Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
Подождите немного. Документ загружается.
200
A.
S.
Fokas
7
Rigorous Considerations
The eigenvalue problem
(4.2)
is considered rigorously by Sung and
the author in
[21]
where it is assumed that
Q
belongs to the Schwartz
class and then it is shown that: (a) The direct problem is always
solvable. Namely the integral equations for
p*
and
po
are always
solvable without
a
small norm assumption. (b)
If
where
then the inverse problem is also solvable. (c) The norm
of
T
can be
estimated from
a
norm
of
Q
as follows:
Thus in general the inverse problem is solvable only under
a
small
norm assumption. However, if
Q
have some symmetry, then the
small norm assumption can be relaxed. In particular if
Q
is skew
Hermitian, i.e.
Q*
=
-9,
then it can be shown that the inverse
problem is always solvable.
The above results can be used to prove that the skew-Hermitian
N
wave interactions
(Q*
=
-Q)
and the “focusing”
DSI
equation
(E
=
-1
in equation
(6.2))
have global solutions for any Schwartz
initial data.
With respect to
KPI:
Rigorous results for the direct problem are
given by Segur
[25].
Solvability is established only if
a
certain norm
is small. Zhou
[26]
has recently shown that the inverse problem is
solvable without
a
small norm assumption.
With respect to
KPII:
Wickerhauser
[27]
has established solv-
ability of the direct problem under
a
certain small norm assumption.
Integrable Nonlinear Equations
201
For
the inverse problem one does not need
a
small norm assumption.
Indeed, for
o
=
-1,
equation
(5.4)
yields
However, if
q
is real then
p(x,
y,
-rc)
=
p(x,
y,
k)
and equation (7.1)
shows that
p
is
a
generalized analytic function
[28].
With respect to DSII:
For
the defocusing case
(Q*
=
Q),
Beals
and Coifman
[29]
have shown solvability without
a
small norm as-
sumption. This is
a
consequence of the following facts: (a) Both the
direct and inverse problems satisfy similar kinds of
2
x
2
matrix
8
problems. (b) These
8
problems are equivalent to scalar
8
problems
for generalized analytic functions.
For
the focusing case
(Q*
=
-Q)
of
DSII rigorous results assuming that
a
certain norm is small, are
given in
[30].
Also this is
a
special case of
a
general
N
x
N
problem
in multidimensions rigorously studied by Sung and the author
[31].
8
Self Similar and Periodic Solutions
8.1
It is well known that similarity reductions of integrable PDE’s in
1
+
1
lead to ODE’S of Painlev6 type.
For
example
[32]
the transformation
The Connection with PainlevC Equations
q(x,T)
=
(3T>-’/’~(t),
t
=
(3T)-’/’x,
(8.1)
maps the KdV equation
qT
+
6qqx
+
qxxx
=
0,
to an ODE for
u(t)
of third order. This ODE in turn can be mapped
to the second Painlev6 equations (PII),
(8.2)
(8.3)
uII
=
u3
+
tu
+
8,
8
constant.
Similarly, the Lie point group
XI
=
x
+
6TA,
TI
=
T
+
A/a,
qI
=
q
+
A,
(8.4)
202
A.
S.
Fokas
leaves equation
(8.2)
invariant and gives rise to the transformation
The transformations
(8.5)
map
(8.2)
to
ulll+ 6uu‘
+
CY
=
0,
i.e. to Painlev6
I
equation,
U’I
=
-3u2
+
at.
(8.6)
Flaschka and Newel1
[33]
and Jimbo, Miwa and Ueno
[34]
have in-
troduced
a
powerful approach for studying the initial value problem
of equations of the Painlev6 type. Their approach can be thought
of as an extension of the inverse scattering transform method for
solving
ODE’S:
They have shown that solving such an initial value
problem is essentially equivalent to solving an inverse problem for
an associated isomonodromic linear equation. This inverse problem
can be formulated in terms of monodromy data which can be calcu-
lated from initial data. The author and Ablowitz
[35]
have shown
that the inverse problem can be formulated as
a
matrix, singular,
discontinuous Riemann-Hilbert (RH) problem defined on
a
compli-
cated contour. Hence techniques from RH theory can be employed to
study the solvability of certain nonlinear
ODE’S.
The above method
is called inverse monodromic transform (IMT), and can be thought
of
as
a
nonlinear analogue of the Laplace’s method for solving linear
ODE’S.
A
rigorous methodology for studying the RH problems asso-
ciated with isomonodromic equations has been recently introduced
by the author and Zhou
[36].
Using this methodology it can be shown
that Painlev6 type equations admit in general global, meromorphic
in
t
solutions. Furthermore,
for
special relations among the mon-
odromy data and for
t
on certain rays, these solutions are bounded
for finite
t.
8.2
Periodic
Solutions
The initial value problem for periodic initial data for several equa-
tions in
l
+
l
have been solved using techniques from algebraic ge-
Integrable Nonlinear Equations
203
ometry (see for example
[37]-[41]).
For
equations in
2
+
1
there exist
only weaker results (except for KPII): Krichever and Novikov
[42]-
[44]
have found an effective way of generating periodic solutions for
solvable equations in
2
+
1.
Their method can be thought of as an
extension of the dressing method from the complex plane to an ar-
bitrary Ftiemann surface. In what follows we summarize the main
construction. The interested reader is refered to
[45].
Let be
a
compact Riemann surface of genus
g
2
1.
It
is
well known
[46]
that on
I'
there exist
g
holomorphic differentials
w1,
-
-,
w,
normalized by the conditions
where
ul,--~,ug
and
bl,...,
b,
denote
a
basis of canonical cycles.
The matrix of b-periods, the
so
called Ftiemann matrix is defined by
Bjk
+
ik
wj,
j,c
=
~,*-*,g.
(8.8)
It can be shown that
B
is symmetric and has negative definite real
parts. Let
J(r)
be the Jacobi variety associated with
r.
The Abel
map
A
:
r
+
J(q,
A(P)
=
(AQ),
- -
-,
A,(P)),
is defined by
P
&(P)
=
Lo
wk,
k
=
1,...,g
.
(8.9)
An important role in the theory of periodic solutions is played by
the Riemann theta function which is defined by:
(8.10)
where
z
=
(21,.
.
-
,
z,)eCg
is
a
complex vector,
B
is
a
Ftiemann matrix
and
(N,
z)
=
C:='=,
Nit;.
The summation in
(8.10)
is
taken over the
lattice of integer vectors
N
=
(N1,.
.
.
,
N,).
204
A.
S.
Fobs
d(z)
is analytic in
439
furthermore it possesses the following im-
portant properties:
(4
B(A(P)
-
A(D)
-
I<)
has exactly
g
zeros
P1,--.,Pg,
(8.12)
where the vector
K
has components
and the vector
A(D)
has components
Using the above notion it is straightforward to construct periodic
solutions of integrable equations. The construction is based on the
Baker-Akhiezer
function
!P(P):
Let
PI,
-
-
-
,
Pg
be
g
generic points on
I?
(more precisely assume that the divisor
D
=
P1
+
.
.
Pg
is non
spatial). Let
Q
be
a
point on
r
corresponding to
00,
i.e.
z
=
z(P)
with
z(Q)
=
0
in the neighborhood of
Q,
and
b
=
$.
Let
q(b)
be an
arbitrary given polynomial. Then there exist
a
unique function
!P(P)
with the following properties:
(a)
!J!(P)
is meromorphic everywhere
on
r,
except at
P
=
Q.
(b) Its only poles on
r/Q
are at the points
PI,..
,
Pg.
(c)
Q(P)
has an essential singularity of the form
!J!(P)
N
cdk)
near
P
=
Q
(c
is
a
constant).
The uniqueness is
a
simple consequence of the fact that the de-
visor is non-special. It is simple to show that
(8.13)
JpP
n
B(A(P)
-
A(D)
+
U
-
I<)
B(A(P)
-
A(D)
-
I<)
'
Q(P)=e
0
Integrable Nonlinear Equations
205
where the vector
U
has components
c
(8.14)
and
R
is
a
normalized differential of the second kind on
r
with prin-
cipal part at
Q
of the form
dq(k),
i.e.
R=
O,j=
1,..-,g
and
R~q(k)
as
P+Q.
(8.15)
Indeed, using
(8.12)
it follows that
!P
has poles at
PI,
- - -
,
Pg.
Also
it is obvious that
!@
has the correct behavior at
P
=
Q.
To show
that
!P
is meromorphic we note that if we choose another path of
integration then
Ji
+
J&
+
&
where
y
is an
N
multiple of the
a
cycles and an
M
multiple
of
the
b
cycles. Thus
Hence, the additional terms in
(8.13)
cancel (using
(8.11))
and the
meromorphicity of
*(P)
is established.
Taking
q(k)
=
kz
+
k2y
+
k3t
and using the dressing method on
!@
it follows that
k~+k~y+k~t
!P=e
satisfies the Lax pair associated with the KP. In particular
-(ay
+
a:
+
q)Q
=
0,
where
q
.I.
-2!Pc)
solves
Acknowledgements
206
A.
S.
Fobs
This work was partially supported by the Office
of
Naval Research
under Grant Number N00014-88K-0447, National Science Founda-
tion under Grant Number DMS-8803471, and Air Force Office
of
Scientific Research under Grant Number 87-0310.
Bibliography
[
11
Symmetries and Nonlinear Phenomena,
World Scientific, eds.
D. Levi and
P.
Winternitz (1988).
[2]
Backlund Transformations,
Lecture Notes in Mathematics 515,
ed.
R.
Miura, Springer Verlag (1976).
[3]
R. Hirota,
Direct Method
of
Finding Exact Solutions
of
Nonlin-
ear Evolution Equations,
Lecture Notes in Mathematics,
515,
40 (1976);
Direct Methods in Soliton Theory,
in
Solitons,
Top-
ics in Current Physics,
17,
eds.
R.
Bullough,
P.
Caudrey,
157,
(1980).
[4] M. Kashiwara and
T.
Miwa, Proc. Japan Acad.
57A,
342 (1981);
E.
Date, M. Kashiwara, and
T.
Miwa, Proc. Japan Acad.
57A,
387 (1981); M. Jimbo,
E.
Date, M. Kashiwara, and
T.
Miwa,
J.
Phys. SOC.
of
Japan,
50,
3806 (1981); and RIMS preprints 359
and 360.
[5]
V.E. Zakharov and P.B. Shabat, Funct. Anal. Appl.
8,
226-235
(1974).
[6] V.E. Zakharov and P.B. Shabat, Funct. Anal. Appl.
13,
166
(1979).
[7] V.E. Zakharov and S.V. Manakov, Funct. Anal. Appl.
19,
No.
2,
11
(1985).
[8]
A.S. Fokas and M.J. Ablowitz, Phys. Rev. Lett.
47,
1096 (1981).
[9] V.E. Zakharov, S.V. Manakov, S.P. Novikov, and L.P.
Pitayevskil,
Theory
of
Solitons. The Method
of
the Inverse Scat-
tering Problem,
Nauka, Moscow (in Russian), 1980.
Integrable Nonlinear Equations
207
[lo]
A.R. Its and V.Y. Novokshenov,
The Isomonodromic Defor-
mation Method in the Theory
of
Painleve' Equation,
Springer-
Verlag, 1191, (1986).
[ll]
R. Rosdes, Stud. Appl. Math.
59,
117-151 (1978).
[12]
P.
Santini, M.J. Ablowitz, and A.S. Fokas, J. Math. Phys.
25,
2614 (1984).
[13] F.W. Nijhoff, G.R.W. Quispel,
J.
Van der Linden, H.W. Capel,
Physica A,
119,
101 (1983); F.W. Nijhoff, H.W. Capel, G.L.
Wiersma, G.R.W. Quispel, Phys. Lett. A,
103,
293 (1984).
[14] V.E. Zakharov, On the Dressing Method, to appear (1990).
[15] F.D. Gakhov,
Boundary Value Problems,
Pergamon, 1966.
[16]
N.I.
Muskhelishvili,
Singular Integral Equations,
Noordhoff,
Groningen, 1953.
[17]
N.P.
Vekua,
Systems
of
Singular Integral Equations,
Gordon and
Breach, 1967.
[18]
A.S. Fokas and V.E. Zakharov, The Dressing Method and Non-
local Riemann-Hilbert Problems, preprint, Clarkson University,
INS #137, (1989).
[19]
P.
Deift and
E.
Trubowitz, Comm. Pure Appl. Math.,
32,
121-
125 (1979).
[20] D.J. Kaup, Physica
3D,
45 (1980).
[21] L.-Y. Sung and A.S. Fokas, Inverse Problems
of
N
x
N
Hyper-
bolic Systems on the Plane and the
N
Wave Interactions, to
appear in Comm. Pure and Appl. Math. (1990).
[22] A.S. Fokas and
M.J.
Ablowitz, Stud. Appl. Math.,
69,
211-228
(1983); M.J. Ablowitz,
D.
Bar Yaacov, and A.S. Fokas, Stud.
Appl. Math.,
69,
135-143 (1983).
[23] M. Boiti,
J.
Leon,
L.
Martina,
F.
Pempinelli, Phys. Lett. A,
132,
432 (1988).
208
A.
S.
Fobs
[24] A.S. Fokas and P.M. Santini, Phys. Rev. Lett.
63,
1329 (1989);
Dromions and
a
Boundary Value Problem for the Davey-
Stewartson I Equation, to appear in Physica D.
[25]
H.
Segur, AIP Conference Proceedings,
88,
211 (1982).
[26]
X.
Zhou, Inverse Scattering Transform for the Time Dependent
Schrodinger Equation and KPI, to appear in Comm. in Math.
Phys.
[27] M.V. Wickerhauser, Comm. Math. Phys.,
108,
67 (1987).
[28]
Y.L.
Rodin, Generalized Analytic Functions on Riemann Sur-
faces, Lecture Notes in Mathematics, (1988), Springer Verlag,
(1987).
[29] R. Beals and R.R. Coifman, “The Spectral Problem for the
Davey-Stewartson and Ishimori Hierarchies”, Proc. Conf. on
Nonlinear Evolution Equations: Integrability and Spectral
Methods, Como, University
of
Manchester, 1988.
[30] R. Beals and
R.R.
Coifman, ”Multidimensional Inverse Scatter-
ing and Nonlinear Partial Differential Equations”, Proc. Symp.
Pure Math.
43,
Amer. Math. SOC. Providence, pp. 45-70 (1985).
[31]
L.-Y. Sung and A.S. Fokas, Inverse Problems in Multidimen-
sions, INS #98, preprint, Clarkson University, (1990).
[32] A.S.
Fokas
and M.J. Ablowitz,
J.
Math. Phys. 23, 2033 (1982).
[33]
H.
Flaschka and A.C. Newell, Commun. Math. Phys.
76,
67
(1980).
[34]
K.
Ueno, Proc. Jpn. Acad.
56A,
97 (1980); M. Jimbo,
T.
Miwa
and
K.
Ueno, Physica
2D,
306 (1981); M. Jimbo and
T.
Miwa,
Physica
2D,
407 (1981);
4D,
47 (1981); M. Jimbo, Prog. Theor.
Phys.
61,
359 (1979).
[35] A.S. Fokas and M.J. Ablowitz, Comm. Math. Phys.
91,
381
(1983).
Integrable Nonlinear Equations
209
[36]
A.S. Fokas and
X.
Zhou, On the Solvability of Painlev4
I1
and
IV, INS
#148,
preprint, Clarkson University
(1990).
[37]
P.D.
Lax,
Comm. Pure Appl. Math.
28,
141 (1975).
[38]
H.P.
McKean and
P.
van Moerbeke, Invent. Math.
30,
217
(1975).
[39]
H.P. McKean and
E.
Trubowitz, Comm. Pure Appl. Math.
29,
143 (1977).
[40]
A.P. Its and V.B. Matveev, Teoret. Mat. Fiz.
23, 51 (1975).
[41]
B.A. Dubrovin, V.B. Matveev and S.P. Novikov, Uspekhi Mat.
Nauk
31,
1,
55 (1976).
Russian Math. Surveys
31,
1,
59 (1976).
[42]
I.M. Krichever and
S.P.
Novikov, Uspekhi Mat. Nauk
35,
6, 47
(1980).
Russian Math. Surveys,
35,
6, 53 (1980).
[43]
I.M. Krichever and
S.P.
Novikov, Uspekhi Mat. Nauk
32,
6, 183
(1977).
Russian Math. Surveys,
32,
6, 185 (1977).
[44]
I.M. Krichever and S.P. Novikov, Funktsional. Anal. i Prilozhen,
11,
1,
15 (1977).
Functional Anal. Appl.
11,
1,
12 (1977).
[45]
B.A. Dubrovin, Uspekhi Mat. Nauk,
36,
2,
11
(1981).
Russian
Math. Surveys,
36,
2,
11
(1981).
[46]
G.
Springer, Introduction to Riemann Surfaces, Addison Wesley,
MA
(1957).
[47]
P.M. Santini, Physica D,
(1990).