.8) and Lemma 4.1 by F,, or F,,.
is uniquely
5. The inverse scattering problem
The inverse scattering problem for the system (1.1) consists in finding the coefficients u1 (x, y), u2(x, y) when the scattering operator or scattering data are given. At the beginning we shall find the connection between the scattering data and the transmutation operators’ kernels. By comparison of the equalities (3.3), (3.4) we obtain assuming a, = 0, bl = F,,a,, b, = a,+F,,a, that B,, (K Y; s)+ :B,2@5
Y; ~3 &I(<,
$d5
= 0,
Y >s,
x
(5.1) 42k
Y; s)+
1 B,,(x, --oo
Y; <)F12(5,
4dt+f’,,(y,
4 = 0,
s > x.
268
L.P.NIZHNIK
Comparing
(3.3), (3.4) and assuming 6, = 0, a, = bI + YI1 b,, a2 = Y2r bI we obtain
&1(x,
Y > s, y;d+ ~822(X,Y;5)Y21(r,S)di:+YZI(X,S)=0, x (5.2)
%,(x9 y;s)+ i &,(x, v:OF,*(5,S)& =o, -CO
s > x.
By setting in the equalities (3.3), (3.4) first b, = 0, a, = Y,, b2, a, = b,+ then a2 =0, bz =F2raI, b, =a,+F,,uI we get
YZZbZ
&,(x,y;s)+ i B,,(x,y;~)F,,(5,s)d5=0, s > Y, -UTJ (5.3)
&,(x,y;s)+ ~B,,(x,Y;e,r,,ce,S)dg+~2(Y,S)=o, x > s, ‘Y
&1(x,
y;s)+ i B*,cGY; ~)F,,(5,S)d~+Fzl(x,S) = 0, -m
s > Y, (5.4)
&,ky;4+
p,,(*,Y:5Ms,s)dc
=o,
x > s.
The systems of equations (5.1), (5.2) are called the system of fundamental equations for the inverse scattering problem with given scattering data F12, Yzl. The systems (5.3), (5.4) are the fundamental equations for the inverse scattering problem when the scattering data F,,, YIz are known. For the Dirac equations (1.1) the systems of fundamental equations (5.1H5.4) are the analogues of usual Gel’fand-LevitanMarchenko equations in the theory of the one-dimensional inverse problems of spectral analysis.
LEMMA 5.1.Let FIz, Y,, or Fzl, Y12 be the scattering data for the system (1.1) with the coefficients satisfying the condition (1.2). Then for any x, y: + 00 the systems of the equations
UI(V)+ j F,,h
- CC < x, y <
rl)u,Wds =fih),
-m
(5.5)
UI(V)+ : L(s, v)u,(s)ds =fi(d, x
uz(~9+ are uniquely
jm FI 6, rl)ul(s) ds= fi(tl) z
solvable in L,( - CO, + co; E2) for
any
fi,f2ELM.
(5.6)
THE INVERSE SCATTERING PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
269
Proof: The solvability of the equations (5.5), (5.6) follows from the factorization properties (4.3) of the scattering operator. The proof is carried out in Appendix III. The solvability of the equations (5.5), (5.6) means solvability of the fundamental equations (5.1H5.4). Together with the formulas (3.6) it gives the solution of the inverse scattering problem for the system (1.1). THEOREM 5.1. The inverse scattering problem for the system (l.l), that is the problem of finding the coefficients ut, u2 of the system (1.1) when the scattering data F 121 YZI or Fx, Y12 are given (hence, when the scattering operator is given), is uniquely solvable. The coefticients u1 (x, y), u2(x, y) can be defined by the formulas (3.6), where Bij(x, y; s) are the solutions of the system of the fundamental equations (5.1)-(5.4).
Remark 5.1. If we consider the then the inverse scattering problem In fact, according to Remark 4.1 uniquely determined by one of the
system (1.1) with u2(x, y) =‘~ui(x, y), v] = q, is uniquely solvable for given F,, or F21. all the elements of the scattering operator are elements F,, or F,, .
6. The description of the scattering data For every Dirac system (1.1) with coefficients ui, u2 ELM there is a scattering operator S with the scattering data F12, .Yzl or F2i, Y12 which are the H.-S. operators with the kernels F,,(x, y), Yi2(x, y), F,,(x, y), Y2i(x, Y)EL~(E’). This defines the mapping of the scattering data in the space L2(E2, E2) (% (x> Y), Uz(JG Y)) z(F(x,
y), Y(x, y)),
(6.1)
where F, Y is one of the pairs F12, Y21 or F2i, Yi2. For definiteness we shall assume F(x, Y) = Fzl (x, Y), Yb, Y) = K2(x, Y). Now we shall examine in detail the mapping 91. We shall describe its range, that is we shall establish the necessary and sufficient conditions for the preassigned pair of H.-S. operators to be the scattering data for the system (1.1). THEOREM 6.1. In order that the pair F,,, Y,, of the H.-S. operators be the scattering data for the system (1 .l) with the coefficients ul, u2 EL, (E2) it is necessary and sufficient that the homogeneous system of the fundamental equations be only trivialy solvable, i.e. for any -CO < x, y < + co the system VI
@I)+
,f
f’2,
(~9
v)
cp2 (s)ds
=
0,
-UZ
cp2 (v)
+
T
(6.2) &
2 6,
rl)
cp1M
ds
must have only the trivial solution in the space L2.
=
0
270
L.P.NIZHNIK
Proof: The necessity has been established in Lemma 5.1. Let the system (6.2) have only the trivial solution. By virtue of the Fredholm property the nonhomogeneous system (5.5) is uniquely solvable at any fi , fi E&. We shall write the solution of the system (5.5) in the form YVf
a =(I+H(x,
=(I+R(x,
Y))L
(6.3)
where H(x, y), and consequently R(x, y), are H.-S. operators uniformly in the H.S. norm depending on x, y on the compact set [ - 00, + m] x [ - co, + co J (here [ - co, + 001 is the extended real axis which is compact with respect to the natural topology). As a consequence the operator function R(x,y) is uniformly bounded in x, y with respect to the H.-S. norm. Therefore the solutions of the systems (5.3), (5.4), as special cases of (5.5), can be represented in the form of (6.3) and
~jlBl& [~I&(x,
Y; 1)12dYd? G
c, k =
Y; 1)12dxdrl G c,
(6.4)
1, 2.
From the systems (5.3), (5.4) and (6.4) we infer existence of the functions 4(x,
Y) = -&,b,
Y; x-0)
= +
Y,,(y,
x)+
;B,,(x,
y; s)
Yi2(S, x)ds,
Y
(6.5)
uz(x, Y)'=B,,(x,y;Y-t-0) = --Fzl(x,y)- j &2(x, Y;s)F21(S,Y)dS --co
and the fact that they belong to the class L,(E’). Now we shall show that the scattering data for the system (1.1) with the coefficients uk(x, y), k = 1, 2, defined by the equalities (6.5) coincide with the initial operators F2i, Yi2. In terms of the solutions Bij(X, y; q) of the equations (5.3), (5.4) we define the functions Aij(x,y; q): A,l(x,Y;rl)=B,,(x,Y;vl)- i -m A,,k
Yi 'I) =
~I,(-%
B,,(<,
Y;5)%1(5,Y;r)&,
Yl rl)---B,,(vl,Y; VI)-pl,(r,Y:
OB22(5,
y:rl)G, (6.6)
A,,(x,Y;y)=B,,(x,Y;tl)--B,,(x,tl;9)-jB21(X15;e)B,,(X,e;ad~, Y
A22(x,
Y; r)=
B,,(x,
y;rl- ;&,(x, 5; t)B,,(x,t;rl)4
THE INVERSE
SCATTERING
It is easy to verify that
A,l(&
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
271
Aii satisfy the equations
y; q)+ j Al2(X, y; s)Fz1(s,tl)ds= 0, -CO
A,,(x, &1(x,
Yi
Y;
v)+
q,Mx7 Yi s)-A,,(vl,
49+ i CA22k Y; s)--‘4*,(x,
Y;
dl Y,,(s, v)ds= 0,
rl; s)lF21h
?)dS= 0,
-CC
Mi
Yi
v)+
~‘h(x, Y; 4 hz(s, rl)4 = 0.
par the functions V,(V) = Ai,(x+a, obtain the system of equations ‘PI
Y; v)--Ai,@,
(vl)+ i cPz(s)Fn(s, rl)ds = -m
Y; tl), k = I, 2, from (6.7) we
x~6A,*~x+S, y; s)F21(s,rl)ds, x
m
cpzh)+ .Fcpl(S)Kz(s, ?)dS= 0. Y
The right sides of the system (6.8) at S --+O are of order o(6). By virtue of (6.3) the solution of the system (6.8) is of order o(6). Then the functions Aii(x, y; Y))do not depend on x. From (6.6) we obtain Al1 (x, y; q) -+O as x --t - co, A,,(x, y; q) =,O. Therefore Aii(x, y; II) E 0, i = 1, 2. Similarly from the system (6.7) we obtain A*i(Xp y;lj) =O, i = 1, 2. By comparing the equations (6.6) with Aij = 0 and the equations (3.8), (3.9) connecting the transmutation operators’ kernels with the coefficients of the system (1.1) we conclude that these equations coincide when uk (x, y) are chosen according to the formulas (6.5). Due to their unique solvability we conclude that the solution Bij(x, y; q) of the system (5.3), (5.4) are the transmutation operators for the system (1.1) with the coefficients uk(x, y) which are given by the formulas (6.5). But the scattering data are uniquely determined by the transmutation operators’ kernels. It leads to the assertion that the initial operators Fzl, Y,, are the scattering data for the system (1.1) with coefficients given according to the formulas (6.5) where Bij are the solutions of the systems (5.3), (5.4). Remark 6.1. The totality of the scattering data forms an open set in the space of the pairs of the H.-S. operators. Proof: If F, 1, Y,, are the scattering data then the operator function R(x, y) of (6.3) exists for x, y EC- co, CCJ].Under small, with respect to the H.-S. norm, variations of the scattering data the function H(x, y) has a small variation and consequently R(x, y) exists, so the conditions of the Theorem 6.1 are fulfilled.
L. P. NIZHNIK
272
Remark 6.2. The pair of the operators F2i, Y,, for which the H.-S. norm is less than 1 are the scattering data. Proof: The conditions of Theorem 6.1 are fulfilled by virtue of the condensed mapping principle. Remark 6.3. If u2 = U, in the system (l.l), then Yi2 = -F$, and totality the elements F,, constitutes the whole space of the H.-S. operators.
of all
Proof: Let F,, be an arbitrary H.-S. operator. We shall show that the system (6.2) with Y,, = -FTi has only the trivial solution. From (6.2) we obtain ~IcMV)l2~V+
5
lcp2(1)12~rl
Y =
i[-
7
Y
-CO
=
i
~,(s;DF,,b,
cW]cp,W~+
j
[-
~rp~(s)Y,,(s,
tt)+~2Wv
-a,
dsqb~2OF,,(s,
rl)rp,(rl)-(~2MF21@,
+~1(tt)l~rl=~-~=0~
-03
Thus (Pi = 0, k = 1, 2, and by virtue of the Theorem 6.1 the operators -F& are the scattering data. From (6.5) we obtain u2 = Ui. Remark rate such
6.4.
For arbitrary
scattering that
llf’2l
data -
scattering
F2i, Yi2 =
data FZ1, Y,, and any E > 0 the degene-
FEZI(x, y) = 2 fi’(x)gt (y),
Yf2(x, y) = 5 pi(x) qi(y)
i=
i=
1
exist
1
GIIH.-S. < E, II q2 - Y~211H.-S. < E.
Proofi According to Remark 6.1 for the scattering data F2i, K2 there exists a neighbourhood consisting exclusively of scattering data. Since an arbitrary H.-S. operator can be arbitrarily well approximated in the H.-S. norm by finite-dimensional operators, this leads to the assertion formulated above. The results of Theorem 6.1 and Remarks 6.1-6.4 can be reformulated in terms of the scattering data operator ‘3. The scattering data operator ‘3 generally defined by the equality (6.1) in the special case at u2 = viii as Y, 2 = - nFT1 is defined by the equality 2% (x, Y) =
F,,
6,
Y)
(6.10)
in the space L2(E2). THEOREM 6.2. The scattering data operator % mapping the coefficients of the system (1.1) into the scattering data is continuous in L2. Its range is an open set. The operator ‘II-’ exists and is continuous and its action can be constructively described by means of the uniquely solvable systems of equations (5.3), (5.4) and the formulas (6.5).
THE INVERSE
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273
7. The exact solutions
We shall give some examples when the inverse scattering problem for the system (1.1) can be explicitly solved. This will occur when the fundamental integral equations of the inverse scattering problem (5.3), (5.4) can be explicitly solved. Let the scattering data be degenerate, that is representable in the form
J-21(XTY) = f:‘(X)fl
= ;
(Y)
n=
fin(X)f 1
(7.1) &2(x, Y) = s:‘Wgz(Y)
= f sl.(X)S*n(Y), “=1
where the components of the vectors f, = col& , *.*,.h~h’), gk = col(gk,, .**> gkh’), k = 1,2, belong to Lz, and tr denotes the transposition operation. In this case the systems (5.3), (5.4) are reduced to the linear algebraic equations and their solutions are of the form
&I (xvY; rl) = s:'WQz(x)C~+Q, (~)Q,(x)l-‘h
(~1,
Biz(x,
Y;
v) =s’;(y)C~+Q,(x)
&,(xv
Y;
rl) = -f~(x)Cz+Ql(~)Qz(x)l-~fi(?),
Q&91-‘g,(rl),
(7.2)
B,, (x, Y; rl) = --f;(x) CI+Ql(~)Qz(x)l-' QI(~)g2(tl)Here Qi, Q2 are the matrices of the form
Ql (Y) = -
is,(s)g:‘(s)b Y
Q2W =
; g2(4f:‘Wds+
(7.3)
--Go
The formulas (6.5) give the following expressions for the coefficients of the system (1.1): ul(x, Y) =s:'(~)Cz+Q,(x)Q,(y)l-'gz(x), uz(x,
Y) = -S~‘(X)C~+Q~(Y)Q~(X)I-‘~~(Y).
(7.4)
The general solution of the system (1.l) with the coefficients of the form (7.4) is th’k
y) = a, (v)+s;(Y)C~+Q,WQ,
(YJI-‘(Q,(x)
ih @Ial b)dsY
-
j 92W2W~)~ -CO
(7.5)
214
L. P. NIZHNIK
tiz(x, Y) = b,(x)--f:'b)Cl+Q, (~>Qz(x)l-‘(Q, (Y) j gz(s)bz(s)ds+ -*
+
7-h 6) 4 Y
(4 ds),
where a,, b, are arbitrary functions belonging to L,. The formulas (7.5), (7.4) can be represented in the operator form for arbitrary scattering data F,, = F, Y, 2 = Y The admissible solutions of the system (1.1) can be represented in the form $1 = Cl -
YQxFP,I- ’ (a(Y) - YQx(b(Y)),
tiz = [I-FP,
YQJ-‘(b(x)-
The coefficients of the system (1.1) are expressed means of the formulas ul(x, Y) = Cl-
FP,(a(x)).
in terms of the scattering
YQ,FP,I-’
uz(x, Y) = -WFP,
(7.6) data by
Y(Y, 4,
YQxl-‘J’k
Y).
(7.7)
The operators in the right-hand sides of the formulas (7.6), depending on x and y as on parameters act on the functions a and b. The arguments of these functions indicate that the result will be considered as a function of the corresponding argument. The right-hand sides of the formulas (7.7) are the kernels of the corresponding operators. It should be noted that the formulas (7.7) can be obtained by substituting (7.6) into the Dirac system (1.1). 8. The non-linear integrable evolution equations in two spatial dimensions We shall consider the non-linear evolution equations which are integrated by the inverse scattering method [12]. For the inverse problem we choose the inverse problem for the Dirac system (1.1). The simplest example of a nonlinear equation is the non-linear Schrbdinger (Davey-Stewartson) equation in two spatial variables which for the first time was obtained in the paper [15] and has been integrated by the inverse scattering method in the papers [S, 9, 43: iu, = -u,,
- ku,, + vwu >
wxy = 2 ($+
k-$),ui2,
(8.1)
where k and q are the real scalars which may. be reduced to the values k = + 1 and r) = & 1 by scaling the independent variables. The quasi-potential w is expressed in terms of the solution of the equation (8.1) by the formula w = -2&&(x, Y
s; t),‘ds+2kd
i lu(s, y; t)12ds. aY4
(8.2)
THE INVERSE
SCATTERING
PROBLEMS
Another example of the non-linear analogue of the modified Korteweg-de
FOR THE HYPERBOLIC
equation in two spatial Vries equation [7]:
EQUATIONS
dimensions
is the
+ uyyy+ @a), + (wu), -$ (0, + w,) u, u, = %cXX u, = 3(U2),,
u(x, Y; %o The equations
(8.1) and (8.3) admit
(8.3)
WY= 3 (U2),.
For the equations (8.1) or (8.3) the Cauchy solutions satisfying the initial condition
problem
= %k
consists
in finding their
Y).
(8.4)
the Lax representation
LP-QL=O, where L is the two-dimensional system (1.1)
275
Dirac operator
(8.5) defined by the left-hand side of the
For the equation (8.1) u1 = U, u2 = VU, for the equation (8.3) aI = -u2 operators P and Q for the equation (8.1) are of the form [4, 6, 71:
= U. The
(8.7)
and for the equation
P=
(8.3) of the form
c
D-&,-w,
b-&l,
-3&uu,
3% a,
.3ayu,
1 (8.8)
D--qw,-v,_,'
D = a,-a;-a;-va,-wax. 9. The evolution of the scattering data Let the coefficients uk of the system (1.1) or the Dirac operator (8.6) depend on and satisfy one of the non-linear equations (8.1), (8.3). Then the scattering data F,,, Y12 corresponding to the coefficients ul, u2 also depend on t as a parameter. We shall study the evolution of the scattering data. t as a parameter
L. P. NIZHNIK’
276
Zf rl/ is the solution of the Dirac system (1.1) with the coefficients
LEMMA 9.1. satisfying
the non-linear
then the function Proof:
account
equations
which are equivalent
to the Lax
cp = P+ will also be a solution of the system (1.1).
Applying the operator equality (8.5) to the function that LIc/ = 0 we have L(PII/) = 0.
LEMMA 9.2.
equation (8.5)
Zf the function
tj and taking into
u(x, y, t) satisfies the non-linear equation (8.1) then
the scattering
data-for the Dirac system (1.1) with u1 = u, u2 = nii are determined the function-F(x, y, t) (F,, = F, Y12 = - nF*) satisfying the linear equation .dF
a2F --
lx-ax2
hd2F
o
JJF=
by
(9.1)
*
Proof: Let Ii/(x, y, t) be the solution of the system (1.1). Then as it follows from the Lemma 9.1, the function cp = L$ is the solution of the system (1.1). If a, = $1 (-00, y), a2 = I,/I~(x, -00) are the asymptotics of the solution $ then the asymptotics of the solution Ltj are expressed in terms of al, a2 by the relations cpl(-03,
y) =
cp2(x, -a)
By virtue of the assumption (9.1) is valid. 10. The inverse scattering
ii+kz C
S’
)
al(y), (9.2)
= cii-$ja2(x).
u1 = U, u2 = v]ii and the condition
(8.2), the equation
method
To solve the Cauchy problem (8.1), (8.2), (8.4) or (8.3), (8.4) by the inverse scattering method it is necessary to construct the scattering data F(x, y, 0) at an initial moment using the initial data uo, to solve the Cauchy problem with these initial data for the Linear equation (9.1) and then to solve the inverse scattering problem to find u(x, y, t). The solutions of the Cauchy problem for the equations ’ (8.1) or (8.3) can be represented in the form u(x, y, t) = W ’ eeiAr Pluo,
where the scattering
A determines +k--
a2 w
data operator
the evolution
for the equation
(10.1)
VI is defined by the equality (6.10), the operator
of the scattering
(8.1) or A = z‘($+-$)
a2
data
and is of the form for the equation
A = Q
(8.3).
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
277
Using the representation (10.1) for the solutions of the non-linear equations (8.1) and (8.3) we can construct the explicit solutions of these equations. In addition, the representation (10.1) can be used to prove the existence of the solution of the Cauchy problem on the basis of the statement converse to Lemma 9.2. THEOREM 10.1. L,et the function function together with its derivatives derivatives with respect to x and y homogeneous equations (6.2) with F,, solution in L,. Then the equation K(x, y; <, r)-t-F(x,
t; t)+q
F(x, y, t) satisfy the equation (9.1) and this with respect to t and its first and second belong to L2(E2). Also let the system of = F and Y12 = - nF* have only the trivial
k.u,
Y; a, t)
i
&S,
ct:
t)F(j?, 5; t)d/3dcc = 0
--Di
Y
(10.2) is uniquely solvable and the function the non-linear equation (8.1).
u(x, y, t) = r]- ’ i? (x, y; y, t) is the solution of
The proof of this Theorem is given in Appendix IV and is actually contained in the paper [9]. The statement of the Theorem 10.1 is equivalent to the assertion that the function u (x, y, t) = OI- ’ e-tAf F
(10.3)
. is the solution of the equation (8.1) if it is determined. Thus the solution of the Cauchy problem (8.1), (8.4) can be represented in the form (10.1) and, conversely, if the formula (10.1) is determined then it gives the solution of the Cauchy problem. To obtain the solutions of the non-linear equations (8.1) and (8.3) in an explicit form we shall use the formulas (7.4) for the explicit solutions of the inverse problem. In this case for the equation (8.1) the functions f = fI = g1 and g = g2 = -qf2 must depend on t and satisfy the equations = 0.
(10.4)
For the equation (8.3) the vector functions f = fi = -gl, and satisfy the equations
g = g2 = f2 depend on t
iJ;+hfyy = 0,
J;-f,,, 11. Investigation
= 0,
ig, +g,,
St-Sxxx
= 0.
(10.5)
of the Cauchy problem by the inverse scattering method
Consider .the Cauchy belonging to L2.
problem
(8.1), (8.4) in the space L2 with the initial data
278
L. P. NIZHNIK
DEFINITION 11.1. The function u(x, y, t) continuous in t, whose values with respect to x and y belong to L,(E’) is called the solution of the Cauchy problem (8.1), (8.4) if at t = 0 it coincides with the initial data U(X, y, 0) = uO(x, y) and if the sequence u,(x, y, t) of the smooth in x, y, t solutions of the equation (8.1) exists and is such that Ilu-u,,IIL2 -+O as n + a uniformly in t, belonging to an arbitrary finite interval [0, T]. The functions u,(x, y, t) and their derivatives respect to x and y up to second order belong to L2(E2).
with
The solution of the Cauchy problem (8.1), (8.4) in the space is unique. For arbitrary initial data from L2 the local in time solution of the
THEOREM 11.1. L,(E’)
Cauchy problem exists. For arbitrary interval of time.
small initial data from
L,
the solution exists
on an
Proof:
The uniqueness of the solution of the Cauchy problem (8.1), (8.4) follows from the possibility of representing it in the form (10.1). The existence of the local in time solution follows from the continuity of the unitary operator eTiar, from the continuity of the operators 2I, ‘9-l and from the openness of the range of the operator ‘+2I (Remark 6.1). By virtue of these circumstances the formula (10.1) has a sense for arbitrary initial data u0 ELM if t is sufficiently small. If the initial data are sufficiently small then by virtue of the continuity of 2I and unitarity of . 1ess than 1. But then formula (10.1) has a sense according to e -iAt, jle-iA’ pIa, is Remark 6.2. For the case r7 = 1 in equation (8.1) the result of the Theorem 11.1 can be refined using Remark 6.3.
Ifq = 1 the Cauchy problem (8.1), (8.4) is uniquely solvable in initial datu u0 ELM on an arbitrary interval of time.
THEOREM 11.2. L, for
arbitrary
Above the exact solutions of the nonlinear equations (8.1), (8.3) were given. The question arises: To what extent is the set of such the solutions rich? THEOREM 11.3. Let u be an arbitrary solution in L, of the Cauchy problem (P.l), (8.4) and E > 0 be arbitrary. Then on an arbitrary interval of time [0, T] there exists a solution u,(x, y, t) of the form (7.4) such that sup
O
Ilu- UT/lL2(E2)
< E.
(11.1)
Proof: Let F, be the scattering data constructed from the initial data u. that is F. = ‘%uo. By virtue of (10.1) u = W’e-iA’ F,. The openness of the range of the
operator ‘%, the compactness of the finite interval [0, T], the unitarity of the operator eeiAz imply that 6 > 0 exists such that for an arbitrary function F satisfying the condition IIF- Foil < 6 the function u = %I-’ emiArF is defined on the interval t E[O, 7’J. For F we shall take the degenerate kernel F = Fdeg. Then uT = a-1 e-iAt Fdeg belongs to the class of the explicit’ solutions of (7.4). On the
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
other hand, due to the continuity of ‘%I-’ and the unitarity we can find &: 0 < 6, < 6 such that IIu-+IILz
= II+%-’ e -iAtFo-~-le-iA’Fdegll
<.5,
EQUATIONS
279
of emiA’, for any E > 0 tE[O, T-J:
Appendix I. The Volterra type equations We shall consider
in a Banach
space B the linear equation u = h+Au.
(A.I.l)
THEOREM A.I. The solution of the equation (A.I.l) for any h EB exists and is unique if with respect to the monotone in TE( - co, + co) set of the semi-norms llullT (IIullm = [lulls) the, inequality
IIA~IT G i a(d lb4 dT
(A.I.2)
--oo
holds for an integrable function
u(z)
5 a(z)dr
< +CO.
(A.I.3)
-a, Proof:
By iterating
the inequality T
lIA”4b c J a(z,)dr, -m
From
(A.I.4) we have
(A.I.2) n times we obtain
Tl
.r ~b,)d~,...
-CC
rn-
1
.r 4dI4lr$n.
-m
(A.I.4)
L. P. NIZHNIK
280
Appendix II. The existence of the transmutation
operators
THEOREM A.11. Let ul, u2 be functions (3.8), (3.9) which satisfy the inequalities
Then the solutions of the systems
~~s~pI~,,(x,y;~)1~dyds
<
+m,
in L,.
J+up~B,,(~,y;s)~~dxds Y
<
(A.II.l)
+m
exist and are unique. Proof: By elimination we can obtain from the systems (3.8), (3.9) four equations for each of the functions B1i, B,,, B2i, B22. The equation for Bzl is of the form Y; d--2(x,4+
B,,(x,
ju2(x,
5)
; -m
u,(vl, <)B,,(rl,
5; dW<
=
0.
(A.II.2)
To prove solvability of equation (A.II.2) we use Theorem AI. For the Banach space B we shall take the space of functions of three variables B,, (x, y; s) with the norm IIB21(x,
Y;
s)lls =
[~,bplB2,(x,
Y; ~)l~dxd$‘~.
Y
For the set of the semi-norms
in B we shall take
IIB21ll~ = [&$B21(x, The fundamental
inequality
AB,,(x,
Y; s)12dxds]1’2.
(A.I.2) is fulfilled by the operator
Y; 4 =
"cuz(x, Y
0
l‘cu,(rl, OB21h, -m
5; s)dvds
with the majorant a(x) = lIuz(x, Therefore Similarly
.)II lb1k .)II.
by Theorem A.1 we obtain unique solvability of equation we prove existence of the operators B,,, B1 1, B22.
(A.II.2).
Appendix III. Solvability of the fundamental equations of the inverse problem In this Appendix we shall prove Lemma 5.1, that is we shall show uniqueness of solution of the fundamental equations (5.5), (5.6) of the inverse problem (see [3, 21). The scattering operator S admits the two-sided factorization S =Z+F
=(I+W+)(Z+k),
S-l
= I+ Y = (I+ V+)(Z+ I’_).
(A.III.1)
THE INVERSE
SCATTERING
We shall consider
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
281
the operator @n =
U+QAFP,+PAYQ,L
(A.III.2)
where Pn is the set of the orthoprojectors on the semi-axis x 3 1 that is gLf(x) = 0 (x- A)f(x) and Qn = I- Pk. The operator @‘1can be represented in the form
=(I+Q,W+)(I+QnW-)(z+I/, QJU+v-Qn). Since the operators (Ii-Q1 W,), (I+ V, QJ are reversible in L,, the operator Gi is reversible in L2. It follows from the reversibility of the operator @, that the homogeneous equation Qnu = 0 has in L, only the trivial solution. Taking into account that the diagonal elements Fi,, Fzz, Yll, Yz2 are Volterra operators, the equation @;Iu = 0 can be written in the form ui (vl)+ J Fiz(s,
q)uz(s)ds
= 0,
uz(rl)+ ,f yzl(S, Vl)Ul(S)dS = 0, --oo
(A.III.3)
u1(ul)+ ,f %2(s,rl)%(S)dS= 0, -00
uz(rl1-t
~F,,(s, v)u(s)ds= 0.
Using the Fredholm property of the system (A.III.3), passing over to the conjugate system, we can obtain the trivial solvability of the systems cpl (rl) + 7 cpz(s) r,l (ST q) ds = 09
(A.III.4) 401 (rl)
+
I’
cpz
(4
f’21b
9)
ds
=
0,
-a,
Thus the solvability
of the system (5.5)-(5.6) for x = y = 1 is shown. Then
the
282
L. P. NIZHNIK
operator TIzSTJ is the scattering operator for the system (1.1) with the coefficients i& = U1(x, y+z), ii2 = uz(x, y+z). The operator TL,ST: has the same factorization properties as the operator S. (TI,S-’ T& = Y,, T,, therefore the system of the But (TI,ST,),, = T_,F,,, form (A.III.4) for the operator TI,Sc is
pfm) YZl(S,VI--Z)ds= 0,
%(?I+
A
“rcP1(4F,,(s-z, ?)dS= 03
a(r)+
--oo
(Pled+ j cPz(s)F,,h rl-z)ds = 05 -‘X3
cpz(d+ This system is equivalent
T’pl(4 Y1z(s-z, v)ds= 0.
to the systems (5.5), (5.6) for x = 1, y = I-z.
Appendix IV. The Cornille Theorem We shall give the proof of Theorem 10.1 (see [9]). The trivial solvability of the homogeneous equations (6.2) due to their Fredholm property implies uniqueness of the solution of the systems (5.3), (5.4) for F,,(x, y) = F(x, y; 0 and Yi2(x, y) = -$(Y, x; 4 We shall differentiate both equations in (5.3) with respect to x, multiply both equations in (5.4) by Bi2(x, y; x, t) and subtract the hrst equations and the second equations in the obtained systems. Then by virtue of the trivial solvability of the system (6.2) we have
d&l (x,
y;
s, 0
8X
--&2(x,
y; x, Oh (x7Y; s>t) = 0. (A.IV.l)
y; s, 0 ----lz(x,
8&,(x,
.
i?X
Similarly
we obtain
y; x, 0&2(X,
s, 0
0.
relations
a&,(x,
t)
+&1(x,
Y; Y,
t)Blz(x,
Y;
s,
= 0, (A.IV.2)
&I
(x, Y; s, G ay
+B,,(x,
Y; Y, t)B~,(x,
y; s, t) = 0.
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
283
2
Applying the operator iG-&
-k
’ to the system (5.3), using (A.IV.l),
(A.IV.2), the trivial solvability of the system (6.2) and taking into account the F,, (x, s; t) = 0,
Y,,(y, s; t) = 0 we
get B,,
k
Y; ST 0
(A.IV.3)
+
2kB,, (x, y; y, t)
B, t (x,y; s, t) =
o.
Assuming s = y in (A.IV.3) for U(X, y, t) = q-i B2i (x, y; y, t) we obtain the equation (8.1) with
The relation (8.2) follows from (A.IV.l), (A.IV.2). From (5.4) we get the equation (10.2) with respect to K(x, y; s, t) = B,, (x, y; s, t) by eliminating B22(~, y; s, t). REFERENCES [l] Nizhnik, L. P.: Vkr. Math. .I. 24 (1972), 110-113. [2] Nizhnik, L. P.: Nonstationary Inverse Scattering Problem, Naukova Dumka, Kiev, 1973. [3] Nizhnik, L. P., Pochinayko, M. D., Tarasov, V. G.: in: Spectral Theory of Operatorsin Problems of Mathematical Physics, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev, 1983, 72-93. [4] Nizhnik, L. P., Pochinayko, M. D.: Funct. Analys. and Appl. 16 (1982), 80-82. [S] Nizhnik, L. P.: Dokl. Acad. Sci. USSR 254 (1980), 332-335. [6] Nizhnik, L. P., Pochinayko, M. ‘D.: in: Spectral Theory of Differential-Operator Equations, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev, 1986, 89-98. [7] Pochinayko, M. D.: (see above), 103-106. [S] Ablowitz, M. J., Haberman R.: Phys. Rev. Left. 35 (18) (1975), 1185-1189. [9] Cornille, H.: J. Math, Phys. 20 (1979), 199-209. Cl01Nizhnik, L. P., Tarasov, V. G.: Dokl. Acad. Sci. USSR 233 (1977), 300-303. El11Kaup, D. J.:Physica 1D (1980), 45-67. WI Ablowitz, M. J., Segur, H.: Solitons and the Inverse Scattering Transform, SIAM Philadelphia, 1981. Cl33 Nachman, A. I., Ablowitz, M. J.: Stud. Appl. Math. 71 (1984), 251-262. Cl41 Fokas, A. S., Ablowitz, M. J.: J. Math. Phys. 25 (1984), 2494-2505. WI Davey, A., Stewartson, K.: Proc. R. Sot. Lond. A338 (1974), 101-110.
THE
REPORTS
ON
MATHEMATICAL
No.
PHYSlCS
2
INVERSE SCATI’ERING PROBLEMS FOR THE HYPERBOLIC EQUATIONS AND THEIR APPLICATION TO NON-LINEAR INTEGRABLE SYSTEMS
L. P. NIZHNIK Institute
of Mathematics,
Academy
of Sciences of the Ukrainian
SSR, Kiev, USSR
(Recez’oed September 9, 1987) The results on two-dimensional inverse scattering problem for the two-component hyperbolic Dirac system of equations are formulated. The complete description of the scattering data is given and the algorithm of reconstructing the coefficients of equations from scattering data is formulated. The results have been applied to integration of the non-linear Schrodinger equation in two spatial dimensions by the inverse scattering method: the Cauchy problem is investigated, the exact soliton-like solutions are represented.
1. Introduction
In this paper the two-dimensional two-component the form
Dirac system of equations of
(1.1)
is considered under the assumption that the coefficients u1 (x, y), uz(x, y) are complex-valued measurable in x and y functions which are square integrable with respect to both variables, that is Ul, u2 E&(E2).
(1.2)
The direct and inverse scattering problems both for the Dirac system (1.1) in characteristic variables and for the two-dimensional nonstationary system
(1.3) *2 -=
w2 -7&+uzk
at
12611
0$1
L. P. NIZHNIK
262
have been studied in detail in [l-3]. In the papers [4-91 the results on the inverse problem for the system (1.1) have been applied to study the non-linear evolution equations in two spatial dimensions. The results of the papers quoted are used essentially in the following text. The inverse scattering problems for hyperbolic systems of n equations have been considered in [lO-151
2. Tbe scattering problem We shall consider the solutions of the system (1.1) such that for ‘any x the function tji (x, Y) is square integrable with respect to Y and for any y the function tiz(x, y) is square integrable with respect to x, that is Il/l(X, .), $2(-y Y)EJ%. Such solutions
(11/i, $J
(2.1)
of the system (1.1) will be called the admissible
solutions.
Let the coefficients of the system (1.1) satisfy the conditions (1.2). the admissible solution (til, ti2) the limits
THEOREM 2.1.
Then
for
$I(--co7
$I(+
$2 (x3 - a) = a2 (x),.
Y) = al(Y),
$zb,
00, Y) = bl (y),
(2.2)
+ ~0) = b,(x)
exist in L,. For one of the pairs (a,, 4, (bIy U, (aI, bJ, (k, e) of arbitrary functions belonging to the class L2 the admissible solution exists and is unique and this pair is the corresponding asymptotics at infinity for this solution. Proof:
If ($1, $2) is the admissible
$1(x, Y) = $l(O, Y)-
solution
of the system (1.1) then
j~~Cs,;M,(s, y)ds, b
(2.3) Ic/z(x, Y) = tizk Applying equations
adx,
s)IcIik
s)ds.
to the system (2.3) the Theorem on the solvability of the Volterra (see Appendix I), we obtain the boundedness of ll~$i(x, *)llL, and
lIti ( ‘9 YJIIL, uniformly $i(+a,
Q-
in x and y. Then (2.3) implies the existence
Y) = tbI(O, Y)-
‘~uI(s,
y)ti~h
Y)~S and $z(x,
of
+I;o)
0
=$,(x5
We now prove the second
part of the theorem.
0) - ‘Jmu,(z
s)$~(x,
s)ds.
Let .&, a2 ELM be given for
THE INVERSE SCATTERING
definiteness.
PROBLEMS
We shall consider
FOR THE HYPERBOLIC
the system
Ii/l@, Y) =%(Y)-
of integral
5 4(s7 -m
Y)$z(s,
EQUATIONS
263
equations y)ds,
(2.4)
According to Theorem A. I the solution of the system (2.4) exists, is unique’and II$1 (x7 *)I1Lz < C, jl$2 ( ., y)llL, d C. It follows from (2.4) that (11/I, 11/J is the admissible solution and 11/i(- CO,y) = a, (y), t,bz(x, - co) = a2 (x). When the functions bi, bz ELM are given the proof of this theorem to the analysis of the system
is reduced
$1 (x, Y) = bl (Y) + 7~1 (s, Y) $2 (sv Y) ds, x
$2(x,
Y) = bz(x)+
7~2(x,
(2.5) s)$I(x,
s)ds.
Y
Similarly we consider the cases when a,, b, EL, and bI, a, ELM are known. Theorem 2.1 gives the solution of the scattering problem for the system (1.1). The vector-function a = (a,, u2) defines the incident waves and b = (b,, b,) defines the scattered waves. The scattering problem consists in finding the solution of the system (1.1) when the incident wave is given. The operator S transforming the incident waves a into the scattered waves b is called the scattering operator b =Su.
We shall study the scattering 3. The transmutation
operator
(2.6)
in the space L, (- 00, + CC; E2).
operators
According to Theorem 2.1 the admissible solution is defined uniquely when the pair of asymptotics $r (+ cc, y), t,b2(x, + co) is given. The transmutation operators allow us to express explicitly the admissible solutions in terms of the asymptotics. LEMMA3.1. fying
The admissible solution of the system (1.1) with coefficients the c on d it’ions (1.2) can be represented in the form GI(x,Y)=~~(Y)+
j A,,(x,y;s)ui(s)ds+ -U3
sutis-
5 A,,(x.y;s)u,(s)ds, -CC
(3.1) 1(/2(x, Y) =
a,(x)+
j
A,,@,
-CO
Y; s)u,(s)ds+
-j -00
A,,@,
Y; s)u,(s)dk
’
264
L. P. NIZHNIK
Il/I(% Y) = b,(y)+
~~‘%I(X, Y; s)bt(s)ds+
~‘Mx,
Y
x
y; s)b2(s)ds, (3.2)
ti2k
Y) = J%(x)+ :A&
Y; s)br(s)ds+
:‘Mx,
$1(&Y)=
at(Y)+
$2(x9 Y) = b,(x)+
OD &(x9 ;
Y; s)a,(s)ds+
~&(x9
Y; s)a,(s)ds+
; &2(x, -52
Y) =b,(x)+
Y; s)&(s)& (3.3)
i &2(x,
y; s)b,(s)k
-03
Y
$I&
y; s)b,(s)&
x
Y
‘s 4,(x, -El
y; s)b,(s)ds+
T&,(x, x
Y; ~)a,(@& (3.4)
$2(x, Y) = az(y)+
j &,(x, -co
Y; s)br(s)ds+
where a,, u2, bl, b2 are the usymptotics (2.2) of Bij(X, y; s), i, j = 1, 2, are determined uniquely system (1.1) and umit the estimates JJ suPlA,j(x, x
u
~B,,(x, x
Y: s)+(s)&
solution. The kernels Aij(X, y; s), by the coeficients ul, u2 of the
JJ sup lA2j (X9y ; <)I2dxd5 < + 00 * (3.5)
Y; 012dyd5 < +m,
Y
The coefficients ul, u2 of the system (1.1) are expressed in terms of the transmutation operators kernels by means of formulas u~(x, Y) =
+A,,(x,
y; x310)
Y) =
+A,,(x,
Y; Y&O) =
=
fB,,(x,y;
xl@,
(3.6) u2(x,
+B,,(x,
Y; Y,O).
Proof: We shall prove the representation (3.3). The admissible solution of the system (1.1) with the given asymptotics u,(y) = tjr( - co, y), b,(x) = I(12(x, +co) satisfies the system of integral equations @1(x, Y) = al(~)-
j ur(s,
y)ti2h
y)ds,
-a,
(3.7)
cc $2(x>
Y) =b,(x)+
.fuz(x, Y
s)@l(x,
s)ds.
Conversely, the solution of the equations (3.7) with the uniformly bounded norms Ik (x9 911L2and lhk2( .9 Y)II~, is the admissible solution of the system (1.1) with the needed asymptotics.
THE INVERSE SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
265
If the solution of the equations (3.7) can be represented in the form of (3.3) for any al, b, ELM, then substituting (3.3) in (3.7) we obtain the system of equations for the kernels
(3.8)
B,l(X,
Y;
s)---2(x, s)-- Lu2(x,O&,(x,
Bl2k
YGs)+u,(s, y)+
r; d&f
jm
y)B,,(t,
= 0,
s 2 y,
y; S)d5 = 0,
s
(3.9)
m
&2(x, y; s)- ,$u2(x, OB,,(x, g; S)d5= 0,
s 6 x.
Conversely, if the kernels B,(x, y; s) satisfy the system (3.8H3.9) and the estimates (3.5) then the representation (3.3) gives the admissible solution for any a,, b, ELM. Thus to prove the representation (3.3) it is sufficient to prove that the systems (3.8), (3.9) have the solution admitting the estimate (3.5). This follows from the Volterra property of the systems (3.8), (3.9) on the basis of Theorem A. I. The equalities (3.6) follow directly from the systems (3.8), (3.9) with s = y, s = x. The proof of the representations (3.1), (3.2), (3.4) is similar to the proof of the representation (3.3). 4. The properties of the scattering operator The scattering operator is defined by the equality (2.6) as the operator connecting the asymptotics t+Gr (- co, a), t+Gz ( a, --co) and tii(+a, *), ti2(*, +a) of the admissible solutions of the system (1.1). If u1 = u2 = 0 then the scattering operator S is an identical operator I. THEOREM4.1. Let S be the scattering operator for the system (1.1) with the coefficients satisfying the conditions (1.2). Then in the space L, ( - co, + 03 ; E2) the operator S-l exists and s-‘=z+r,
S=I+F,
(4.1)
where F = IIFijll?j= 1) Y = Ilxjllilljfj= 1 are the matrix Hilbert-Schmidt operators S.). The diagonal elements of F and Y are the integral Volterra operators F,,
= W+,
F22
=
W2+,
51
=
w-9
Y22 = w,_
with the variable upper W, + , W2+ and lower W, _ , W, _ limits of integration. scattering operator S admits the two-sided factorization s = (I+AJ’(Z+A+)
= (z+B+)(I+B_)-‘,
(H.-
(4.2) The
(4.3)
L. P. NIZHNIK
266 where A-, A+, B-, B, corresponding polarity. Proof:
are the matrix
Comparing
Volterra Hilbert-Schmidt (3.1),
the representations
(3.2)
and
operators
assuming
of the s
y = x we
obtain (Z+A+)a
=(Z+A_)b,
(4.4)
where the kernels of the integral Volterra operators A+, A- are expressed in terms of the transmutation operators’ kernels by means of the equality A* (x, s) = llAij(X, x; s)ll~j=1. Due to the estimates (3.5), the operators A+, A_ are HilbertSchmidt operators. Taking into account the definition (2.6) of the scattering operator, we obtain from (4.4) the left factorization of the scattering operator. The right factorization (4.3) is obtained by comparing the representations (3.3), (3.4) with the operators B+, B_ which are connected with the transmutation operators by means of the equalities
B+(x, 4 =
f-41(x, x; 4 -&I (x, x; 4
--4,(x, x; 4 B,,(x, x; $ 1 ’
x 2s.
the first equalities in (3.1) and (3.4) and assuming y = x, a, = 0, b, = Sllal, we obtain Sii = (ZfB,l+)-‘(Z+A,l+) =Z+WI+. From the second equalities in (3.1), (3.3) it follows that Sz2 = Z+ W,, . The Volterra properties of Y,, and Yz2 follow correspondingly from the first equalities in (3.2), (3.3) and from the second equalities in (3.2), (3.4). The Volterra properties (4.22) of the diagonal elements of the scattering operator S and operator S-l lead to the important relations between their elements. LEMMA 4.1. The scattering operator is uniquely determined by one of the collecComparing
tions of the operators
F12,
Y,,
or F21,
Proof:
From the equality SS-’ + Y+ YF = 0. Hence we have
Y12.
= S-i S = Z we obtain
Z-J’12
L
=
U+J’,JU+
&A
I--
L
F12
Z-F,,
Y12
=
U+J’,,)U+
Y22L
I-
Y12F21
F+
Y+ FY = 0 and F
=
U+
Y,,)U+F22)7
(4.5)
=
(I+
Y,lIU+f’,,).
(4.6)
As the diagonal elements of S and S-l are the Volterra operators, these equalities are the right factorization of the operators I- F12 Yzl and Z-F,, Y12 and the left factorization of the operators I- Yzl F12 and I- Y,, Fzl. If the pair F12, Y21 or F 213 Yl2 is given, then all diagonal elements F, 1, F22, Y, 1, Y,; are found uniquely from the factorizations (4.5), (4.6). The remaining elements are found from the equalities f’12
=
-(I+
YlJ’
YlzU+F,,),
which follow from the identities
F21
=
-V+F22)
r,,U+
F + Y+ FY = F + Y+ YF = 0.
&,I-’
THE INVERSE
SCATTERING
DEFINITION 4.1.
PROBLEMS
The pair of operators
data for the system (1.1) connected
called the scattering Remark
4.1.
FOR THE HYPERBOLIC
267
Fi 2, Y,, will be called the scattering wave; the pair Fzl, YIz will be
with the jrst
data connected
If the
EQUATIONS
coefficients
with the second
wave.
of the
(1.1) are connected
system
by the
relation
then the elements
of S and S-’
Let the relation (1.1) the identity
a
this identity
(4.8)
of the system
a
(x, YV = -W2k
ay
with respect
vlcIlII/I(+~> ~~llt,-ll11/1(-~~9ll2,l The equality
Yz2 = F;,.
(4.7) be fulfilled. Then for the solution
‘l&h holds. Integrating
by the relations r,, = FT,,
Y,, = -YIFT,,
r,, = -?FZ,, Proof:
are connected
(4.7)
rl = ul,
%(X, Y) = vu1 (x9 Y),
Y)12
to x and y, we obtain
= lb/92( .? + 4112,
- w2 ( .Y -
4112,.
(4.9)
(4.9) means that r) [l/bill2 - lla1112] = llb2112- lla2112, i.e.
It leads to the equalities (4.8). Note that in the case u2 = vii,, 7 = if, the scattering operator determined according to (4.8) and Lemma 4.1 by F,, or F,,.
is uniquely
5. The inverse scattering problem
The inverse scattering problem for the system (1.1) consists in finding the coefficients u1 (x, y), u2(x, y) when the scattering operator or scattering data are given. At the beginning we shall find the connection between the scattering data and the transmutation operators’ kernels. By comparison of the equalities (3.3), (3.4) we obtain assuming a, = 0, bl = F,,a,, b, = a,+F,,a, that B,, (K Y; s)+ :B,2@5
Y; ~3 &I(<,
$d5
= 0,
Y >s,
x
(5.1) 42k
Y; s)+
1 B,,(x, --oo
Y; <)F12(5,
4dt+f’,,(y,
4 = 0,
s > x.
268
L.P.NIZHNIK
Comparing
(3.3), (3.4) and assuming 6, = 0, a, = bI + YI1 b,, a2 = Y2r bI we obtain
&1(x,
Y > s, y;d+ ~822(X,Y;5)Y21(r,S)di:+YZI(X,S)=0, x (5.2)
%,(x9 y;s)+ i &,(x, v:OF,*(5,S)& =o, -CO
s > x.
By setting in the equalities (3.3), (3.4) first b, = 0, a, = Y,, b2, a, = b,+ then a2 =0, bz =F2raI, b, =a,+F,,uI we get
YZZbZ
&,(x,y;s)+ i B,,(x,y;~)F,,(5,s)d5=0, s > Y, -UTJ (5.3)
&,(x,y;s)+ ~B,,(x,Y;e,r,,ce,S)dg+~2(Y,S)=o, x > s, ‘Y
&1(x,
y;s)+ i B*,cGY; ~)F,,(5,S)d~+Fzl(x,S) = 0, -m
s > Y, (5.4)
&,ky;4+
p,,(*,Y:5Ms,s)dc
=o,
x > s.
The systems of equations (5.1), (5.2) are called the system of fundamental equations for the inverse scattering problem with given scattering data F12, Yzl. The systems (5.3), (5.4) are the fundamental equations for the inverse scattering problem when the scattering data F,,, YIz are known. For the Dirac equations (1.1) the systems of fundamental equations (5.1H5.4) are the analogues of usual Gel’fand-LevitanMarchenko equations in the theory of the one-dimensional inverse problems of spectral analysis.
LEMMA 5.1.Let FIz, Y,, or Fzl, Y12 be the scattering data for the system (1.1) with the coefficients satisfying the condition (1.2). Then for any x, y: + 00 the systems of the equations
UI(V)+ j F,,h
- CC < x, y <
rl)u,Wds =fih),
-m
(5.5)
UI(V)+ : L(s, v)u,(s)ds =fi(d, x
uz(~9+ are uniquely
jm FI 6, rl)ul(s) ds= fi(tl) z
solvable in L,( - CO, + co; E2) for
any
fi,f2ELM.
(5.6)
THE INVERSE SCATTERING PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
269
Proof: The solvability of the equations (5.5), (5.6) follows from the factorization properties (4.3) of the scattering operator. The proof is carried out in Appendix III. The solvability of the equations (5.5), (5.6) means solvability of the fundamental equations (5.1H5.4). Together with the formulas (3.6) it gives the solution of the inverse scattering problem for the system (1.1). THEOREM 5.1. The inverse scattering problem for the system (l.l), that is the problem of finding the coefficients ut, u2 of the system (1.1) when the scattering data F 121 YZI or Fx, Y12 are given (hence, when the scattering operator is given), is uniquely solvable. The coefticients u1 (x, y), u2(x, y) can be defined by the formulas (3.6), where Bij(x, y; s) are the solutions of the system of the fundamental equations (5.1)-(5.4).
Remark 5.1. If we consider the then the inverse scattering problem In fact, according to Remark 4.1 uniquely determined by one of the
system (1.1) with u2(x, y) =‘~ui(x, y), v] = q, is uniquely solvable for given F,, or F21. all the elements of the scattering operator are elements F,, or F,, .
6. The description of the scattering data For every Dirac system (1.1) with coefficients ui, u2 ELM there is a scattering operator S with the scattering data F12, .Yzl or F2i, Y12 which are the H.-S. operators with the kernels F,,(x, y), Yi2(x, y), F,,(x, y), Y2i(x, Y)EL~(E’). This defines the mapping of the scattering data in the space L2(E2, E2) (% (x> Y), Uz(JG Y)) z(F(x,
y), Y(x, y)),
(6.1)
where F, Y is one of the pairs F12, Y21 or F2i, Yi2. For definiteness we shall assume F(x, Y) = Fzl (x, Y), Yb, Y) = K2(x, Y). Now we shall examine in detail the mapping 91. We shall describe its range, that is we shall establish the necessary and sufficient conditions for the preassigned pair of H.-S. operators to be the scattering data for the system (1.1). THEOREM 6.1. In order that the pair F,,, Y,, of the H.-S. operators be the scattering data for the system (1 .l) with the coefficients ul, u2 EL, (E2) it is necessary and sufficient that the homogeneous system of the fundamental equations be only trivialy solvable, i.e. for any -CO < x, y < + co the system VI
@I)+
,f
f’2,
(~9
v)
cp2 (s)ds
=
0,
-UZ
cp2 (v)
+
T
(6.2) &
2 6,
rl)
cp1M
ds
must have only the trivial solution in the space L2.
=
0
270
L.P.NIZHNIK
Proof: The necessity has been established in Lemma 5.1. Let the system (6.2) have only the trivial solution. By virtue of the Fredholm property the nonhomogeneous system (5.5) is uniquely solvable at any fi , fi E&. We shall write the solution of the system (5.5) in the form YVf
a =(I+H(x,
=(I+R(x,
Y))L
(6.3)
where H(x, y), and consequently R(x, y), are H.-S. operators uniformly in the H.S. norm depending on x, y on the compact set [ - 00, + m] x [ - co, + co J (here [ - co, + 001 is the extended real axis which is compact with respect to the natural topology). As a consequence the operator function R(x,y) is uniformly bounded in x, y with respect to the H.-S. norm. Therefore the solutions of the systems (5.3), (5.4), as special cases of (5.5), can be represented in the form of (6.3) and
~jlBl& [~I&(x,
Y; 1)12dYd? G
c, k =
Y; 1)12dxdrl G c,
(6.4)
1, 2.
From the systems (5.3), (5.4) and (6.4) we infer existence of the functions 4(x,
Y) = -&,b,
Y; x-0)
= +
Y,,(y,
x)+
;B,,(x,
y; s)
Yi2(S, x)ds,
Y
(6.5)
uz(x, Y)'=B,,(x,y;Y-t-0) = --Fzl(x,y)- j &2(x, Y;s)F21(S,Y)dS --co
and the fact that they belong to the class L,(E’). Now we shall show that the scattering data for the system (1.1) with the coefficients uk(x, y), k = 1, 2, defined by the equalities (6.5) coincide with the initial operators F2i, Yi2. In terms of the solutions Bij(X, y; q) of the equations (5.3), (5.4) we define the functions Aij(x,y; q): A,l(x,Y;rl)=B,,(x,Y;vl)- i -m A,,k
Yi 'I) =
~I,(-%
B,,(<,
Y;5)%1(5,Y;r)&,
Yl rl)---B,,(vl,Y; VI)-pl,(r,Y:
OB22(5,
y:rl)G, (6.6)
A,,(x,Y;y)=B,,(x,Y;tl)--B,,(x,tl;9)-jB21(X15;e)B,,(X,e;ad~, Y
A22(x,
Y; r)=
B,,(x,
y;rl- ;&,(x, 5; t)B,,(x,t;rl)4
THE INVERSE
SCATTERING
It is easy to verify that
A,l(&
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
271
Aii satisfy the equations
y; q)+ j Al2(X, y; s)Fz1(s,tl)ds= 0, -CO
A,,(x, &1(x,
Yi
Y;
v)+
q,Mx7 Yi s)-A,,(vl,
49+ i CA22k Y; s)--‘4*,(x,
Y;
dl Y,,(s, v)ds= 0,
rl; s)lF21h
?)dS= 0,
-CC
Mi
Yi
v)+
~‘h(x, Y; 4 hz(s, rl)4 = 0.
par the functions V,(V) = Ai,(x+a, obtain the system of equations ‘PI
Y; v)--Ai,@,
(vl)+ i cPz(s)Fn(s, rl)ds = -m
Y; tl), k = I, 2, from (6.7) we
x~6A,*~x+S, y; s)F21(s,rl)ds, x
m
cpzh)+ .Fcpl(S)Kz(s, ?)dS= 0. Y
The right sides of the system (6.8) at S --+O are of order o(6). By virtue of (6.3) the solution of the system (6.8) is of order o(6). Then the functions Aii(x, y; Y))do not depend on x. From (6.6) we obtain Al1 (x, y; q) -+O as x --t - co, A,,(x, y; q) =,O. Therefore Aii(x, y; II) E 0, i = 1, 2. Similarly from the system (6.7) we obtain A*i(Xp y;lj) =O, i = 1, 2. By comparing the equations (6.6) with Aij = 0 and the equations (3.8), (3.9) connecting the transmutation operators’ kernels with the coefficients of the system (1.1) we conclude that these equations coincide when uk (x, y) are chosen according to the formulas (6.5). Due to their unique solvability we conclude that the solution Bij(x, y; q) of the system (5.3), (5.4) are the transmutation operators for the system (1.1) with the coefficients uk(x, y) which are given by the formulas (6.5). But the scattering data are uniquely determined by the transmutation operators’ kernels. It leads to the assertion that the initial operators Fzl, Y,, are the scattering data for the system (1.1) with coefficients given according to the formulas (6.5) where Bij are the solutions of the systems (5.3), (5.4). Remark 6.1. The totality of the scattering data forms an open set in the space of the pairs of the H.-S. operators. Proof: If F, 1, Y,, are the scattering data then the operator function R(x, y) of (6.3) exists for x, y EC- co, CCJ].Under small, with respect to the H.-S. norm, variations of the scattering data the function H(x, y) has a small variation and consequently R(x, y) exists, so the conditions of the Theorem 6.1 are fulfilled.
L. P. NIZHNIK
272
Remark 6.2. The pair of the operators F2i, Y,, for which the H.-S. norm is less than 1 are the scattering data. Proof: The conditions of Theorem 6.1 are fulfilled by virtue of the condensed mapping principle. Remark 6.3. If u2 = U, in the system (l.l), then Yi2 = -F$, and totality the elements F,, constitutes the whole space of the H.-S. operators.
of all
Proof: Let F,, be an arbitrary H.-S. operator. We shall show that the system (6.2) with Y,, = -FTi has only the trivial solution. From (6.2) we obtain ~IcMV)l2~V+
5
lcp2(1)12~rl
Y =
i[-
7
Y
-CO
=
i
~,(s;DF,,b,
cW]cp,W~+
j
[-
~rp~(s)Y,,(s,
tt)+~2Wv
-a,
dsqb~2OF,,(s,
rl)rp,(rl)-(~2MF21@,
+~1(tt)l~rl=~-~=0~
-03
Thus (Pi = 0, k = 1, 2, and by virtue of the Theorem 6.1 the operators -F& are the scattering data. From (6.5) we obtain u2 = Ui. Remark rate such
6.4.
For arbitrary
scattering that
llf’2l
data -
scattering
F2i, Yi2 =
data FZ1, Y,, and any E > 0 the degene-
FEZI(x, y) = 2 fi’(x)gt (y),
Yf2(x, y) = 5 pi(x) qi(y)
i=
i=
1
exist
1
GIIH.-S. < E, II q2 - Y~211H.-S. < E.
Proofi According to Remark 6.1 for the scattering data F2i, K2 there exists a neighbourhood consisting exclusively of scattering data. Since an arbitrary H.-S. operator can be arbitrarily well approximated in the H.-S. norm by finite-dimensional operators, this leads to the assertion formulated above. The results of Theorem 6.1 and Remarks 6.1-6.4 can be reformulated in terms of the scattering data operator ‘3. The scattering data operator ‘3 generally defined by the equality (6.1) in the special case at u2 = viii as Y, 2 = - nFT1 is defined by the equality 2% (x, Y) =
F,,
6,
Y)
(6.10)
in the space L2(E2). THEOREM 6.2. The scattering data operator % mapping the coefficients of the system (1.1) into the scattering data is continuous in L2. Its range is an open set. The operator ‘II-’ exists and is continuous and its action can be constructively described by means of the uniquely solvable systems of equations (5.3), (5.4) and the formulas (6.5).
THE INVERSE
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273
7. The exact solutions
We shall give some examples when the inverse scattering problem for the system (1.1) can be explicitly solved. This will occur when the fundamental integral equations of the inverse scattering problem (5.3), (5.4) can be explicitly solved. Let the scattering data be degenerate, that is representable in the form
J-21(XTY) = f:‘(X)fl
= ;
(Y)
n=
fin(X)f 1
(7.1) &2(x, Y) = s:‘Wgz(Y)
= f sl.(X)S*n(Y), “=1
where the components of the vectors f, = col& , *.*,.h~h’), gk = col(gk,, .**> gkh’), k = 1,2, belong to Lz, and tr denotes the transposition operation. In this case the systems (5.3), (5.4) are reduced to the linear algebraic equations and their solutions are of the form
&I (xvY; rl) = s:'WQz(x)C~+Q, (~)Q,(x)l-‘h
(~1,
Biz(x,
Y;
v) =s’;(y)C~+Q,(x)
&,(xv
Y;
rl) = -f~(x)Cz+Ql(~)Qz(x)l-~fi(?),
Q&91-‘g,(rl),
(7.2)
B,, (x, Y; rl) = --f;(x) CI+Ql(~)Qz(x)l-' QI(~)g2(tl)Here Qi, Q2 are the matrices of the form
Ql (Y) = -
is,(s)g:‘(s)b Y
Q2W =
; g2(4f:‘Wds+
(7.3)
--Go
The formulas (6.5) give the following expressions for the coefficients of the system (1.1): ul(x, Y) =s:'(~)Cz+Q,(x)Q,(y)l-'gz(x), uz(x,
Y) = -S~‘(X)C~+Q~(Y)Q~(X)I-‘~~(Y).
(7.4)
The general solution of the system (1.l) with the coefficients of the form (7.4) is th’k
y) = a, (v)+s;(Y)C~+Q,WQ,
(YJI-‘(Q,(x)
ih @Ial b)dsY
-
j 92W2W~)~ -CO
(7.5)
214
L. P. NIZHNIK
tiz(x, Y) = b,(x)--f:'b)Cl+Q, (~>Qz(x)l-‘(Q, (Y) j gz(s)bz(s)ds+ -*
+
7-h 6) 4 Y
(4 ds),
where a,, b, are arbitrary functions belonging to L,. The formulas (7.5), (7.4) can be represented in the operator form for arbitrary scattering data F,, = F, Y, 2 = Y The admissible solutions of the system (1.1) can be represented in the form $1 = Cl -
YQxFP,I- ’ (a(Y) - YQx(b(Y)),
tiz = [I-FP,
YQJ-‘(b(x)-
The coefficients of the system (1.1) are expressed means of the formulas ul(x, Y) = Cl-
FP,(a(x)).
in terms of the scattering
YQ,FP,I-’
uz(x, Y) = -WFP,
(7.6) data by
Y(Y, 4,
YQxl-‘J’k
Y).
(7.7)
The operators in the right-hand sides of the formulas (7.6), depending on x and y as on parameters act on the functions a and b. The arguments of these functions indicate that the result will be considered as a function of the corresponding argument. The right-hand sides of the formulas (7.7) are the kernels of the corresponding operators. It should be noted that the formulas (7.7) can be obtained by substituting (7.6) into the Dirac system (1.1). 8. The non-linear integrable evolution equations in two spatial dimensions We shall consider the non-linear evolution equations which are integrated by the inverse scattering method [12]. For the inverse problem we choose the inverse problem for the Dirac system (1.1). The simplest example of a nonlinear equation is the non-linear Schrbdinger (Davey-Stewartson) equation in two spatial variables which for the first time was obtained in the paper [15] and has been integrated by the inverse scattering method in the papers [S, 9, 43: iu, = -u,,
- ku,, + vwu >
wxy = 2 ($+
k-$),ui2,
(8.1)
where k and q are the real scalars which may. be reduced to the values k = + 1 and r) = & 1 by scaling the independent variables. The quasi-potential w is expressed in terms of the solution of the equation (8.1) by the formula w = -2&&(x, Y
s; t),‘ds+2kd
i lu(s, y; t)12ds. aY4
(8.2)
THE INVERSE
SCATTERING
PROBLEMS
Another example of the non-linear analogue of the modified Korteweg-de
FOR THE HYPERBOLIC
equation in two spatial Vries equation [7]:
EQUATIONS
dimensions
is the
+ uyyy+ @a), + (wu), -$ (0, + w,) u, u, = %cXX u, = 3(U2),,
u(x, Y; %o The equations
(8.1) and (8.3) admit
(8.3)
WY= 3 (U2),.
For the equations (8.1) or (8.3) the Cauchy solutions satisfying the initial condition
problem
= %k
consists
in finding their
Y).
(8.4)
the Lax representation
LP-QL=O, where L is the two-dimensional system (1.1)
275
Dirac operator
(8.5) defined by the left-hand side of the
For the equation (8.1) u1 = U, u2 = VU, for the equation (8.3) aI = -u2 operators P and Q for the equation (8.1) are of the form [4, 6, 71:
= U. The
(8.7)
and for the equation
P=
(8.3) of the form
c
D-&,-w,
b-&l,
-3&uu,
3% a,
.3ayu,
1 (8.8)
D--qw,-v,_,'
D = a,-a;-a;-va,-wax. 9. The evolution of the scattering data Let the coefficients uk of the system (1.1) or the Dirac operator (8.6) depend on and satisfy one of the non-linear equations (8.1), (8.3). Then the scattering data F,,, Y12 corresponding to the coefficients ul, u2 also depend on t as a parameter. We shall study the evolution of the scattering data. t as a parameter
L. P. NIZHNIK’
276
Zf rl/ is the solution of the Dirac system (1.1) with the coefficients
LEMMA 9.1. satisfying
the non-linear
then the function Proof:
account
equations
which are equivalent
to the Lax
cp = P+ will also be a solution of the system (1.1).
Applying the operator equality (8.5) to the function that LIc/ = 0 we have L(PII/) = 0.
LEMMA 9.2.
equation (8.5)
Zf the function
tj and taking into
u(x, y, t) satisfies the non-linear equation (8.1) then
the scattering
data-for the Dirac system (1.1) with u1 = u, u2 = nii are determined the function-F(x, y, t) (F,, = F, Y12 = - nF*) satisfying the linear equation .dF
a2F --
lx-ax2
hd2F
o
JJF=
by
(9.1)
*
Proof: Let Ii/(x, y, t) be the solution of the system (1.1). Then as it follows from the Lemma 9.1, the function cp = L$ is the solution of the system (1.1). If a, = $1 (-00, y), a2 = I,/I~(x, -00) are the asymptotics of the solution $ then the asymptotics of the solution Ltj are expressed in terms of al, a2 by the relations cpl(-03,
y) =
cp2(x, -a)
By virtue of the assumption (9.1) is valid. 10. The inverse scattering
ii+kz C
S’
)
al(y), (9.2)
= cii-$ja2(x).
u1 = U, u2 = v]ii and the condition
(8.2), the equation
method
To solve the Cauchy problem (8.1), (8.2), (8.4) or (8.3), (8.4) by the inverse scattering method it is necessary to construct the scattering data F(x, y, 0) at an initial moment using the initial data uo, to solve the Cauchy problem with these initial data for the Linear equation (9.1) and then to solve the inverse scattering problem to find u(x, y, t). The solutions of the Cauchy problem for the equations ’ (8.1) or (8.3) can be represented in the form u(x, y, t) = W ’ eeiAr Pluo,
where the scattering
A determines +k--
a2 w
data operator
the evolution
for the equation
(10.1)
VI is defined by the equality (6.10), the operator
of the scattering
(8.1) or A = z‘($+-$)
a2
data
and is of the form for the equation
A = Q
(8.3).
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
277
Using the representation (10.1) for the solutions of the non-linear equations (8.1) and (8.3) we can construct the explicit solutions of these equations. In addition, the representation (10.1) can be used to prove the existence of the solution of the Cauchy problem on the basis of the statement converse to Lemma 9.2. THEOREM 10.1. L,et the function function together with its derivatives derivatives with respect to x and y homogeneous equations (6.2) with F,, solution in L,. Then the equation K(x, y; <, r)-t-F(x,
t; t)+q
F(x, y, t) satisfy the equation (9.1) and this with respect to t and its first and second belong to L2(E2). Also let the system of = F and Y12 = - nF* have only the trivial
k.u,
Y; a, t)
i
&S,
ct:
t)F(j?, 5; t)d/3dcc = 0
--Di
Y
(10.2) is uniquely solvable and the function the non-linear equation (8.1).
u(x, y, t) = r]- ’ i? (x, y; y, t) is the solution of
The proof of this Theorem is given in Appendix IV and is actually contained in the paper [9]. The statement of the Theorem 10.1 is equivalent to the assertion that the function u (x, y, t) = OI- ’ e-tAf F
(10.3)
. is the solution of the equation (8.1) if it is determined. Thus the solution of the Cauchy problem (8.1), (8.4) can be represented in the form (10.1) and, conversely, if the formula (10.1) is determined then it gives the solution of the Cauchy problem. To obtain the solutions of the non-linear equations (8.1) and (8.3) in an explicit form we shall use the formulas (7.4) for the explicit solutions of the inverse problem. In this case for the equation (8.1) the functions f = fI = g1 and g = g2 = -qf2 must depend on t and satisfy the equations = 0.
(10.4)
For the equation (8.3) the vector functions f = fi = -gl, and satisfy the equations
g = g2 = f2 depend on t
iJ;+hfyy = 0,
J;-f,,, 11. Investigation
= 0,
ig, +g,,
St-Sxxx
= 0.
(10.5)
of the Cauchy problem by the inverse scattering method
Consider .the Cauchy belonging to L2.
problem
(8.1), (8.4) in the space L2 with the initial data
278
L. P. NIZHNIK
DEFINITION 11.1. The function u(x, y, t) continuous in t, whose values with respect to x and y belong to L,(E’) is called the solution of the Cauchy problem (8.1), (8.4) if at t = 0 it coincides with the initial data U(X, y, 0) = uO(x, y) and if the sequence u,(x, y, t) of the smooth in x, y, t solutions of the equation (8.1) exists and is such that Ilu-u,,IIL2 -+O as n + a uniformly in t, belonging to an arbitrary finite interval [0, T]. The functions u,(x, y, t) and their derivatives respect to x and y up to second order belong to L2(E2).
with
The solution of the Cauchy problem (8.1), (8.4) in the space is unique. For arbitrary initial data from L2 the local in time solution of the
THEOREM 11.1. L,(E’)
Cauchy problem exists. For arbitrary interval of time.
small initial data from
L,
the solution exists
on an
Proof:
The uniqueness of the solution of the Cauchy problem (8.1), (8.4) follows from the possibility of representing it in the form (10.1). The existence of the local in time solution follows from the continuity of the unitary operator eTiar, from the continuity of the operators 2I, ‘9-l and from the openness of the range of the operator ‘+2I (Remark 6.1). By virtue of these circumstances the formula (10.1) has a sense for arbitrary initial data u0 ELM if t is sufficiently small. If the initial data are sufficiently small then by virtue of the continuity of 2I and unitarity of . 1ess than 1. But then formula (10.1) has a sense according to e -iAt, jle-iA’ pIa, is Remark 6.2. For the case r7 = 1 in equation (8.1) the result of the Theorem 11.1 can be refined using Remark 6.3.
Ifq = 1 the Cauchy problem (8.1), (8.4) is uniquely solvable in initial datu u0 ELM on an arbitrary interval of time.
THEOREM 11.2. L, for
arbitrary
Above the exact solutions of the nonlinear equations (8.1), (8.3) were given. The question arises: To what extent is the set of such the solutions rich? THEOREM 11.3. Let u be an arbitrary solution in L, of the Cauchy problem (P.l), (8.4) and E > 0 be arbitrary. Then on an arbitrary interval of time [0, T] there exists a solution u,(x, y, t) of the form (7.4) such that sup
O
Ilu- UT/lL2(E2)
< E.
(11.1)
Proof: Let F, be the scattering data constructed from the initial data u. that is F. = ‘%uo. By virtue of (10.1) u = W’e-iA’ F,. The openness of the range of the
operator ‘%, the compactness of the finite interval [0, T], the unitarity of the operator eeiAz imply that 6 > 0 exists such that for an arbitrary function F satisfying the condition IIF- Foil < 6 the function u = %I-’ emiArF is defined on the interval t E[O, 7’J. For F we shall take the degenerate kernel F = Fdeg. Then uT = a-1 e-iAt Fdeg belongs to the class of the explicit’ solutions of (7.4). On the
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
other hand, due to the continuity of ‘%I-’ and the unitarity we can find &: 0 < 6, < 6 such that IIu-+IILz
= II+%-’ e -iAtFo-~-le-iA’Fdegll
<.5,
EQUATIONS
279
of emiA’, for any E > 0 tE[O, T-J:
Appendix I. The Volterra type equations We shall consider
in a Banach
space B the linear equation u = h+Au.
(A.I.l)
THEOREM A.I. The solution of the equation (A.I.l) for any h EB exists and is unique if with respect to the monotone in TE( - co, + co) set of the semi-norms llullT (IIullm = [lulls) the, inequality
IIA~IT G i a(d lb4 dT
(A.I.2)
--oo
holds for an integrable function
u(z)
5 a(z)dr
< +CO.
(A.I.3)
-a, Proof:
By iterating
the inequality T
lIA”4b c J a(z,)dr, -m
From
(A.I.4) we have
(A.I.2) n times we obtain
Tl
.r ~b,)d~,...
-CC
rn-
1
.r 4dI4lr$n.
-m
(A.I.4)
L. P. NIZHNIK
280
Appendix II. The existence of the transmutation
operators
THEOREM A.11. Let ul, u2 be functions (3.8), (3.9) which satisfy the inequalities
Then the solutions of the systems
~~s~pI~,,(x,y;~)1~dyds
<
+m,
in L,.
J+up~B,,(~,y;s)~~dxds Y
<
(A.II.l)
+m
exist and are unique. Proof: By elimination we can obtain from the systems (3.8), (3.9) four equations for each of the functions B1i, B,,, B2i, B22. The equation for Bzl is of the form Y; d--2(x,4+
B,,(x,
ju2(x,
5)
; -m
u,(vl, <)B,,(rl,
5; dW<
=
0.
(A.II.2)
To prove solvability of equation (A.II.2) we use Theorem AI. For the Banach space B we shall take the space of functions of three variables B,, (x, y; s) with the norm IIB21(x,
Y;
s)lls =
[~,bplB2,(x,
Y; ~)l~dxd$‘~.
Y
For the set of the semi-norms
in B we shall take
IIB21ll~ = [&$B21(x, The fundamental
inequality
AB,,(x,
Y; s)12dxds]1’2.
(A.I.2) is fulfilled by the operator
Y; 4 =
"cuz(x, Y
0
l‘cu,(rl, OB21h, -m
5; s)dvds
with the majorant a(x) = lIuz(x, Therefore Similarly
.)II lb1k .)II.
by Theorem A.1 we obtain unique solvability of equation we prove existence of the operators B,,, B1 1, B22.
(A.II.2).
Appendix III. Solvability of the fundamental equations of the inverse problem In this Appendix we shall prove Lemma 5.1, that is we shall show uniqueness of solution of the fundamental equations (5.5), (5.6) of the inverse problem (see [3, 21). The scattering operator S admits the two-sided factorization S =Z+F
=(I+W+)(Z+k),
S-l
= I+ Y = (I+ V+)(Z+ I’_).
(A.III.1)
THE INVERSE
SCATTERING
We shall consider
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
281
the operator @n =
U+QAFP,+PAYQ,L
(A.III.2)
where Pn is the set of the orthoprojectors on the semi-axis x 3 1 that is gLf(x) = 0 (x- A)f(x) and Qn = I- Pk. The operator @‘1can be represented in the form
=(I+Q,W+)(I+QnW-)(z+I/, QJU+v-Qn). Since the operators (Ii-Q1 W,), (I+ V, QJ are reversible in L,, the operator Gi is reversible in L2. It follows from the reversibility of the operator @, that the homogeneous equation Qnu = 0 has in L, only the trivial solution. Taking into account that the diagonal elements Fi,, Fzz, Yll, Yz2 are Volterra operators, the equation @;Iu = 0 can be written in the form ui (vl)+ J Fiz(s,
q)uz(s)ds
= 0,
uz(rl)+ ,f yzl(S, Vl)Ul(S)dS = 0, --oo
(A.III.3)
u1(ul)+ ,f %2(s,rl)%(S)dS= 0, -00
uz(rl1-t
~F,,(s, v)u(s)ds= 0.
Using the Fredholm property of the system (A.III.3), passing over to the conjugate system, we can obtain the trivial solvability of the systems cpl (rl) + 7 cpz(s) r,l (ST q) ds = 09
(A.III.4) 401 (rl)
+
I’
cpz
(4
f’21b
9)
ds
=
0,
-a,
Thus the solvability
of the system (5.5)-(5.6) for x = y = 1 is shown. Then
the
282
L. P. NIZHNIK
operator TIzSTJ is the scattering operator for the system (1.1) with the coefficients i& = U1(x, y+z), ii2 = uz(x, y+z). The operator TL,ST: has the same factorization properties as the operator S. (TI,S-’ T& = Y,, T,, therefore the system of the But (TI,ST,),, = T_,F,,, form (A.III.4) for the operator TI,Sc is
pfm) YZl(S,VI--Z)ds= 0,
%(?I+
A
“rcP1(4F,,(s-z, ?)dS= 03
a(r)+
--oo
(Pled+ j cPz(s)F,,h rl-z)ds = 05 -‘X3
cpz(d+ This system is equivalent
T’pl(4 Y1z(s-z, v)ds= 0.
to the systems (5.5), (5.6) for x = 1, y = I-z.
Appendix IV. The Cornille Theorem We shall give the proof of Theorem 10.1 (see [9]). The trivial solvability of the homogeneous equations (6.2) due to their Fredholm property implies uniqueness of the solution of the systems (5.3), (5.4) for F,,(x, y) = F(x, y; 0 and Yi2(x, y) = -$(Y, x; 4 We shall differentiate both equations in (5.3) with respect to x, multiply both equations in (5.4) by Bi2(x, y; x, t) and subtract the hrst equations and the second equations in the obtained systems. Then by virtue of the trivial solvability of the system (6.2) we have
d&l (x,
y;
s, 0
8X
--&2(x,
y; x, Oh (x7Y; s>t) = 0. (A.IV.l)
y; s, 0 ----lz(x,
8&,(x,
.
i?X
Similarly
we obtain
y; x, 0&2(X,
s, 0
0.
relations
a&,(x,
t)
+&1(x,
Y; Y,
t)Blz(x,
Y;
s,
= 0, (A.IV.2)
&I
(x, Y; s, G ay
+B,,(x,
Y; Y, t)B~,(x,
y; s, t) = 0.
THE INVERSE
SCATTERING
PROBLEMS
FOR THE HYPERBOLIC
EQUATIONS
283
2
Applying the operator iG-&
-k
’ to the system (5.3), using (A.IV.l),
(A.IV.2), the trivial solvability of the system (6.2) and taking into account the F,, (x, s; t) = 0,
Y,,(y, s; t) = 0 we
get B,,
k
Y; ST 0
(A.IV.3)
+
2kB,, (x, y; y, t)
B, t (x,y; s, t) =
o.
Assuming s = y in (A.IV.3) for U(X, y, t) = q-i B2i (x, y; y, t) we obtain the equation (8.1) with
The relation (8.2) follows from (A.IV.l), (A.IV.2). From (5.4) we get the equation (10.2) with respect to K(x, y; s, t) = B,, (x, y; s, t) by eliminating B22(~, y; s, t). REFERENCES [l] Nizhnik, L. P.: Vkr. Math. .I. 24 (1972), 110-113. [2] Nizhnik, L. P.: Nonstationary Inverse Scattering Problem, Naukova Dumka, Kiev, 1973. [3] Nizhnik, L. P., Pochinayko, M. D., Tarasov, V. G.: in: Spectral Theory of Operatorsin Problems of Mathematical Physics, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev, 1983, 72-93. [4] Nizhnik, L. P., Pochinayko, M. D.: Funct. Analys. and Appl. 16 (1982), 80-82. [S] Nizhnik, L. P.: Dokl. Acad. Sci. USSR 254 (1980), 332-335. [6] Nizhnik, L. P., Pochinayko, M. ‘D.: in: Spectral Theory of Differential-Operator Equations, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev, 1986, 89-98. [7] Pochinayko, M. D.: (see above), 103-106. [S] Ablowitz, M. J., Haberman R.: Phys. Rev. Left. 35 (18) (1975), 1185-1189. [9] Cornille, H.: J. Math, Phys. 20 (1979), 199-209. Cl01Nizhnik, L. P., Tarasov, V. G.: Dokl. Acad. Sci. USSR 233 (1977), 300-303. El11Kaup, D. J.:Physica 1D (1980), 45-67. WI Ablowitz, M. J., Segur, H.: Solitons and the Inverse Scattering Transform, SIAM Philadelphia, 1981. Cl33 Nachman, A. I., Ablowitz, M. J.: Stud. Appl. Math. 71 (1984), 251-262. Cl41 Fokas, A. S., Ablowitz, M. J.: J. Math. Phys. 25 (1984), 2494-2505. WI Davey, A., Stewartson, K.: Proc. R. Sot. Lond. A338 (1974), 101-110.