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The Nonlinear Schrödinger Equation
G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Conpany (1978)
THE NONLINEAR SCHR~DINGER EQUATION
WALTER A. STRAUSS Mathematics Department Brown University
A survey is presented o f some physical applications and recent mathematical results on the Schr6dinger equation with a power nonlinearity.
1. Trapping and focusing of laser beams.
Consider the electromagnetic wave equation
i 3a c
2 (gi)E)
at
4
-4
CIE = 0.
Assume a linearly polarized wave (E vector
-4
e)
parallel to a fixed unit
which is monochromatic with frequency
which propagates along the z-axis.
ul
and
Thus
and (1) reduces to (2)
2ik
3 Z aU
Au
+
2 (k
-
gw2/c2)u
= 0.
The high intensity of a laser beam can produce significant local changes in the density of the medium and hence i n the dielectric constant
g .
This work was supported by the National Science Foundation under Grant MCS75-08827.
453
THE N O N L I N E A R SCIIRdDINGER E Q U A T I O N
C h i a o , Garmire and Townes [ 2 ] l i n e a r dependence
+ c 2 1312
= €,
6
assume t h e s i m p l e non-
They show how t h e r e s u l t -
.
i n g n o n l i n e a r t e r m may g i v e r i s e t o a n e l e c t r o m a g n e t i c beam which p r o d u c e s i t s own waveguide and p r o p a g a t e s w i t h o u t spreading.
T h i s phenomenon i s c a l l e d " s e l f - t r a p p i n g " .
corresponds t o a s o l u t i o n of
-2-av b2u
(3)
a2u
Kelley
p a r t of called
( k2
2
-
c
IuI2 u = 0.
C
t h e beam a s a f u n c t i o n o f
Z.
W
~
~
~
/
(2)
b l o w s up a t a c e r t a i n v a l u e of
IuI2
If t h e " p a r a x i a l t t approximation k =
T h i s phenomenon i s
I t c o r r e s p o n d s t o a s o l u t i o n of
'+self-focusing".
and we c h o o s e
2
-
w c oW ~ ) u e 2 2
z:
can produce a b u i l d - u p i n t h e i n t e n s i t y o f
i n which t h e i n t e n s i t y z.
( 2 ) which i s i n d e p e n d e n t of
[8] and T a l a n o v [ 1 6 ] show how a n o n l i n e a r
c
dependehce o f
+
It
luzzl
<< \ k u Z l
i s used
eCq u, a t i o n ( 2 ) r e d u c e s t o
(4) Under a p p r o p r i a t e a s s u m p t i o n s a b o u t t h e n a t u r e of t h e a b s o r b i n g medium,
o r bent instead of blooming". Aitken
[71
t h e beam c a n b e s p r e a d o u t , d i s t o r t e d These phenomena a r e c a l l e d t l t h e r m a l
focused.
G e b h a r d t and Smith [ 31 and H a y e s , U l r i c h and t a k e i n t o account t h e e f f e c t s o f a s t e a d y t r a n s -
v e r s e wind a l o n g t h e y - a x i s . intensity
(5)
IuI
2
upstream.
-
So
-
€
co
d e p e n d s on t h e beam
I n t h e s i m p l e s t c a s e they have Y
c0
= const
T a n i u t i and Washimi
[ I 7 1 have found a s i m i l a r
e q u a t i o n t o d e s c r i b e t h e long-time b e h a v i o r of
one-dimensional
4 54
WALTER A .
STRAUSS
hydromagnetic waves i n a plasma:
-
ax where
a, b, c
are constants.
(a1u12
+
b)u =
o
Such an equation also occurs
in the Ginzburg-Landau theory of superconductivity [ ? I ] .
It
can also b e regarded as a simple non-relativistic quantum field equation.
2. Existence and blow-up.
We consider the problem
(7) where
@
p > 1
and
(0 E 8
is a nice function is real.
if not stated otherwise),
T h e two standard conservation laws
(7) by
are obtained by multiplying parts, and by multiplying by
Gt
ii
and taking imaginary
and taking real parts. They
are IuI E =
and
d x = constant
p+l x lulP+']dx
\(y 1 louIz +
= constant.
They would lead us to believe that solutions ought to exist globally and be stable i f
7
0, but that if
X < 0
instability may be possible. Theorem 1 val at
~x E
u = 0
-
There exists a unique solution in some time intelr
I R ~ , It of
integer it is
< tl].
the function
ern
It i s as smooth as the singularity /U(~-'U
allows; i f
p
is an odd
455
THE NONLINEAR SCIIRdDINGER EQUATION
This local existence result follows easily by the standard Picard method. Vela [
51
proof may be found in Ginibre and
A
.
Theorem 2
-
Let
X < 0
and
p
2
1 +
4/n.
Let
I$
satisfy
the condition
(7) can exist for all
Then no smooth solution o f
t.
This result is a variation o f one of Glassey C61, 0< T <
who shows under certain conditions that there exists
<
a
such that
n = 2,
[ lvu(t) I
2
dx + =
t
as
7
T.
p = 3,
In case
(4) and Theorem 2 provides a
the equation reduces to
precise initial condition for the self-focusing of an electromagnetic beam.
The opposite situation is described i n the
next theorem. Theorem 3 n > 2
-
If
assume
1 < 0 p < 1
p < 1
assume
+ 4/(n-2).
a unique solution o f
(7)
continuous function of regularity statement
If
f o r all
t
t
+ 4/n. @
If
E Hs(IRn),
h > 0
and
there exists
which is a bounded
with values in
H1(Rn).
of Theorem 1 is valid for all
The t.
In particular, equation (6) has global smooth solutions.
Theorem 3 has been proved by Ginibre and Velo [51
except for the regularity.
Particular cases have also been
proved by Baillon, Cazenave and Figueira [11, by Glassey [61 and by Lin and Strauss [ 9 ] .
A l l four proofs were done
independently. In the case
p=3,
n=2,
Baillon et al. also prove
the global existence and regularity provided
11 I
1 @ I2dx
< 2.
456
WALTER A .
STRAUSS
T h e i r r e s u l t d o e s n o t c o n t r a d i c t Theorem 2 b e c a u s e t h e inequality
\\@lL'
dx S
,f lv@l2
1]@I2
dx
dx
p < 1
+
4/n
Theorem
4 -
p
2
1
+ 4/n
>
3 , we c a l l
I n v i e w o f Theorems 2 and
1 < 0,
implies
and
0
< 0,
t h e s t a b l e c a s e s and we c a l l
the unstable case. Let
1
7
0
and
p < m.
If
@
E H1(!Rn) ,
there
e x i s t s a s o l u t i o n which i s a bounded, weakly c o n t i n u o u s f u n c -
t
tion of
with values i n
H1(Rn).
T h i s theorem i s p r o v e d below.
We d o n o t know
whether t h i s weak s o l u t i o n i s u n i q u e or smooth. B a i l l o n e t a l . [l] h a v e a l s o proved t h e f o l l o w i n g r e s u l t for equation
(5).
The i n t e g r a l i n t h e n o n l i n e a r term
p r e v e n t s blow-up, Theorem 5
-
F o r any
1 , t h e r e e x i s t s a u n i q u e smooth s o l u t i o n
_.
of
with
u(x,y,O) = @ ( x , y ) .
Proof
of Theorem 2 :
= Xlul
P+1 /(p+l).
r = 1x1 ) d
Write
F ( u ) = Xlul
u
M u l t i p l y e q u a t i o n ( 7 ) by
G(u) =
and
2r; r
+
n;
where
and t a k e r e a l p a r t s :
I m Jru,;
dx = -2 (lvuI2
dx
[(Zn+4)G(u)-n~F(u)]dx 2 Now multiply
(7)
by
2-
r u
+ -4E
n
I
[2G(u)-;F(u)ldx
since
p > 1
and t a k e imaginary p a r t s :
+
4/n.
THE N O N L I N E A R SC1lRe)DINGER EQUATION
457
Hence
the l a s t i n t e g r a l i s p o s i t i v e .
T h i s i s nonsense because
Proof p
o f Theorem
< 1 +lr/(n-Z),
3: We m e r e l y s k e t c h t h e i n g r e d i e n t s .
H1 c LP+'.
S o b o l e v ' s Theorem s t a t e s t h a t
( i i ) The e v o l u t i o n o p e r a t o r
Uo(t):
@I
( i )I f
-+ ~ ( t f) o r t h e f r e e
( l i n e a r ) Schrddinger equation s a t i s f i e s t h e e s t i m a t e
2 S
for
q S
+
l/q
m,
6 = n/2
= 1,
l/q'
o b t a i n e d by i n t e r p o l a t i o n f r o m t h e c a s e s Y
= p+l,
t = 0.
E
t-'
so that
a r e non-negative
X < 0,
q=2
dx 5:
C
+
n
2a = ( 2 - n ) p + 2
a >
we h a v e
and
q=m.
i s integrable a t
1 > 0 , both
If
and h e n c e bounded f o r a l l
uniqueness,
H1
u
let
and
ft
l l u ~ t ~ - v ( t ~ l l p + cl
\,I2
([
1
dxIa
B
and
$ < 1.
and
0
p r o v i d e a bound i n
and i n
v
= n(p-l)/h.
If
p
< 1 + 4/13,
Hence t h e c o n s e r v a t i o n l a w s
LP+'.
( i v ) To p r o v e t h e
be t w o s o l u t i o n s .
(t-sP
ft s c
II l u l p - l u -
IvIp-lvlI
By ( i i )
( p + l )(/s p )ds
I
(t-s)
-b
~ ~ u - ~ l ~ ~ + ~ ( s ) d s
'0
implies
u
t.
( 1VuI2 dx]'
0
since
If
Sobolev's inequality s t a t e s t h a t
\lulp+l where
6 < 1
as c a n b e
n/q,
( i i i ) We u s e t h e c o n s e r v a t i o n l a w s .
terms i n If
0 <
then
-
and u = v
v
a r e bounded i n because
6 < 1.
LP+',
by ( i i i ) . T h i s
( v ) The e x i s t e n c e and
458
WALTER A .
STRAUSS
r e g u l a r i t y a r e p r o v e d by combining ( i ) - ( i i iw ) i t h Theorem 1. Proof
4: L e t
of Theorem
approximate such t h a t g N ( s ) -+
g ( s ) = 1sP+ 1/ ( p + l
s 2 0.
by a s e q u e n c e of smooth f u n c t i ons
g(s)
gN(s) = c o n s t s 2
g(s)
for
for a l l
s.
= g~([u])u/lul. L e t
for Let
u,(x,t)
s
l a r g e and
0 5
F(U) = Xlu P- 1 u
We
gNb) g N (s )
5
4.1
FN(u)=
and
be t h e u n i q u e s o l u t i o n of
the
problem
-
iu
(8)
N t
riuN
It e x i s t s because
+
F ~ ( u = ~ 0 ) ,
FN(u)
= ~ ( x ) .
u,(x,o)
u
i s linear for large
e a s y v a r i a t i o n o f Theorem 3 i s a p p l i c a b l e .
s o t h a t an
Then we h a v e t h e
o b v i o u s a p r i o r i bounds
By c o m p a c t n e s s ,
t h e r e i s a subsequence ( s t i l l c a l l e d
which c o n v e r g e s
t o some
where.
Hence
L
locally i n
,
weakly i n
l u N l ) + g( I u I )
g,(
1
u
and
u
FN(uN) + F N ( u )
by Theorem 1.1 o f S t r a u s s [ 1 2 ] .
pass t o t h e l i m i t i n each term i n ( 8 ) . of
and a l m o s t e v e r y -
H1
a.e.
UN)
We may
The o t h e r p r o p e r t i e s
f o l l o w e a s i l y a s i n C121.
3. S o l i t o n s , I n t h i s s e c t i o n w e assume ion i n t o equation
( 7 ) shows t h a t ,
parameter f a m i l y o f
if
1 <
Direct substitut-
0.
n=l,
t h e r e i s a four-
ttsolitary-wavett solutions
459
THE N O N L I N E A R SCEIRbDINGEK E Q U A T I O N
u ( x , ~ )= f ( x - c t ) e x p ( i g ( x - b t ) )
I n any d i m e n s i o n
n,
u = @ ( x ) exp(iult)
s t a n d i n g wave s o l u t i o n s
( 7 ) s a t i s f y a n e l l i p t i c e q u a t i o n which i s t h e same as ( 3 )
of
except f o r c o n s t a n t s .
V a r i a t i o n a l methods
p < 1
show t h e e x i s t e n c e of s u c h s o l u t i o n s p r o v i d e d p <
n=l
if
m
2.
or
t r a p p e d l a s e r beams
T h i s proves
(the case
[l51)
(see Strauss
+
4/(n-2);
the existence of s e l f -
n = 2,
p =
3).
These s o l u t i o n s o b v i o u s l y do n o t d e c a y t o z e r o a s
t +
m.
O n t h e o t h e r hand,
if
t h e i n i t i a l datum
s m a l l enough ( i n some Sobolev norm) and
p 2
u(x,O)
3,
n
s o l u t i o n c a n b e shown t o d e c a y t o z e r o u n i f o r m l y
[l3]
or [ l 4 ] ) .
2
is
3,
the
( s e e Strauss
I n t h e u n s t a b l e c a s e w e t h u s have t h e
s i t u a t i o n t h a t c e r t a i n s o l u t i o n s go t o z e r o ,
others maintain
t h e i r a m p l i t u d e s and s t i l l o t h e r s blow up i n a f i n i t e t i m e . The s t a b l e c a s e
p =
n = 1,
3,
1 < 0
has been
s t u d i e d i n g r e a t d e t a i l s i n c e t h e i m p o r t a n t work of Zakharov and S h a b a t [ I S ] . properties.
We now o u t l i n e some o f i t s s t r i k i n g
The e q u a t i o n i s
(9) The s o l i t a r y waves m e n t i o n e d above a r e c a l l e d s o l i t o n s i n this case. f
u = feig,
They c a n b e w r i t t e n a s
= s e c h [ 211 (x-ll )
+ 8$ tl
and
I t i s e x p o n e n t i a l l y s m a l l as
g = -25x
1x1
-b
m
-
where
4(5 2-q 2 ) t +
for each
t.
V.
Its
460
WALTER A. STRAUSS
*Z.&'q
amplitude oscillates between
while traveling along
45.
the x-axis at speed
Zakharov and Shabat found that the solitons interact almost independently in a manner highly analogous t o that of The general solution of ( 9 )
the Korteweg de Vries equation.
breaks up into a finite number of solitons having a set of four parameters
(s,q,b,V),
component whose amplitude decays like
Ill,.
.. , uN ,
each
plus a spreading
It
as
I tI
+
m .
Certain solutions, the pure combinations o r a finite number of solitons
ul,
...,uN, can be
Suppose the speeds
They are the I1multisolitons". are distinct.
t
As
-b
written explicitly.
..
4C1,. ,4%N
the multisoliton breaks up into
m,
(single) solitons arranged s o that the fastest soliton is in front and the slowest is at the reararrangement is reversed,
As
t +
I n the passage from
each soliton is completely unchanged In shape.
Ej,
a phase shift:
The changes
:l.t
q j , CI-
j'
- b yJ
J
and
v: V+
j
J
-
goes into V-
j
-m,
t =
the to
-m
+m,
There is only
tj, n j ,
-b
b j , v:.
are given by a simple
formula which shows that the shifts are the same as i f the solitons collided only pairwise,
Thus triple collisions have
n o effect!
On the other hand, if some of the speeds coincide, the solitons do not separate.
u
multisoliton
5,
= I2 = 0
depends on
formed by a pair of solitons with parameters
and
t
F o r instance, consider the
ql
#
q2.
The formula shows that
only through the expression
lu(x,t)l
461
THE N O N L I N E A R S C I I R d D I N G E R E Q U A T I O N
and s o i t i s p e r i o d i c i n t i m e .
These s o l u t i o n s a r e c a l l e d
" b r e a t h e r s If. A couple o f important f e a t u r e s o f
u n d e r l i e t h e s e s t r i k i n g phenomena.
One i s t h e e x i s t e n c e o f
an i n f i n i t e number o f c o n s e r v a t i o n laws. association of
u,
The o t h e r i s t h e
( 9 ) w i t h a f a m i l y o f l i n e a r o p e r a t o r s , namely, Lt
Here
(9)
equation
[
= i
the eigenvalues o f and t h e o p e r a t o r s
[ 11.
i
( 9 ) , p l a y s t h e role o f a time-
t h e s o l u t i o n of
dependent p e r t u r b a t i o n .
-
A f t e r a minor change o f v a r i a b l e s , t u r n o u t t o b e i n d e p e n d e n t of
Lt Lt
t o be u n i t a r i l y e q u i v a l e n t .
time
A general
r e f e r e n c e on s o l i t o n s i s [11].
4.
Scattering. W e t u r n now t o t h e a s y m p t o t i c b e h a v i o r o f
k >
s o l u t i o n s i n the case
0.
A s with the case
s o l u t i o n decays t o z e r o u n i f o r m l y i f s m a l l enough i n some norm and
k <
the the
0,
the i n i t i a l data are
p 1 3
(n 2
3).
For
X > 0,
t h i s i s the general situation. Theorem
6
-
It1
4
m.
as
( b ) If
(a) If
n = 3
and
p
2
1
8/3 <
+
4/n,
'then
p
< 5,
then
dx = O(t-')
max
IuI
= O(lt1
- 3/'
)
X
as
It1
m.
Part
( a ) comes from t h e llpseudo-conformalll conserv-
a t i o n law o f G i n i b r e and Velo
L-51
which may b e w r i t t e n a s
4 62
WALTER A. STRAUSS
I t i s a c o m b i n a t i o n o f t h e i d e n t i t i e s i n t h e p r o o f of Theorem 2.
Note t h a t t h i s i s a n e x a c t c o n s e r v a t i o n law i n t h e c a s e
p =
3,
other
n = 2.
They have a l s o p r o v e d s t r o n g e r d e c a y r a t e s i n
Lq-norms
for
q
< 2 + 4/(n-2)
and f o r more g e n e r a l n o n l i n e a r t e r m s . of L i n and S t r a u s s Green's
p < 1
and Part
+
4/(n-2),
(h) is a result
C91, who u s e e x p l i c i t e s t i m a t e s o f t h e m
f u n c t i o n t o j a c k a n L P + l - e s t i m a t e up t o an L - e s t i m a t e .
U s i n g ( a ) , s i m i l a r r e s u l t s can h e d e r i v e d i n o t h e r d i m e n s i o n s . By a f r e e s o l u t i o n i s meant a s o l u t i o n o f t h e l i n e a r Schr6dinger equation
ivt
a s k s whether a s o l u t i o n o f free solution
as
t +
-a.
u
+
as
t +
= Av,
I n s c a t t e r i n g t h e o r y , one
( 7 ) looks a s y m p t o t i c a l l y l i k e a +m
u
and l i k e a f r e e s o l u t i o n
U s u a l l y one u s e s t h e e n e r g y norm
(L2
or
-
HI)
t o express the l i m i t . Theorem
7
-
(a) If
1
+ 2/n < p , u(t)
f r e e s o l u t i o n s such that
(b) If
1
+
4/n < p < 1
pair of f r e e solutions
+ u*
-
u,(t)
4/(n-2),
+ 0
weakly i n
uk of H1(fRn).
t h e r e e x i s t s a unique
such t h a t
where t h e norm i s t h e LP+l-norm, Uo(t)
there exists a p a i r
8 = n/2
-
n/(P+l),
and
i s the f r e e evolution operator.
( c ) If
n =
3 and
i n the
H1
norm.
8/3 < p < 5 ,
we h a v e
lIu(t)-uk(t)ll
+ 0
463
THE NONLINEAR SCHRdDINGER E Q U A T I O N
( a ) f o l l o w s f r o m the ideas o f I.E.
Part
Part
Matsumura [lo].
[5].
and Velo
Part
( b ) i s one o f
Remark:
vu
-
[91.
p < 1
2
+ ; ,
t h e r e do n o t
exist
u
+ 0.
u,(t)\l
The c o n s e r v a t i o n l a w a l s o i m p l i e s a r e g u l a r i z i n g
property: then
IIu(t)
t h e r e s u l t s of Ginibre
( c ) i s a r e s u l t o f L i n and S t r a u s s
I t c a n b e shown, t h a t , i f such t h a t
S e g a l and
if
p 2
1
is locally
+ 4/n, k > L
2 -
for t
0
and
$,
x@
are
L2,
> 0.
References
[l]
J.B.
B a i l l o n , T.
Cazenave and M.
F i g u e i r a , Equation de
SchrGdinger non l i n z a i r e , C.R.Acad.Sci. 284 (19771, 869-872. [2) R . Y .
Chiao, E.
G r m i r e and C . H .
o p t i c a l beams,
[ 3 ] F.G.
G e b h a r d t and D.C.
Phys.
Rev.
Tomes, Lett.
Self-trapping of
1 3 (1964), 479-482.
Smith, S e l f - i n d u c e d thermal d i s t o r -
t i o n i n t h e n e a r f i e l d f o r a l a s e r beam i n a moving medium IEEE, J .
Quantum E l e c t r o n i c s QS-7
(1971),
63-73.
[43
P.G.
deGennes, S u p e r c o n d u c t i v i t y of M e t a l s and A l l o y s , B e n j a m i n , New York,
1966, ch. 6.
[ 5 ] J . G i n i b r e and G , V e l o , On a c l a s s o f n o n l i n e a r Schrddinger e q u a t i o n s , I , I1 and 111, t o a p p e a r .
[ 6 ] R.T.
G l a s s e y , On t h e blowing-up
o f s o l u t i o n s t o t h e Cauchy
problem f o r n o n l i n e a r S c h r d d i n g e r e q u a t i o n s , Math.
Phys. 1 8 ( 1 9 7 7 ) ,
1794-7.
J.
464
C 71
WALTER A .
J.N. Hayes, P . B .
STRAUSS
Ulrich and A.H. Aitken, Effects o f the
atmosphere on the propagation o f 10.6 laser beams, Applied Optics 11 (1972), 257-260.
c 81
P.L.
Kelley, Self-focusing of optical beams, Phys. Rev. Lett. 15 (1965), 1005-1012.
[ 91
J.E. Lin and W.A.
Strauss, Decay and scattering o f
solutions o f a nonlinear Schr8dinger equation,
J. Funct. Anal., to appear.
r 101
A.
Matsumura, On the asymptotic behavior o f solutions o f semi-linear wave equations, Publ. Res. Inst. Math. Sci. Kyoto 12 (1976), 169-189.
I: 111
A.C.
Scott, F . Y . F .
Chu and D.W. McLaughlin, The soliton:
a new concept i n applied science, Proc. IEEE 61
(1973)9 1443-1483. [12] W.A.
Stranss, On weak solutions o f semi-linear hyperbolic equations, Anais Acad. Brasil. CiGncias 4 2 (1970) , 645-651.
[l3] W.A.
Strauss, Dispersion o f low-energy waves for two conservative equations, Arch. Rat. Mech. Anal. (1974)
55
86-92.
[14] W.A. Strauss, Nonlinear scattering theory, i n Scattering Theory i n Math. Physics, Reidel Publ. C o . ,
Dordrecht,
1974, 53-78. [l5] W.A.
Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 ( 1 9 7 7 ) ,
149-162.
THE N O N L I N E A R S C H R d D I N G E R E Q U A T I O N
[16] V . I .
465
Talanov, Self-focusing o f wave beams in nonlinear media, JETP Lett. 2 (1965), 138-141.
T. Taniuti and H . Washimi, Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma, Phys. R e v I Lett. 21 (1968), 209-212. [IS]
V.E. Zakharov and A.B.
Shabat, Exact theory of t w o -
dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media, JETP 34 (1972)
62-69.
THE NONLINEAR SCHR~DINGER EQUATION
WALTER A. STRAUSS Mathematics Department Brown University
A survey is presented o f some physical applications and recent mathematical results on the Schr6dinger equation with a power nonlinearity.
1. Trapping and focusing of laser beams.
Consider the electromagnetic wave equation
i 3a c
2 (gi)E)
at
4
-4
CIE = 0.
Assume a linearly polarized wave (E vector
-4
e)
parallel to a fixed unit
which is monochromatic with frequency
which propagates along the z-axis.
ul
and
Thus
and (1) reduces to (2)
2ik
3 Z aU
Au
+
2 (k
-
gw2/c2)u
= 0.
The high intensity of a laser beam can produce significant local changes in the density of the medium and hence i n the dielectric constant
g .
This work was supported by the National Science Foundation under Grant MCS75-08827.
453
THE N O N L I N E A R SCIIRdDINGER E Q U A T I O N
C h i a o , Garmire and Townes [ 2 ] l i n e a r dependence
+ c 2 1312
= €,
6
assume t h e s i m p l e non-
They show how t h e r e s u l t -
.
i n g n o n l i n e a r t e r m may g i v e r i s e t o a n e l e c t r o m a g n e t i c beam which p r o d u c e s i t s own waveguide and p r o p a g a t e s w i t h o u t spreading.
T h i s phenomenon i s c a l l e d " s e l f - t r a p p i n g " .
corresponds t o a s o l u t i o n of
-2-av b2u
(3)
a2u
Kelley
p a r t of called
( k2
2
-
c
IuI2 u = 0.
C
t h e beam a s a f u n c t i o n o f
Z.
W
~
~
~
/
(2)
b l o w s up a t a c e r t a i n v a l u e of
IuI2
If t h e " p a r a x i a l t t approximation k =
T h i s phenomenon i s
I t c o r r e s p o n d s t o a s o l u t i o n of
'+self-focusing".
and we c h o o s e
2
-
w c oW ~ ) u e 2 2
z:
can produce a b u i l d - u p i n t h e i n t e n s i t y o f
i n which t h e i n t e n s i t y z.
( 2 ) which i s i n d e p e n d e n t of
[8] and T a l a n o v [ 1 6 ] show how a n o n l i n e a r
c
dependehce o f
+
It
luzzl
<< \ k u Z l
i s used
eCq u, a t i o n ( 2 ) r e d u c e s t o
(4) Under a p p r o p r i a t e a s s u m p t i o n s a b o u t t h e n a t u r e of t h e a b s o r b i n g medium,
o r bent instead of blooming". Aitken
[71
t h e beam c a n b e s p r e a d o u t , d i s t o r t e d These phenomena a r e c a l l e d t l t h e r m a l
focused.
G e b h a r d t and Smith [ 31 and H a y e s , U l r i c h and t a k e i n t o account t h e e f f e c t s o f a s t e a d y t r a n s -
v e r s e wind a l o n g t h e y - a x i s . intensity
(5)
IuI
2
upstream.
-
So
-
€
co
d e p e n d s on t h e beam
I n t h e s i m p l e s t c a s e they have Y
c0
= const
T a n i u t i and Washimi
[ I 7 1 have found a s i m i l a r
e q u a t i o n t o d e s c r i b e t h e long-time b e h a v i o r of
one-dimensional
4 54
WALTER A .
STRAUSS
hydromagnetic waves i n a plasma:
-
ax where
a, b, c
are constants.
(a1u12
+
b)u =
o
Such an equation also occurs
in the Ginzburg-Landau theory of superconductivity [ ? I ] .
It
can also b e regarded as a simple non-relativistic quantum field equation.
2. Existence and blow-up.
We consider the problem
(7) where
@
p > 1
and
(0 E 8
is a nice function is real.
if not stated otherwise),
T h e two standard conservation laws
(7) by
are obtained by multiplying parts, and by multiplying by
Gt
ii
and taking imaginary
and taking real parts. They
are IuI E =
and
d x = constant
p+l x lulP+']dx
\(y 1 louIz +
= constant.
They would lead us to believe that solutions ought to exist globally and be stable i f
7
0, but that if
X < 0
instability may be possible. Theorem 1 val at
~x E
u = 0
-
There exists a unique solution in some time intelr
I R ~ , It of
integer it is
< tl].
the function
ern
It i s as smooth as the singularity /U(~-'U
allows; i f
p
is an odd
455
THE NONLINEAR SCIIRdDINGER EQUATION
This local existence result follows easily by the standard Picard method. Vela [
51
proof may be found in Ginibre and
A
.
Theorem 2
-
Let
X < 0
and
p
2
1 +
4/n.
Let
I$
satisfy
the condition
(7) can exist for all
Then no smooth solution o f
t.
This result is a variation o f one of Glassey C61, 0< T <
who shows under certain conditions that there exists
<
a
such that
n = 2,
[ lvu(t) I
2
dx + =
t
as
7
T.
p = 3,
In case
(4) and Theorem 2 provides a
the equation reduces to
precise initial condition for the self-focusing of an electromagnetic beam.
The opposite situation is described i n the
next theorem. Theorem 3 n > 2
-
If
assume
1 < 0 p < 1
p < 1
assume
+ 4/(n-2).
a unique solution o f
(7)
continuous function of regularity statement
If
f o r all
t
t
+ 4/n. @
If
E Hs(IRn),
h > 0
and
there exists
which is a bounded
with values in
H1(Rn).
of Theorem 1 is valid for all
The t.
In particular, equation (6) has global smooth solutions.
Theorem 3 has been proved by Ginibre and Velo [51
except for the regularity.
Particular cases have also been
proved by Baillon, Cazenave and Figueira [11, by Glassey [61 and by Lin and Strauss [ 9 ] .
A l l four proofs were done
independently. In the case
p=3,
n=2,
Baillon et al. also prove
the global existence and regularity provided
11 I
1 @ I2dx
< 2.
456
WALTER A .
STRAUSS
T h e i r r e s u l t d o e s n o t c o n t r a d i c t Theorem 2 b e c a u s e t h e inequality
\\@lL'
dx S
,f lv@l2
1]@I2
dx
dx
p < 1
+
4/n
Theorem
4 -
p
2
1
+ 4/n
>
3 , we c a l l
I n v i e w o f Theorems 2 and
1 < 0,
implies
and
0
< 0,
t h e s t a b l e c a s e s and we c a l l
the unstable case. Let
1
7
0
and
p < m.
If
@
E H1(!Rn) ,
there
e x i s t s a s o l u t i o n which i s a bounded, weakly c o n t i n u o u s f u n c -
t
tion of
with values i n
H1(Rn).
T h i s theorem i s p r o v e d below.
We d o n o t know
whether t h i s weak s o l u t i o n i s u n i q u e or smooth. B a i l l o n e t a l . [l] h a v e a l s o proved t h e f o l l o w i n g r e s u l t for equation
(5).
The i n t e g r a l i n t h e n o n l i n e a r term
p r e v e n t s blow-up, Theorem 5
-
F o r any
1 , t h e r e e x i s t s a u n i q u e smooth s o l u t i o n
_.
of
with
u(x,y,O) = @ ( x , y ) .
Proof
of Theorem 2 :
= Xlul
P+1 /(p+l).
r = 1x1 ) d
Write
F ( u ) = Xlul
u
M u l t i p l y e q u a t i o n ( 7 ) by
G(u) =
and
2r; r
+
n;
where
and t a k e r e a l p a r t s :
I m Jru,;
dx = -2 (lvuI2
dx
[(Zn+4)G(u)-n~F(u)]dx 2 Now multiply
(7)
by
2-
r u
+ -4E
n
I
[2G(u)-;F(u)ldx
since
p > 1
and t a k e imaginary p a r t s :
+
4/n.
THE N O N L I N E A R SC1lRe)DINGER EQUATION
457
Hence
the l a s t i n t e g r a l i s p o s i t i v e .
T h i s i s nonsense because
Proof p
o f Theorem
< 1 +lr/(n-Z),
3: We m e r e l y s k e t c h t h e i n g r e d i e n t s .
H1 c LP+'.
S o b o l e v ' s Theorem s t a t e s t h a t
( i i ) The e v o l u t i o n o p e r a t o r
Uo(t):
@I
( i )I f
-+ ~ ( t f) o r t h e f r e e
( l i n e a r ) Schrddinger equation s a t i s f i e s t h e e s t i m a t e
2 S
for
q S
+
l/q
m,
6 = n/2
= 1,
l/q'
o b t a i n e d by i n t e r p o l a t i o n f r o m t h e c a s e s Y
= p+l,
t = 0.
E
t-'
so that
a r e non-negative
X < 0,
q=2
dx 5:
C
+
n
2a = ( 2 - n ) p + 2
a >
we h a v e
and
q=m.
i s integrable a t
1 > 0 , both
If
and h e n c e bounded f o r a l l
uniqueness,
H1
u
let
and
ft
l l u ~ t ~ - v ( t ~ l l p + cl
\,I2
([
1
dxIa
B
and
$ < 1.
and
0
p r o v i d e a bound i n
and i n
v
= n(p-l)/h.
If
p
< 1 + 4/13,
Hence t h e c o n s e r v a t i o n l a w s
LP+'.
( i v ) To p r o v e t h e
be t w o s o l u t i o n s .
(t-sP
ft s c
II l u l p - l u -
IvIp-lvlI
By ( i i )
( p + l )(/s p )ds
I
(t-s)
-b
~ ~ u - ~ l ~ ~ + ~ ( s ) d s
'0
implies
u
t.
( 1VuI2 dx]'
0
since
If
Sobolev's inequality s t a t e s t h a t
\lulp+l where
6 < 1
as c a n b e
n/q,
( i i i ) We u s e t h e c o n s e r v a t i o n l a w s .
terms i n If
0 <
then
-
and u = v
v
a r e bounded i n because
6 < 1.
LP+',
by ( i i i ) . T h i s
( v ) The e x i s t e n c e and
458
WALTER A .
STRAUSS
r e g u l a r i t y a r e p r o v e d by combining ( i ) - ( i i iw ) i t h Theorem 1. Proof
4: L e t
of Theorem
approximate such t h a t g N ( s ) -+
g ( s ) = 1sP+ 1/ ( p + l
s 2 0.
by a s e q u e n c e of smooth f u n c t i ons
g(s)
gN(s) = c o n s t s 2
g(s)
for
for a l l
s.
= g~([u])u/lul. L e t
for Let
u,(x,t)
s
l a r g e and
0 5
F(U) = Xlu P- 1 u
We
gNb) g N (s )
5
4.1
FN(u)=
and
be t h e u n i q u e s o l u t i o n of
the
problem
-
iu
(8)
N t
riuN
It e x i s t s because
+
F ~ ( u = ~ 0 ) ,
FN(u)
= ~ ( x ) .
u,(x,o)
u
i s linear for large
e a s y v a r i a t i o n o f Theorem 3 i s a p p l i c a b l e .
s o t h a t an
Then we h a v e t h e
o b v i o u s a p r i o r i bounds
By c o m p a c t n e s s ,
t h e r e i s a subsequence ( s t i l l c a l l e d
which c o n v e r g e s
t o some
where.
Hence
L
locally i n
,
weakly i n
l u N l ) + g( I u I )
g,(
1
u
and
u
FN(uN) + F N ( u )
by Theorem 1.1 o f S t r a u s s [ 1 2 ] .
pass t o t h e l i m i t i n each term i n ( 8 ) . of
and a l m o s t e v e r y -
H1
a.e.
UN)
We may
The o t h e r p r o p e r t i e s
f o l l o w e a s i l y a s i n C121.
3. S o l i t o n s , I n t h i s s e c t i o n w e assume ion i n t o equation
( 7 ) shows t h a t ,
parameter f a m i l y o f
if
1 <
Direct substitut-
0.
n=l,
t h e r e i s a four-
ttsolitary-wavett solutions
459
THE N O N L I N E A R SCEIRbDINGEK E Q U A T I O N
u ( x , ~ )= f ( x - c t ) e x p ( i g ( x - b t ) )
I n any d i m e n s i o n
n,
u = @ ( x ) exp(iult)
s t a n d i n g wave s o l u t i o n s
( 7 ) s a t i s f y a n e l l i p t i c e q u a t i o n which i s t h e same as ( 3 )
of
except f o r c o n s t a n t s .
V a r i a t i o n a l methods
p < 1
show t h e e x i s t e n c e of s u c h s o l u t i o n s p r o v i d e d p <
n=l
if
m
2.
or
t r a p p e d l a s e r beams
T h i s proves
(the case
[l51)
(see Strauss
+
4/(n-2);
the existence of s e l f -
n = 2,
p =
3).
These s o l u t i o n s o b v i o u s l y do n o t d e c a y t o z e r o a s
t +
m.
O n t h e o t h e r hand,
if
t h e i n i t i a l datum
s m a l l enough ( i n some Sobolev norm) and
p 2
u(x,O)
3,
n
s o l u t i o n c a n b e shown t o d e c a y t o z e r o u n i f o r m l y
[l3]
or [ l 4 ] ) .
2
is
3,
the
( s e e Strauss
I n t h e u n s t a b l e c a s e w e t h u s have t h e
s i t u a t i o n t h a t c e r t a i n s o l u t i o n s go t o z e r o ,
others maintain
t h e i r a m p l i t u d e s and s t i l l o t h e r s blow up i n a f i n i t e t i m e . The s t a b l e c a s e
p =
n = 1,
3,
1 < 0
has been
s t u d i e d i n g r e a t d e t a i l s i n c e t h e i m p o r t a n t work of Zakharov and S h a b a t [ I S ] . properties.
We now o u t l i n e some o f i t s s t r i k i n g
The e q u a t i o n i s
(9) The s o l i t a r y waves m e n t i o n e d above a r e c a l l e d s o l i t o n s i n this case. f
u = feig,
They c a n b e w r i t t e n a s
= s e c h [ 211 (x-ll )
+ 8$ tl
and
I t i s e x p o n e n t i a l l y s m a l l as
g = -25x
1x1
-b
m
-
where
4(5 2-q 2 ) t +
for each
t.
V.
Its
460
WALTER A. STRAUSS
*Z.&'q
amplitude oscillates between
while traveling along
45.
the x-axis at speed
Zakharov and Shabat found that the solitons interact almost independently in a manner highly analogous t o that of The general solution of ( 9 )
the Korteweg de Vries equation.
breaks up into a finite number of solitons having a set of four parameters
(s,q,b,V),
component whose amplitude decays like
Ill,.
.. , uN ,
each
plus a spreading
It
as
I tI
+
m .
Certain solutions, the pure combinations o r a finite number of solitons
ul,
...,uN, can be
Suppose the speeds
They are the I1multisolitons". are distinct.
t
As
-b
written explicitly.
..
4C1,. ,4%N
the multisoliton breaks up into
m,
(single) solitons arranged s o that the fastest soliton is in front and the slowest is at the reararrangement is reversed,
As
t +
I n the passage from
each soliton is completely unchanged In shape.
Ej,
a phase shift:
The changes
:l.t
q j , CI-
j'
- b yJ
J
and
v: V+
j
J
-
goes into V-
j
-m,
t =
the to
-m
+m,
There is only
tj, n j ,
-b
b j , v:.
are given by a simple
formula which shows that the shifts are the same as i f the solitons collided only pairwise,
Thus triple collisions have
n o effect!
On the other hand, if some of the speeds coincide, the solitons do not separate.
u
multisoliton
5,
= I2 = 0
depends on
formed by a pair of solitons with parameters
and
t
F o r instance, consider the
ql
#
q2.
The formula shows that
only through the expression
lu(x,t)l
461
THE N O N L I N E A R S C I I R d D I N G E R E Q U A T I O N
and s o i t i s p e r i o d i c i n t i m e .
These s o l u t i o n s a r e c a l l e d
" b r e a t h e r s If. A couple o f important f e a t u r e s o f
u n d e r l i e t h e s e s t r i k i n g phenomena.
One i s t h e e x i s t e n c e o f
an i n f i n i t e number o f c o n s e r v a t i o n laws. association of
u,
The o t h e r i s t h e
( 9 ) w i t h a f a m i l y o f l i n e a r o p e r a t o r s , namely, Lt
Here
(9)
equation
[
= i
the eigenvalues o f and t h e o p e r a t o r s
[ 11.
i
( 9 ) , p l a y s t h e role o f a time-
t h e s o l u t i o n of
dependent p e r t u r b a t i o n .
-
A f t e r a minor change o f v a r i a b l e s , t u r n o u t t o b e i n d e p e n d e n t of
Lt Lt
t o be u n i t a r i l y e q u i v a l e n t .
time
A general
r e f e r e n c e on s o l i t o n s i s [11].
4.
Scattering. W e t u r n now t o t h e a s y m p t o t i c b e h a v i o r o f
k >
s o l u t i o n s i n the case
0.
A s with the case
s o l u t i o n decays t o z e r o u n i f o r m l y i f s m a l l enough i n some norm and
k <
the the
0,
the i n i t i a l data are
p 1 3
(n 2
3).
For
X > 0,
t h i s i s the general situation. Theorem
6
-
It1
4
m.
as
( b ) If
(a) If
n = 3
and
p
2
1
8/3 <
+
4/n,
'then
p
< 5,
then
dx = O(t-')
max
IuI
= O(lt1
- 3/'
)
X
as
It1
m.
Part
( a ) comes from t h e llpseudo-conformalll conserv-
a t i o n law o f G i n i b r e and Velo
L-51
which may b e w r i t t e n a s
4 62
WALTER A. STRAUSS
I t i s a c o m b i n a t i o n o f t h e i d e n t i t i e s i n t h e p r o o f of Theorem 2.
Note t h a t t h i s i s a n e x a c t c o n s e r v a t i o n law i n t h e c a s e
p =
3,
other
n = 2.
They have a l s o p r o v e d s t r o n g e r d e c a y r a t e s i n
Lq-norms
for
q
< 2 + 4/(n-2)
and f o r more g e n e r a l n o n l i n e a r t e r m s . of L i n and S t r a u s s Green's
p < 1
and Part
+
4/(n-2),
(h) is a result
C91, who u s e e x p l i c i t e s t i m a t e s o f t h e m
f u n c t i o n t o j a c k a n L P + l - e s t i m a t e up t o an L - e s t i m a t e .
U s i n g ( a ) , s i m i l a r r e s u l t s can h e d e r i v e d i n o t h e r d i m e n s i o n s . By a f r e e s o l u t i o n i s meant a s o l u t i o n o f t h e l i n e a r Schr6dinger equation
ivt
a s k s whether a s o l u t i o n o f free solution
as
t +
-a.
u
+
as
t +
= Av,
I n s c a t t e r i n g t h e o r y , one
( 7 ) looks a s y m p t o t i c a l l y l i k e a +m
u
and l i k e a f r e e s o l u t i o n
U s u a l l y one u s e s t h e e n e r g y norm
(L2
or
-
HI)
t o express the l i m i t . Theorem
7
-
(a) If
1
+ 2/n < p , u(t)
f r e e s o l u t i o n s such that
(b) If
1
+
4/n < p < 1
pair of f r e e solutions
+ u*
-
u,(t)
4/(n-2),
+ 0
weakly i n
uk of H1(fRn).
t h e r e e x i s t s a unique
such t h a t
where t h e norm i s t h e LP+l-norm, Uo(t)
there exists a p a i r
8 = n/2
-
n/(P+l),
and
i s the f r e e evolution operator.
( c ) If
n =
3 and
i n the
H1
norm.
8/3 < p < 5 ,
we h a v e
lIu(t)-uk(t)ll
+ 0
463
THE NONLINEAR SCHRdDINGER E Q U A T I O N
( a ) f o l l o w s f r o m the ideas o f I.E.
Part
Part
Matsumura [lo].
[5].
and Velo
Part
( b ) i s one o f
Remark:
vu
-
[91.
p < 1
2
+ ; ,
t h e r e do n o t
exist
u
+ 0.
u,(t)\l
The c o n s e r v a t i o n l a w a l s o i m p l i e s a r e g u l a r i z i n g
property: then
IIu(t)
t h e r e s u l t s of Ginibre
( c ) i s a r e s u l t o f L i n and S t r a u s s
I t c a n b e shown, t h a t , i f such t h a t
S e g a l and
if
p 2
1
is locally
+ 4/n, k > L
2 -
for t
0
and
$,
x@
are
L2,
> 0.
References
[l]
J.B.
B a i l l o n , T.
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THE N O N L I N E A R S C H R d D I N G E R E Q U A T I O N
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