c. besse and c.h. bruneau- numerical study of elliptic-hyperbolic davey-stewartson system: dromions...
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M a t h e m a t i c a l M o d e l s a n d M e t h o d s i n A p p l i e d S c i e n c e s
f
c W o r l d S c i e n t i c P u b l i s h i n g C o m p a n y
N U M E R I C A L S T U D Y O F E L L I P T I C - H Y P E R B O L I C
D A V E Y - S T E W A R T S O N S Y S T E M :
D R O M I O N S S I M U L A T I O N A N D B L O W - U P
C . B e s s e
y
, C . H . B r u n e a u
y
y : M a t h e m a t i q u e s A p p l i q u e e s d e B o r d e a u x
U n i v e r s i t e B o r d e a u x I
3 3 4 0 5 T a l e n c e C e d e x .
T h i s p a p e r i s d e v o t e d t o t h e n u m e r i c a l a p p r o x i m a t i o n o f t h e e l l i p t i c - h y p e r b o l i c f o r m o f
t h e D a v e y - S t e w a r t s o n e q u a t i o n s . A w e l l s u i t e d n i t e d i e r e n c e s s c h e m e t h a t p r e s e r v e s
t h e e n e r g y i s d e r i v e d . T h i s s c h e m e i s t e s t e d t o c o m p u t e t h e f a m o u s d r o m i o n 1 ? 1 a n d
d r o m i o n 2 ? 2 s o l u t i o n s . T h e a c c u r a c y o f C r a n k - N i c o l s o n s c h e m e i s d i s c u s s e d a n d i t i s
s h o w n t h a t i t i n d u c e s a p h a s e e r r o r . T h e n , t h e q u a l i t a t i v e b e h a v i o r o f t h e s o l u t i o n s i s
s t u d i e d ; i n p a r t i c u l a r t h e i n u e n c e o f t h e i n i t i a l d a t u m a n d o f t h e v a r i o u s p a r a m e t e r s
i s p o i n t e d o u t . F i n a l l y , n u m e r i c a l e x p e r i m e n t s s h o w t h e e x i s t e n c e o f b l o w - u p s o l u t i o n s
1 . I n t r o d u c t i o n
T h e a i m o f t h i s p a p e r i s t o s t u d y n u m e r i c a l l y t h e b e h a v i o r o f t h e s o l u t i o n s o f
D a v e y - S t e w a r t s o n s y s t e m
( D S )
i u
t
+ u
x x
+ u
y y
= j u j
2
u + b u '
x
;
'
x x
+ m '
y y
= ( j u j
2
)
x
;
w h e r e t h e c o n s t a n t s , , b , m a n d a r e r e a l . T h i s s y s t e m d e s c r i b e s t h e e v o l u t i o n o f
w a t e r s u r f a c e w a v e s i n p r e s e n c e o f g r a v i t y a n d c a p i l l a r i t y
1
,
2
. F o l l o w i n g G h i d a g l i a -
S a u t
3
, w e c l a s s i f y t h e s e s y s t e m s a s e l l i p t i c - e l l i p t i c ( E - E ) , e l l i p t i c - h y p e r b o l i c ( E - H ) ,
h y p e r b o l i c - e l l i p t i c ( H - E ) a n d h y p e r b o l i c - h y p e r b o l i c ( H - H ) a c c o r d i n g t o t h e s i g n o f
( ; m ) : ( + ; + ) , ( + ; ? ) , ( ? ; + ) a n d ( ? ; ? ) . I n t h i s p a p e r , w e r e s t r i c t o u r s e l v e s t o
t h e ( E - H ) c a s e . A s d e s c r i b e d i n
4
, t h e ( E - H ) m o d e n e e d s a p p r o p r i a t e b o u n d a r y
c o n d i t i o n s . F o l l o w i n g
4
, w e t a k e
l i m
! ? 1
' ( x ; y ; t ) = '
1
( ; t ) ;
l i m
! ? 1
' ( x ; y ; t ) = '
2
( ; t ) ;
w h e r e = c x ? y , = c x + y r e p r e s e n t t h e c h a r a c t e r i s t i c v a r i a b l e s , m = ? c
2
a n d
'
1
a n d '
2
a r e g i v e n f u n c t i o n s .
1
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2 . E x a c t s o l u t i o n s o f D S I
U s i n g i n v e r s e s c a t t e r i n g m e t h o d s , F o k a s a n d S a n t i n i
8 ; 1 0
s h o w f o r D S I t h a t t h e r e
e x i s t s l o c a l i z e d c o h e r e n t s t r u c t u r e s w h i c h a r e g o v e r n e d b y t h e n o n t r i v i a l b o u n d a r i e s
'
1
a n d '
2
, a n d c a l l t h e m \ d r o m i o n s " s i n c e t h e y t r a v e l o n t h e t r a c k s ( i n a n c i e n t
g r e e k \ d r o m o s " ) g e n e r a t e d b y t h e b o u n d a r i e s a n d a r e d r i v e n b y t h e m . T h e y a r e
l o c a l i z e d t r a v e l i n g s o l u t i o n s w h i c h d e c a y e x p o n e n t i a l l y i n b o t h a n d , a n d c a n
i n t e r a c t u p o n t h e m o v e m e n t . C o n t r a r y t o s o l i t o n s , t h e y d o n o t p r e s e r v e t h e i r f o r m
u p o n i n t e r a c t i o n a n d c a n e x c h a n g e e n e r g y . T o o b t a i n t h e m , F o k a s a n d S a n t i n i w r i t e
( D S I ) i n t h e f o r m
8
>
>
>
<
>
>
>
:
i u
t
+ u + u U
1
+ U
2
] = 0 ;
'
= ? U
1
+
2
j u j
2
;
'
= ? U
2
+
2
j u j
2
;
w h e r e
U
1
= ?
2
Z
? 1
( j u j
2
)
d
0
+ u
1
( ; t ) ;
U
2
= ?
2
Z
? 1
( j u j
2
)
d
0
+ u
2
( ; t ) :
T h e r e f o r e , u
1
( ; t ) c o r r e s p o n d s t o ? '
2
( ; t ) a n d u
2
( ; t ) t o ? '
1
( ; t ) . A s u s u a l
b y i n v e r s e s c a t t e r i n g t e c h n i q u e s , t h e a n a l y t i c f o r m o f d r o m i o n s s o l u t i o n s i s v e r y
h a r d t o o b t a i n . H o w e v e r , t h e ( M ; N ) d r o m i o n s o l u t i o n , d e s c r i b i n g t h e i n t e r a c t i o n
o f N M l o c a l i z e d l u m p s , t a k e s t h e f o l l o w i n g f o r m
u ( ; ; t ) = 2 X
t
Z Y ( 2 . 1 )
w i t h
X = ( C
x
+ I )
? 1
V i s a v e c t o r o f s i z e N ;
Y = ( C
y
+ I )
? 1
W i s a v e c t o r o f s i z e M ;
Z = ( A ? I )
? 1
a n d a N M m a t r i x ;
A = ( C
y
+ I )
? 1
( C
x
+ I )
? 1
]
t
i s a N N m a t r i x ;
w h e r e t h e s u p e r s c r i p t t d e n o t e s t h e t r a n s p o s e o f a m a t r i x . T h e N N m a t r i x C
x
i s g i v e n b y
( C
x
)
j k
=
m
j
m
k
j
+
k
e x p ? (
j
+
k
) ( ? i (
j
?
k
) t ) ] ;
a n d t h e M M m a t r i x C
y
b y
( C
y
)
j k
=
l
j
l
k
j
+
k
e x p ? (
j
+
k
) ( ? i (
j
?
k
) t ) ] :
F i n a l l y ,
( V )
j
= l
j
e x p
?
j
(
? i
j
t ) ] ; 1
j
N ;
( W )
j
= m
j
e x p ?
j
( ? i
j
t ) ] ; 1 j M ;
3
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w h e r e
j
;
j
; l
j
; m
j
a n d 2 C a n d R e (
i
) ; R e (
i
) 2 R
+
. B e s i d e s , t h e b o u n d a r y
c o n d i t i o n s a r e
u
1
( ; t ) = 2 @
( Y
t
W ) ; ( 2 . 2 )
u
2
( ; t ) = 2 @
( X
t
V ) ; ( 2 . 3 )
w h i c h c a n b e w r i t t e n a s
u
1
( ; t ) = ? 2 @
M
X
k = 1
m
k
e x p ?
k
( + i
k
t ) ] Y
k
( ; t ) ;
u
2
( ; t ) = ? 2 @
L
X
j = 1
l
j
e x p ?
j
( + i
j
t ) ] X
j
( ; t ) :
T h e n , a s t ! 1 , u
1
( ; t ) ( r e s p . u
2
( ; t ) ) c o n s i s t s o f M ( r e s p . L ) s o l i t o n s e a c h
t r a v e l i n g w i t h v e l o c i t y ? 2 I m (
k
) ( r e s p . ? 2 I m (
j
) ) . M o r e o v e r , a l w a y s a s t ! 1 ,
u ( ; ; t ) c o n s i s t s o f M t i m e s N w i d e l y s e p a r a t e d l u m p s , n a m e d
k j
, k = 1 : : M ; j =
1 : : N , e a c h t r a v e l i n g w i t h v e l o c i t y ( ? 2 I m (
k
) ; ? 2 I m (
j
) ) . I n t h e s p e c i a l c a s e o f
k j
= 0 f o r k 6= j , t h e n u m b e r o f l u m p s i s m i n ( M ; N ) .
T o i l l u s t r a t e t h e s e f o r m u l a , w e g i v e h e r e t h e ( 1 ? 1 ) d r o m i o n e x p r e s s i o n . L e t =
R
+ i
I
, =
R
+ i
I
,
̂
= + 2
I
t , ̂ = + 2
I
t , =
1
R
l n
j j
p
2
R
,
=
1
R
l n
j j
p
2
R
,
R
u
=
R
( ̂ ? ) +
R
(
̂
?
) a n d I
u
= ? (
I
̂ +
I
̂
) + ( j j
2
+ j j
2
) t + a r g ( l m ) , t h e n
u
1
( ; t ) =
2
R
2
c o s h (
R
( ̂ ? ) )
2
; ( 2 . 4 )
u
2
( ; t ) =
2
R
2
c o s h (
R
(
̂
?
) )
2
; ( 2 . 5 )
a n d
u =
4
p
R
R
e x p f ? R
u
+ i I
u
g
( 1 + e x p ( ? 2
R
( ̂ ? ) ) ) ( 1 + e x p ( ? 2
R
(
̂
?
) ) ) + j j
2
: ( 2 . 6 )
T h e d r o m i o n s o l u t i o n s a r e n o t t h e o n l y e x p l i c i t s o l u t i o n s o f D S I . I n d e e d , r e c e n t l y ,
H i e t a r i n t a a n d H i r o t a
1 2
a n d J a u l e n t e t a l .
1 3
o b t a i n e d a b r o a d e r c l a s s o f d r o m i o n
s o l u t i o n s i n t e r m s o f W r o n s k i a n d e t e r m i n a n t s . F i n a l l y , G i l s o n a n d N i m m o
1 4
c o n -
s i d e r e d a n a l t e r n a t i v e d i r e c t a p p r o a c h w h i c h u s e s a f o r m u l a t i o n o f t h e s o l u t i o n s a s
g r a m m i a n d e t e r m i n a n t s t o o b t a i n a m u c h b r o a d e r c l a s s o f s o l u t i o n s ( p l a n e - w a v e
s o l i t o n s , d r o m i o n s , s o l i t o s ) .
3 . N u m e r i c a l s c h e m e
I n t h i s s e c t i o n , w e i n t r o d u c e t h e n u m e r i c a l m e t h o d u s e d t o c o m p u t e s o l u t i o n s o f
t h e s l i g h t l y m o d i e d ( D S ) ( E - H ) s y s t e m t o s e e t h e i n u e n c e o f e a c h d e r i v a t i v e t e r m
4
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o n t h e b e h a v i o r o f t h e s o l u t i o n
8
>
>
>
>
>
<
>
>
>
>
>
:
i u
t
+ u
x x
+ u
y y
= j u j
2
u + b u '
x
; ( a )
'
x x
? c
2
'
y y
= ( j u j
2
)
x
; ( b )
l i m
! ? 1
' ( x ; y ; t ) = '
1
( ; t ) ;
l i m
! ? 1
' ( x ; y ; t ) = '
2
( ; t ) ;
u ( t = 0 ; x ; y ) = u
0
( x ; y ) :
( 3 . 7 )
w i t h , > 0 . A p p r o x i m a t i n g t h i s s y s t e m , w e f a c e s e v e r a l d i c u l t i e s ; n a m e l y , t h e
s i z e o f t h e d o m a i n , t h e c o u p l i n g o f e q u a t i o n s a n d t h e h y p e r b o l i c t y p e o f t h e s e c o n d
e q u a t i o n . T h e n , w e h a v e t w o p o s s i b i l i t i e s f o r t h e n u m e r i c a l t r e a t m e n t o f ( 3 . 7 ) . O n
o n e h a n d , w e c a n u s e t h e s t r u c t u r e o f t h e s y s t e m a s i n
1 1
a n d w r i t e ( 3 . 7 ) a s
i u
t
+ u
x x
+ u
y y
=
j u
j
2
u + b u V
w h e r e
V =
4 c
Z
? 1
( j u j
2
)
d
0
+
Z
? 1
( j u j
2
)
d
0
!
+ '
1
+ '
2
h o w e v e r , w i t h o u t t h e u s e o f s p e c t r a l m e t h o d s , t h e i n t e g r a l s a r e d i c u l t t o c o m p u t e
a n d , w i t h s u c h a f o r m u l a t i o n , i t i s n o t p o s s i b l e t o c o m p u t e ' . O n t h e o t h e r h a n d ,
w e c a n a p p r o x i m a t e s e p a r a t e l y b o t h e q u a t i o n s o f ( 3 . 7 ) w h i c h i s m o r e a p p r o p r i a t e
f o r n i t e d i e r e n c e s c h e m e s .
M o r e o v e r , w e w a n t t o p r e s e r v e t h e e n e r g y
M ( u ) =
Z
R
2
j u j
2
d x d y ( 3 . 8 )
3 . 1 . S e m i - d i s c r e t i z a t i o n i n t i m e
T h e m a i n i d e a i s t o u s e t h e C r a n k - N i c o l s o n s c h e m e p r o p o s e d b y D e l f o u r - F o r t i n -
P a y r e
1 5
f o r t h e n o n l i n e a r S c h r o d i n g e r e q u a t i o n ( N L S ) :
( N L S ) i u
t
+ u = j u j
2
u ; x 2 R
d
; t > 0 ;
u ( x ; t = 0 ) = u
0
( x ) ; x 2 R
d
:
T h i s s c h e m e i s s t u d i e d i n
1 6 ; 1 7 ; 1 8
. I t i s f u l l y i m p l i c i t a n d e x a c t l y p r e s e r v e s b o t h
i n v a r i a n t s o f ( N L S ) . I t t a k e s t h e s e m i - d i s c r e t i z e d f o r m
i
u
n + 1
? u
n
t
+
u
n + 1
+ u
n
2
=
j u
n + 1
j
2
+ j u
n
j
2
2
u
n + 1
+ u
n
2
w h e r e u
n
i s t h e a p p r o x i m a t i o n o f u a t t i m e t
n
= n t . I n f a c t , t h i s e q u a t i o n i s
w r i t t e n a t t i m e t
n +
1
2
= ( n + 1 = 2 ) t . I n o r d e r t o u s e t h i s s c h e m e f o r D S ( E - H ) , w e
j u s t h a v e t o a d d t h e d i s c r e t i z a t i o n o f t h e t e r m u '
x
. W e w r i t e i t a s
u
n + 1
+ u
n
2
'
n +
1
2
x
5
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a n d r e p l a c e t h e o p e r a t o r b y D = @
x x
+ @
y y
. N o w , t h e p r i n c i p l e o f r e l a x a t i o n
i s t o w r i t e ( 3 . 7 b ) a t t h e d i e r e n t t i m e t
n
= n t , a n d p r o c e e d i n g i n t h e s a m e w a y ,
w e o b t a i n
L (
'
n +
1
2
+ '
n ?
1
2
2
) = ( j u
n
j
2
)
x
w h e r e L = @
x x
? c
2
@
y y
. W e t a k e a s i n i t i a l c o n d i t i o n u
0
= u
0
. F i n a l l y , w e g e t
i
u
n + 1
? u
n
t
+ D
u
n + 1
+ u
n
2
=
j u
n + 1
j
2
+ j u
n
j
2
2
u
n + 1
+ u
n
2
+
b
u
n + 1
+ u
n
2
'
n +
1
2
x
( 3 . 9 )
L
'
n +
1
2
+ '
n ?
1
2
2
!
= ( j u
n
j
2
)
x
: ( 3 . 1 0 )
T h e n , t h e r e i s c o n s e r v a t i o n o f t h e e n e r g y ( 3 . 8 ) . I n d e e d , m u l t i p l y i n g ( 3 . 9 ) b y
u
n + 1
+ u
n
, i n t e g r a t i n g i n s p a c e a n d t a k i n g t h e i m a g i n a r y p a r t , w e g e t
Z
R
2
j u
n + 1
j
2
d x d y =
Z
R
2
j u
n
j
2
d x d y : ( 3 . 1 1 )
M o r e o v e r , a s t h e f u n c t i o n s '
1
a n d '
2
a r e d e n e d o n t h e c h a r a c t e r i s t i c v a r i a b l e s , w e
h a v e t o r e w r i t e ( 3 . 9 ) a n d ( 3 . 1 0 ) i n t h e ( ; ) p l a n e
8
>
>
>
>
>
<
>
>
>
>
>
:
S (
'
n +
1
2
+ '
n ?
1
2
2
) =
4 c
T ( j u
n
j
2
) ( a )
i
u
n + 1
? u
n
t
+ ( c
2
+ ) (
u
n + 1
+ u
n
2
) + 2 ( c
2
? ) S (
u
n + 1
+ u
n
2
) = ( b )
(
j u
n + 1
j
2
+ j u
n
j
2
2
+ b c T ( '
n +
1
2
) ) (
u
n + 1
+ u
n
2
)
( 3 . 1 2 )
w h e r e S = @
a n d T = @
+ @
.
R e m a r k 0 . 1 I n t h i s p a p e r , w e d o n o t p r o v e n e i t h e r e x i s t e n c e o f a s o l u t i o n n o r
t h e c o n v e r g e n c e o f t h e s c h e m e . H o w e v e r , t h e s a m e s c h e m e i s u s e d i n
1 9
f o r t h e
( N L S ) , D S ( E - E ) a n d D S ( E - H ) e q u a t i o n s a n d r e s u l t s o f e x i s t e n c e a n d c o n v e r g e n c e
o f s o l u t i o n s a r e p r o v e d .
3 . 2 . F u l l d i s c r e t i z a t i o n
F o r t h e s p a t i a l a p p r o x i m a t i o n , w e r e s t r i c t t h e i n n i t e d o m a i n t o a l a r g e e n o u g h
b o u n d e d o n e a n d t a k e h o m o g e n e o u s D i r i c h l e t c o n d i t i o n s o n t h e b o u n d a r y . W e
c o n s i d e r t h e a p p r o x i m a t e d d o m a i n i s l a r g e e n o u g h w h e n t h e v a l u e o f t h e i n i t i a l
d a t u m a t t h e b o u n d a r y i s l e s s t h a n 1 0
? 6
. T h e n u m e r i c a l t e s t s s h o w t h a n t h i s i s
e n o u g h t o a v o i d a d a m a g e o f t h e q u a l i t a t i v e b e h a v i o r o f t h e s o l u t i o n .
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y
0
0
x
y
ξ
η
A B
Cx
F i g . 1 . D o m a i n
T h e n o n t r i v i a l b o u n d a r y c o n d i t i o n s o n t h e m e a n o w ' a r e w e l l a d a p t e d t o
s u c h a d o m a i n . I n d e e d , i f w e r e p r e s e n t i t o n g u r e ( 1 ) , w e j u s t h a v e t o p r e s c r i b e
t h e m o n ( A B ) a n d ( A C ) . O b v i o u s l y , w e w i l l i m p o s e t h a t l i m
! ? 1
'
1
( ; t ) =
l i m
! ? 1
'
2
( ; t ) = 0 , s o t h a t , t h e p o i n t A w i l l h a v e t o b e c h o s e n s u c i e n t l y
f a r f r o m t h e o r i g i n . W e m e s h t h e b o u n d e d d o m a i n w i t h a ( J ? 1 ) ( K ? 1 )
g r i d . W e w i l l d e n o t e b y x a n d y t h e s p a c e s t e p s , a n d u
n
l
t h e v a l u e a t t h e p o i n t
( x
0
+ ( j ? 1 ) x ; y 0 + ( k ? 1 ) y ) , ( j = 1 : : J ; k = 1 : : K ) , l = J ( k ? 1 ) + j , a n d '
n +
1
2
l
t h e v a l u e o f ' a t t h e s a m e p o i n t . A l w a y s u s i n g c e n t e r e d n i t e d i e r e n c e s , t h e f u l l y
d i s c r e t e s c h e m e r e a d s
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
1
4 x y
2
6
6
6
4
( '
n +
1
2
l + 1 + J
? '
n +
1
2
l + 1 ? J
? '
n +
1
2
l ? 1 + J
+ '
n +
1
2
l ? 1 ? J
)
2
+
( '
n ?
1
2
l + 1 + J
? '
n ?
1
2
l + 1 ? J
? '
n ?
1
2
l ? 1 + J
+ '
n ?
1
2
l ? 1 ? J
)
2
3
7
7
7
5
=
4 c
j u
n
l + 1
j
2
? j u
n
l ? 1
j
2
2 x
+
j u
n
l + J
j
2
? j u
n
l ? J
j
2
2 y
]
i
u
n + 1
l
? u
n
l
t
+ ( c
2
+ ) (
u
n + 1
l + 1
? 2 u
n + 1
l
+ u
n + 1
l ? 1
+ u
n
l + 1
? 2 u
n
l
+ u
n
l ? 1
2 x
2
)
+ (
u
n + 1
l + J
? 2 u
n + 1
l
+ u
n + 1
l ? J
+ u
n
l + J
? 2 u
n
l
+ u
n
l ? J
2 y
2
) ]
+
2 ( c
2
? )
4 x y
2
6
4
u
n + 1
l + 1 + J
? u
n + 1
l + 1 ? J
? u
n + 1
l ? 1 + J
2
+
u
n + 1
l ? 1 ? J
+ u
n
l + 1 + J
? u
n
l + 1 ? J
? u
n
l ? 1 + J
+ u
n
l ? 1 ? J
2
3
7
5
= (
j u
n + 1
l
j
2
+ j u
n
l
j
2
2
) + b c (
'
n +
1
2
l + 1
? '
n +
1
2
l ? 1
2 x
+
'
n +
1
2
l + J
? '
n +
1
2
l ? J
2 x
) ] (
u
n + 1
l
+ u
n
l
2
)
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W e c a n n o w s e e t h a t t h e b o u n d a r y c o n d i t i o n s a r e w e l l a d a p t e d . I n d e e d , a s
'
n +
1
2
l + 1 + J
= '
n +
1
2
l + 1 ? J
+ '
n +
1
2
l ? 1 + J
? '
n +
1
2
l ? 1 ? J
+ f ( '
n ?
1
2
; u
n
; x ; y )
w e s e e c l e a r l y o n g u r e ( 2 ) t h a t '
n +
1
2
l + 1 + J
i s t h e o n l y u n k n o w n f o r t h e h y p e r b o l i c
dependencel
l+1-J
l+1+J
l-1-J
l-1+J
known row or column
known data
unknown data
F i g . 2 . M e s h
e q u a t i o n o n ' a t p o i n t l . A l l o t h e r t e r m s a r e k n o w n f r o m t h e b o u n d a r y c o n d i t i o n s
o r f r o m t h e p r e v i o u s s t e p i n a n e x p l i c i t w a y g o i n g a l o n g t h e x - d i r e c t i o n a n d t h e n
a l o n g t h e y - d i r e c t i o n o r c o n v e r s e l y .
T h e n , t h e r s t s t e p i s t o c o m p u t e '
1
2
w i t h S '
1
2
=
4 c
T ( j u
0
j
2
) . A t e a c h t i m e
s t e p , w e s o l v e a l t e r n a t i v e l y t h e h y p e r b o l i c e q u a t i o n ( 3 . 1 2 - ( a ) )
S '
n +
1
2
= ? S '
n ?
1
2
+
2 c
T ( j u
n
j
2
)
a n d t h e e q u a t i o n ( 3 . 1 2 - ( b ) ) u s i n g a s t a n d a r d i t e r a t i o n p r o c e d u r e b a s e d o n a x e d -
p o i n t a l g o r i t h m . S o , w e g e t s u c c e s s i v e a p p r o x i m a t i o n s o f u
n + 1
b y s o l v i n g l i n e a r
s y s t e m s u n t i l w e e s t i m a t e t h a t t h e r e i s e n o u g h p r e c i s i o n .
4 . N u m e r i c a l e x p e r i m e n t s
4 . 1 . C o m p u t a t i o n o f d r o m i o n s o l u t i o n s
A s w e m e n t i o n e d i n s e c t i o n 2 , t h e r e a r e e x p l i c i t s o l u t i o n s o f t h e s u b c a s e ( D S I ) . S o ,
w e r s t t e s t o u r s c h e m e o n ( D S I ) s y s t e m a n d d r o m i o n s o l u t i o n s . W e s t a r t w i t h t h e
1 - 1 d r o m i o n s o l u t i o n
u ( ; ; t ) =
4 i e x p ( ? ( + + 4 t ) ? i ( + ) )
( 1 + e x p (
? 2
? 4 t ) ) ( 1 + e x p (
? 2
? 4 t ) ) + 1
c o r r e s p o n d i n g t o = ? 1 , = = l = m = 1 + i a n d = 1 i n ( 2 . 6 ) . I t r e p r e s e n t s
a s o l i t o n m o v i n g w i t h s p e e d 2
p
2 i n t h e n e g a t i v e d i r e c t i o n o n t h e l i n e = . T h e
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n o n t r i v i a l b o u n d a r y c o n d i t i o n s a s s o c i a t e d w i t h ' a r e
'
1
( ; t ) = ? 2 t a n h ( + 2 t ) + A
'
2
( ; t ) = ? 2 t a n h ( + 2 t ) + B
w h e r e A = B = ? 2 i n o r d e r t h a t l i m
! ? 1
'
1
( ; t ) = l i m
! ? 1
'
2
( ; t ) = 0 . M o r e -
o v e r , a s
' ( ; ; t ) =
2
Z
? 1
j u j
2
( ; s ; t ) d s +
Z
? 1
j u j
2
( s ; ; t ) d s
!
+ '
1
( ; t ) + '
2
( ; t )
a n e a s y c o m p u t a t i o n g i v e s
' ( ; ; t ) = ? 4
1
1 + e x p ( 2 + 4 t )
+
1
1 + e x p ( 2 + 4 t )
( 1 + e x p ( ? 2 ? 4 t ) ) ( 1 + e x p ( ? 2 ? 4 t ) ) + 1
+ '
1
( ; t ) + '
2
( ; t )
W e t a k e t h e i n i t i a l d a t u m a t t = ? 3 . T h e d o m a i n i s t h e s q u a r e ? 1 2 ; 1 2 ] ? 1 2 ; 1 2 ]
w i t h a 1 2 8 1 2 8 m e s h , t = 1 0
? 3
a n d t 2 ? 3 ; 3 ] . O n t h e g u r e s ( 3 ) , ( 4 ) , w e p l o t
t h e m o d u l e o f t h e t h e o r e t i c a l a n d t h e n u m e r i c a l s o l u t i o n a t t = 0 .
−10
−5
0
5
10
−10
−5
0
5
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
η ξ
F i g . 3 . N u m e r i c a l d r o m i o n 1 - 1
−10
−5
0
5
10
−10
−5
0
5
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
η ξ
F i g . 4 . E x a c t d r o m i o n 1 - 1
W e c a n n o t s e e a n y r e v e l a n t d i e r e n c e s . T o e m p h a s i z e i t , w e l o o k a t t h e e v o l u t i o n
o f t h e c o n t o u r o f t h e l o c a l i z e d s o l u t i o n f o r t =
? 3 ; 0 ; 3 o n g u r e ( 5 ) .
9
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−10 −5 0 5 10
−10
−5
0
5
10
t=−3
t=0
t=3
η
ξ
F i g . 5 . C o n t o u r o f n u m e r i c a l d r o m i o n 1 - 1
T h e s t r u c t u r e o f t h e d r o m i o n i s p e r f e c t l y p r e s e r v e d . M o r e o v e r , w e s h o w t h e
e v o l u t i o n o f t h e L
1
- n o r m a n d s e e t h a t i t i s w e l l c o n s e r v e d o n g u r e ( 6 ) .
−3.0 −1.0 1.0 3.00.0
0.5
1.0
F i g . 6 . L
1
n o r m
N e x t , t o u n d e r s t a n d b e t t e r w h a t d r i v e n b y t h e t r a c k s m e a n s , w e p l o t ' f o r t h e
s a m e v a l u e s o f t o n ( g ( 7 , 8 , 9 ) .
1 0
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−10
−5
0
5
10
−10
−5
0
5
10
−8
−6
−4
−2
0
2
4
ξ
η
F i g . 7 . ' a t t = - 3
−10
−5
0
5
10
−10
−5
0
5
10
−8
−6
−4
−2
0
2
4
ξ
η
F i g . 8 . ' a t t = 0
−10
−5
0
5
10
−10
−5
0
5
10
−8
−6
−4
−2
0
2
4
ξ
η
F i g . 9 . ' a t t = 3
I n f a c t , i f w e c o m p a r e p o s i t i o n o f u o n g ( 5 ) w i t h c r o s s s e c t i o n p o s i t i o n o f t h e
c o n t o u r o f ' o n g ( 1 0 , 1 1 , 1 2 ) , w e c a n s e e t h a t u i s e x a c t l y l o c a l i z e d o n t h e t r a c k s
l e f t b y ' .
−10 −5 0 5 10
−10
−5
0
5
10
ξ
η
F i g . 1 0 . ' a t t = - 3
−10 −5 0 5 10
−10
−5
0
5
10
ξ
η
F i g . 1 1 . ' a t t = 0
−10 −5 0 5 10
−10
−5
0
5
10
ξ
η
F i g . 1 2 . ' a t t = 3
I n o r d e r t o a n a l y z e b e t t e r t h e e r r o r w e m a d e , w e p l o t o n g u r e ( 1 3 , 1 4 , 1 5 )
N
1
= j j u
e x
j j
2
? j j u
n u m
j j
2
, N
2
= j j u
e x
? u
n u m
j j
2
a n d N
3
= j j j u
e x
j ? j u
n u m
j j j
2
w h e r e
u
e x
i s t h e e x a c t s o l u t i o n a n d u
n u m
t h e n u m e r i c a l o n e .
1 1
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−3.0 −1.0 1.0 3.0−0.25
−0.15
−0.05
0.05
0.15
F i g . 1 3 . N
1
= j j u
e x
j j
2
? j j u
n u m
j j
2
−3.0 −1.0 1.0 3.00.00
0.02
0.04
0.06
0.08
0.10
F i g . 1 4 . N
2
= j j u
e x
? u
n u m
j j
2
−3.0 −1.0 1.0 3.00.0
0.02
0.04
0.06
0.08
0.10
F i g . 1 5 . N
3
= j j j u
e x
j ? j u
n u m
j j j
2
T h e N
1
q u a n t i t y i s l e s s t h a n 1 0
? 5
a t a n y t i m e a s e x p e c t e d f r o m ( 3 . 1 1 ) . H o w e v e r ,
t h e p h a s e e r r o r i s q u i t e h i g h a s r e v e a l e d b y N
2
.
T h i s p h a s e e r r o r c o m e s f r o m t h e C r a n k - N i c o l s o n s c h e m e . I n d e e d , l e t u s c o n s i d e r
t h e l i n e a r S c h r o d i n g e r e q u a t i o n
( L S )
i u
t
+ u = 0 ;
u ( 0 ; x ) = u
0
( x ) ;
a n d t h e s e m i - d i s c r e t e C r a n k - N i c o l s o n s c h e m e s
8
<
:
i
u
n + 1
? u
n
t
+
u
n + 1
+ u
n
2
= 0 ;
u
0
( x ) = u
0
( x ) :
( 4 . 1 3 )
1 2
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T h e s o l u t i o n o f ( L S ) i s g i v e n b y
u ( x ; t ) = S ( t ) u
0
( x )
w h e r e
d
S ( t ) ( ) = e x p (
? i
2
t ) a n d
c
( : ) d e n o t e s t h e F o u r i e r t r a n s f o r m . W r i t i n g ( 4 . 1 3 )
a s
u
n + 1
= ( O p
1
)
? 1
O p
2
u
n
w i t h O p
1
= (
i
t
+
2
) a n d O p
2
= (
i
t
?
2
) . W e g e t
u
n + 1
= e x p ( i )
c
u
n
= e x p ( i ( n + 1 ) )
c
u
0
w i t h = ?
2
t +
6
t
3
1 2
+ o (
1 0
t
5
) . T h u s , i f t = n t ,
j j u
e x
( t ) ? u
n
j j
2
= 2 j s i n (
t t
2
6
2 4
) j j j u
0
j j
2
S o , t h e e r r o r p h a s e c o u l d g r o w u p t o 2 j j u
0
j j
2
, w h i c h i s n o t n e g l i g i b l e . O n g u r e ( 1 6 ) ,
w e p l o t t h e r e l a t i v e p h a s e e r r o r ( N
2
= j j u
0
j j
2
) i n o n e d i m e n s i o n f o r u
0
( x ) = s i n ( 6 x ) ,
w h i c h i s t h e e i g e n v e c t o r o f L a p l a c i a n o p e r a t o r a s s o c i a t e d t o t h e e i g e n v a l u e k
2
= 3 6
c o r r e s p o n d i n g t o
6
i n t h e a b o v e f o r m u l a . W e t a k e t = 1 0
? 2
, x = 5 : 1 0
? 1
a n d
x 2 0 ; ] .
0.0 10.0 20.0 30.0 40.0 50.0
time
0.0
0.5
1.0
1.5
2.0
e r r o r p h a s e
Crank−Nicolson scheme error phase
k=6
Theoretical error
Numerical error
F i g . 1 6 . P h a s e e r r o r f o r ( L S ) e q u a t i o n
O b v i o u s l y , w e p r e s e n t h e r e t h e s i m p l e e x a m p l e t o a n a l y z e t h e e r r o r p h a s e o f
C r a n k - N i c o l s o n s c h e m e . I f w e w a n t t o u n d e r s t a n d w h y N
3
i s b a d t o o , w e m u s t
p l o t j j j u
e x
j ? j u
n u m
j j j
2
f o r a s u p e r p o s i t i o n o f t w o s o l u t i o n s o f ( L S ) . T h e r e s u l t f o r
u
0
( x ) = s i n ( 6 x ) + s i n ( 8 x ) i s p l o t t e d o n g ( 1 7 ) .
1 3
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0.0 10.0 20.0 30.0 40.00.0
0.5
1.0
1.5
2.0
2.5
N3/||U0||N2/||U0||
F i g . 1 7 . N
2
= j j u
0
j j
2
a n d N
3
= j j u
0
j j
2
f o r ( L S ) e q u a t i o n
W e s e e t h a t t h e e r r o r f o r ( L S ) i s q u i t e h i g h . I n f a c t , j u
e x
j ? j u
n u m
j c o u p l e s t h e
p h a s e a s s o o n a s t h e n u m b e r o f p h a s e i s m o r e t h a n t w o .
A p o s s i b l e c o r r e c t i o n f o r ( L S ) c o n s i s t s i n s o l v i n g
i
u
n + 1
? u
n
t
+
u
n + 1
+ u
n
2
+
t
2
1 2
3
u
n + 1
+ u
n
2
= 0
b u t , a l t h o u g h t h i s i s b e t t e r f o r t h e l i n e a r c a s e , i t i s n o t s o g o o d f o r t h e n o n l i n e a r
S c h r o d i n g e r e q u a t i o n .
W e c o n t i n u e o u r t e s t s w i t h t h e 2 - 2 d r o m i o n a n d a d i a g o n a l m a t r i x , w h i c h
c o r r e s p o n d s t o t h e i n t e r a c t i o n o f t w o l u m p s . W e t a k e
1
= 2
? 2 i ,
2
= 4
? 0 : 5 i ,
l
1
= 2 + 1 , l
2
= 1 + 2 i ,
1
= 1 ? 2 i ,
2
= 3 ? 0 : 5 i , m
1
= 1 + i , m
2
= 2 + 3 i ,
1 1
= 1 + i ,
2 2
= 2 + 3 i a n d
1 2
=
2 1
= 0 i n ( 2 . 1 ) t o ( 2 . 3 ) . T h e r e s u l t i n g a l g e b r a i c
e q u a t i o n s a r e s o l v e d b y m a k i n g u s e o f M a p l e V . T h u s , w e g e t t h e e x p l i c i t e x p r e s s i o n
o f u ( ; ; t ) , u
1
( ; t ) a n d u
2
( ; t ) w h i c h a r e t o o l a r g e t o b e p r i n t e d i n t h i s p a p e r .
T h e r e f o r e , w e c a n c o m p a r e t h e n u m e r i c a l s o l u t i o n c o m p u t e d f r o m u ( ; ; ? 1 ) w i t h
t h e e x a c t s o l u t i o n a b o v e . T h e s p a c e d o m a i n i s ? 1 0 ; 1 0 ] ? 1 0 ; 1 0 ] , w i t h 2 5 7 p o i n t s
i n e v e r y d i r e c t i o n a n d t = 1 0
? 3
. W e p l o t t h e c o n t o u r o f t h e e x a c t s o l u t i o n a n d t h e
n u m e r i c a l o n e f o r t =
? 1 ; 0 ; 1 ( g ( 1 8 ) ) .
1 4
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−10 0 10−10
−5
0
5
10Numerical solution, t=0
−10 0 10−10
−5
0
5
10Numerical solution, t=1
−10 0 10−10
−5
0
5
10Numerical solution, t=−1
−10 0 10−10
−5
0
5
10Exact solution, t=1
−10 0 10−10
−5
0
5
10Exact solution, t=0
−10 0 10−10
−5
0
5
10Exact solution, t=−1
F i g . 1 8 . C o n t o u r o f d r o m i o n 2 - 2
T h e t w o s o l u t i o n s a r e v e r y c l o s e t o e a c h o t h e r a n d w e n o t e o n l y a s m a l l d i s c r e p -
a n c y f o r t = 1 b e h i n d t h e u p p e r b u m p . W e a l s o d r a w t h e m o d u l e o f t h e s o l u t i o n
( g ( 1 9 , 2 0 , 2 1 ) ) a t t h e s a m e t i m e s t o s h o w t h a t i t i s r e a l l y t h e i n t e r a c t i o n o f t w o
l o c a l i z e d l u m p s .
−10
−5
0
5
10
−10
−5
0
5
100
1
2
3
4
5
F i g . 1 9 . N u m e r i c a l d r o m i o n
2 - 2 a t t = - 1
−10
−5
0
5
10
−10
−5
0
5
100
1
2
3
4
5
F i g . 2 0 . N u m e r i c a l d r o m i o n
2 - 2 a t t = 0
−10
−5
0
5
10
−10
−5
0
5
100
1
2
3
4
5
F i g . 2 1 . N u m e r i c a l d r o m i o n
2 - 2 a t t = 1
W e p r e c i s e t h e m o v e m e n t o f t h e t r a c k s l e f t b y ' w i t h t h e r e p r e s e n t a t i o n o f t h e
c o n t o u r o f ' ( g ( 2 2 , 2 3 , 2 4 ) ) .
1 5
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−10 −8 − 6 − 4 − 2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
F i g . 2 2 . ' a t t = - 1
−10 −8 − 6 − 4 − 2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
F i g . 2 3 . ' a t t = 0
−10 −8 − 6 − 4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
F i g . 2 4 . ' a t t = 1
A s t h e m a t r i x i s d i a g o n a l , t h e n u m b e r o f l u m p s i s o n l y t w o i n s t e a d o f f o u r .
H o w e v e r , t h e r e a r e f o u r c r o s s - p o i n t s o n t h e t r a c k s . T w o o f t h e m l o c a l i z e t h e e x i s t i n g
l u m p s a n d t h e t w o o t h e r s g i v e t h e l o c a t i o n o f t h e t w o m i s s i n g o n e s a s s t a t e d i n
8
.
T h e p h a s e e r r o r ( g ( 2 5 ) ) i s b i g g e r t h a n t h e o n e o f d r o m i o n 1 - 1 , b u t t h e d y n a m i c o f
m o v e m e n t i s m o r e s o p h i s t i c a t e d .
−1.0 −0.5 0.0 0.5 1.0
time
0.0
0.5
1.0
1.5
2.0
F i g . 2 5 . j j u
e x
? u
n u m
j j
2
4 . 2 . R o l e o f t h e i n i t i a l d a t u m a n d p a r a m e t e r s
N o w t h a t w e h a v e t e s t e d o u r s c h e m e w i t h e x a c t s o l u t i o n s , w e c a n e x a m i n e t h e
a c t i o n o f D S o n o t h e r i n i t i a l d a t a . F o r t h a t , w e p u t t h e g a u s s i a n d a t u m u
0
=
4 e x p ? ( x
2
+ y
2
) ] i n D S I w i t h '
1
= '
2
= 0 . L i k e i t i s s t a t e d i n F o k a s - S a n t i n i
8
, a l l
i n i t i a l d a t a w i t h '
1
= '
2
= 0 s h o u l d d i s p e r s e a t i n n i t y a n d t h i s i s e x a c t l y w h a t
w e g e t ( g ( 2 6 , 2 7 , 2 8 , 2 9 , 3 0 ) ) .
1 6
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−10
−5
0
5
10 −10−5
05
10
0
1
2
3
4
F i g . 2 6 . G a u s s i a n d a t u m a t t = 0
−10
−5
0
5
10 −10−5
05
10
0
1
2
3
4
F i g . 2 7 . G a u s s i a n d a t u m a t t = 0 . 3 2 1
−10
−5
0
5
10 −10−5
05
10
0
1
2
3
4
F i g . 2 8 . G a u s s i a n d a t u m a t t = 0 . 6 4 2
−10
−5
0
5
10 −10−5
05
10
0
1
2
3
4
F i g . 2 9 . G a u s s i a n d a t u m a t t = 0 . 9 6 3
W e s e e t h e e e c t o f d i s p e r s i o n o n t h e L
1
- n o r m w h i c h d e c r e a s e s w i t h t i m e
( g ( 3 1 ) .
1 7
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−10 −5 0 5 10
−10
−5
0
5
10
t=0.
−10 −5 0 5 10
−10
−5
0
5
10
t=0.321
−10 −5 0 5 10
−10
−5
0
5
10
t=0.642
−10 −5 0 5 10
−10
−5
0
5
10
t=0.963
F i g . 3 0 . C o n t o u r o f g a u s s i a n d a t u m
0.0 0.3 0.5 0.8 1.00.0
1.0
2.0
3.0
4.0
5.0
F i g . 3 1 . L
1
n o r m o f g a u s s i a n d a t u m
T h e n , w e g o o n b y c h a n g i n g s o m e c o e c i e n t s w h i l e k e e p i n g t h e o t h e r s t o t h e
v a l u e s t a k e n f o r D S I . T h e i n i t i a l d a t u m a n d t h e b o u n d a r y c o n d i t i o n s a r e t h e s a m e
t h a n t h o s e o f d r o m i o n 1 - 1 t e s t . I t i s d i c u l t t o m a k e a n e x h a u s t i v e r e v i e w o f
t h e e e c t s o f e a c h p a r a m e t e r d u e t o t h e i r n u m b e r . H o w e v e r , w e c a n i m a g i n e t h e
i n u e n c e o f s o m e c o e c i e n t s . F o r e x a m p l e , a n d s h o u l d m a n a g e t h e d i s p e r s i o n .
S o m e t e s t s s h o w t h a t a n d d o n o t h a v e t h e s a m e i n u e n c e . a c t s d i r e c t l y o n
t h e x - d i r e c t i o n , w h e r e a s a c t s o n t h e y - d i r e c t i o n , b u t w i t h o u t t h e s a m e s t r e n g t h .
I n d e e d , x a n d y d o n o t h a v e t h e s a m e r o l e i n D S . F o r e x a m p l e , w e p r e s e n t h e r e t h e
t e s t f o r = = 1 . T h e n , t h e e q u a t i o n s b e c o m e
i u
t
+ u = j u j
2
u + u '
x
;
'
x x
? '
y y
= ? 2 ( j u j
2
)
x
:
W e s e e t h a t t h e i n i t i a l l u m p d i s p e r s e s a w a y m u c h f a s t e r i n t h e x - d i r e c t i o n t h a n i n
t h e y - d i r e c t i o n g ( 3 2 , 3 3 ) .
1 8
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−100
10
−10
0
100
0.5
1
t=−1.5
−100
10
−10
0
100
0.5
1
t=−1.072
−100
10
−10
0
100
0.5
1
t=−0.640
−100
10
−10
0
100
0.5
1
t=−0.216
F i g . 3 2 . = = 1
−10 −5 0 5 10
−10
−5
0
5
10
t=−0.216
−10 −5 0 5 10
−10
−5
0
5
10
t=−0.640
−10 −5 0 5 10
−10
−5
0
5
10
t=−1.072
−10 −5 0 5 10
−10
−5
0
5
10
t=−1.5
F i g . 3 3 . = = 1
1 9
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T h e n , w e c h o o s e t o s t u d y t h e e e c t s o f t h e b p a r a m e t e r o n t h e b e h a v i o r o f
d r o m i o n 1 - 1 . W e p l u g i n t o ( D S I ) r e s p e c t i v e l y b = 0 : 5 a n d b = 1 : 5 . W e n o t e
( g ( 3 4 , 3 5 , 3 6 , 3 7 ) ) t h a t d r o m i o n 1 - 1 i s n o t s t a b l e a t a l l w i t h r e s p e c t t o b .
−100
10
−10
0
100
0.5
1
t=1.5
−100
10
−10
0
100
0.5
1
t=0.640
−100
10
−10
0
100
0.5
1
t=−0.430
−100
10
−10
0
100
0.5
1
t=−1.5
F i g . 3 4 . b = 0 5
−10 −5 0 5 10
−10
−5
0
5
10
t=−1.5
−10 −5 0 5 10
−10
−5
0
5
10
t=−0.430
−10 −5 0 5 10
−10
−5
0
5
10
t=0.640
−10 −5 0 5 10
−10
−5
0
5
10
t=1.5
F i g . 3 5 . b = 0 5
2 0
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−100
10
−10
0
100
0.5
1
t=−1.5
−100
10
−10
0
100
0.5
1
t=−0.430
−100
10
−10
0
100
0.5
1
t=0.640
−100
10
−10
0
100
0.5
1
t=1.5
F i g . 3 6 . b = 1 5
−10 −5 0 5 10
−10
−5
0
5
10
t=1.5
−10 −5 0 5 10
−10
−5
0
5
10
t=0.640
−10 −5 0 5 10
−10
−5
0
5
10
t=−0.430
−10 −5 0 5 10
−10
−5
0
5
10
t=−1.5
F i g . 3 7 . b = 1 5
2 1
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O t h e r t e s t s s h o w t h a t t h e r e s u l t s a r e a b o u t t h e s a m e w h e n m o d i f y i n g t h e v a l u e
o f a n d . S o , d r o m i o n 1 - 1 i s n o t s t a b l e w i t h r e s p e c t t o t h e c o e c i e n t s o f ( D S I ) .
5 . B l o w - u p o f D S ( E - H )
I t i s w e l l k n o w n t h a t ( N L S ) a d m i t s b l o w - u p s o l u t i o n s ( s e e G i n i b r e - V e l o
2 0
, K a t o
2 1
o r G l a s s e y
2 2
f o r i n s t a n c e ) . N o w , t h e d y n a m i c o f e x p l o s i o n i s b e t t e r u n d e r s t o o d
(
1 7
,
2 3
) . M o r e o v e r , a s p r e d i c t e d b y G h i d a g l i a a n d S a u t
3
, P a p a n i c o l a o u - S u l e m - S u l e m
a n d W a n g
2 4
s h o w n u m e r i c a l l y t h a t t h e b l o w - u p o c c u r s i n t h e e l l i p t i c - e l l i p t i c c a s e
o f ( D S ) . U n f o r t u n a t e l y , n o r e s u l t s a r e k n o w n t o v a l i d a t e o r n o t t h e b l o w - u p i n t h e
( E - H ) m o d e .
F r o m n o w o n , w e s e t = = 1 = 2 , c = 1 a n d '
1
( ; t ) = '
2
( ; t ) = 0 ; 8 t . T h e n ,
i n t h e ( ; ) p l a n e , D S ( E - H ) b e c o m e s
(
i u
t
+ u = j u j
2
u + b u '
+
; ( a )
'
=
4
( j u j
2
+ j u j
2
) : ( b )
( 5 . 1 4 )
I f w e a s s u m e t h a t = ? 1 , b l o w - u p s h o u l d a r i s e w h e n b g o e s t o z e r o , a s t h e r s t
e q u a t i o n t e n d s t o t h e f o c u s i n g ( N L S ) e q u a t i o n . I n t h e s a m e w a y , a s ! 0 , ( 5 . 1 4 b )
g i v e s ' ( ; ; t ) =
1
( ; t ) +
2
( ; t ) . U s i n g t h e h y p o t h e s i s '
1
( ; t ) = '
2
( ; t ) = 0 ; 8 t ,
w e g e t ' ( ; ; t ) = r , r 2 R . P u t t i n g i t o n ( 5 . 1 4 a ) , w e g e t e x a c t l y ( N L S ) . T h e r e f o r e ,
w e w i l l w o r k n a l l y o n t h e s y s t e m
(
i u
t
+ u = ? j u j
2
u + b u '
+
;
'
=
4
( j u j
2
+ j u j
2
) :
( 5 . 1 5 )
I n o r d e r t o v e r i f y o u r s t a t e m e n t , w e t a k e a s i n i t i a l c o n d i t i o n t h e o n e u s e d i n
1 7
a n d
2 3
f o r b l o w - u p o f ( N L S ) . S o , u
0
= 4 e x p ? ( x
2
+ y
2
) ] . I n t h e l a s t r e f e r e n c e
1 7
, t h e p r e s u m e d b l o w - u p t i m e f o r t h i s i n i t i a l d a t u m i s c o m p u t e n u m e r i c a l l y a n d
i s t
= 0 : 1 4 2 5 . T h i s r e f e r e n c e t i m e a l l o w s u s t o v a l i d a t e o u r s u p p o s i t i o n s . F o r
t h e n u m e r i c a l t e s t s , w e t a k e a 2 5 6 2 5 6 m e s h , t = 1 0
? 4
a n d t h e d o m a i n i s
? 4 ; 4 ] ? 4 ; 4 ] .
W e b e g i n b y t h e ? t e s t c o n s i s t i n g t o c o m p u t e t h e a p p r o x i m a t e s o l u t i o n s o f
e q u a t i o n s ( 5 . 1 5 ) , w i t h t h e g a u s s i a n i n i t i a l d a t u m , f o r ! 0 a n d b = 1 . W e p l o t o n
g ( 3 8 ) s u p
;
j u ( ; ; t ) j f o r d i e r e n t v a l u e s o f .
2 2
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.0
25.0
50.0
75.0
100.0 −0.10.01
0.1 0.2
σ= −0.3
σ=
t*
−0.2σ=
σ=σ=
σ=
F i g . 3 8 . E v o l u t i o n o f b l o w - u p t i m e f o r b = 1 a n d d i e r e n t v a l u e s o f
W e s e e c l e a r l y t h a t b l o w - u p s e e m s t o a r i s e a n d t h a t t h e s i g n o f c h a n g e s t h e
r e l a t i v e p o s i t i o n o f t h e b l o w - u p t i m e f o r ( 5 . 1 5 ) c o m p a r e d t o t
.
N e x t , w e g o o n w i t h b ! 0 , s e t t i n g = ? 2 . N o w , w e p l o t t h e r e s u l t s o n g ( 3 9 )
f o r a w i d e r r a n g e o f v a l u e s o f b .
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.0
25.0
50.0
75.0
100.0
b=2
b=1
b=0.5
b=0.3
b=0.1
b=−0.1 b=−0.15
b=−0.25
b=−0.3
t*
F i g . 3 9 . E v o l u t i o n o f b l o w - u p t i m e f o r = ? 2 a n d d i e r e n t v a l u e s o f b
I f b i s s m a l l e n o u g h , t h e r e s u l t s a r e t h e s a m e t h a n f o r t h e ? t e s t . B u t w e s e e
i n a d d i t i o n t h a t a s t r o n g c o n c e n t r a t i o n a n d m a y b e a b l o w - u p o c c u r s f o r v a l u e s o f
b i n O ( 1 ) w h e n b i s p o s i t i v e . T h e r e f o r e , t h i s f a c t s e e m s t o i n d i c a t e t h a t D S ( E - H )
s y s t e m h a s a n i n n e r b l o w - u p m e c h a n i s m . F o r n e g a t i v e v a l u e s o f b , a s t a b i l i z a t i o n
2 3
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a p p e a r s . T h e r s t v a l u e o f b l e a d i n g t o t h i s s t a b i l i z a t i o n i s h a r d t o c o m p u t e b e c a u s e
o f r e l a t i v e i n s t a b i l i t y . W e p l o t o n g ( 4 0 ) a m o r e c o m p l e t e s t u d y o n a 5 1 2 5 1 2
n e r m e s h f o r t h o s e v a l u e s l e a d i n g t o s t a b i l i z a t i o n .
0.00 0.05 0.10 0.15 0.20 0.250.0
50.0
100.0
150.0
200.0
b=−0.18
b=−0.19
b=−0.20
b=−0.21
b−0.22
b=−0.25
F i g . 4 0 . S t a b i l i z a t i o n o f b l o w - u p f o r = ? 2
W e s h o w n o w t h a t t h e s i g n o f h a s a n i n u e n c e o n t h e p o s i t i o n o f t h e s u p p o s e d
b l o w - u p t i m e o f D S w i t h r e s p e c t t o t h e b l o w - u p t i m e o f ( N L S ) . T h e g ( 3 9 ) a n d
g ( 4 1 ) i l l u s t r a t e t h i s f a c t w i t h =
? 2 a n d = + 2 .
2 4
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.0
25.0
50.0
75.0
100.0b=−0.1
b=0.1
b=0.2
t
*
F i g . 4 1 . E v o l u t i o n o f b l o w - u p t i m e f o r = + 2 a n d v a r y i n g b
F i n a l l y , a l t h o u g h w e d o n o t h a v e a r e n e m e n t p r o c e d u r e , w e s h o w o n v a r i o u s
m e s h e s t h e v a l i d i t y o f t h e t e s t s a b o v e a s t h e L
1
- n o r m i n c r e a s e s t w o f o l d e a c h t i m e
t h e m e s h s i z e i s d i v i d e d b y t w o . S o , w e t h i n k t h a t b l o w - u p f o r D S ( E - H ) r e a l l y
e x i s t s . W e p l o t o n ( g ( 4 2 ) ) s u p
;
j u ( ; ; t ) j f o r d i e r e n t m e s h a n d j u j ( g ( 4 3 , 4 4 ) ) a t
t = 0 : 1 7 0 4 f o r = ? 2 a n d b = ? 0 : 1 5 .
0.00 0.05 0.10 0.15 0.200.0
50.0
100.0
150.0
200.0
Mesh=128x128
Mesh=256x256
Mesh=512x512
F i g . 4 2 . E v o l u t i o n o f b l o w - u p t i m e f o r b = ? 0 1 5 a n d d i e r e n t g r i d m e s h
2 5
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−2
−1
0
1
2
−2
−1
0
1
20
5
10
15
20
25
F i g . 4 3 . j u j a t t = 0
−2
−1
0
1
2
−2
−1
0
1
20
5
10
15
20
25
F i g . 4 4 . j u j a t t = 0 . 1 7 0 4
6 . C o n c l u s i o n
W e d e v e l o p a n e w s c h e m e i n o r d e r t o s o l v e D S ( E - H ) s y s t e m s . T h i s t o o l a l l o w s t o
s h o w t h a t d r o m i o n s 1 - 1 a r e n o t s t a b l e w i t h r e s p e c t t o c o e c i e n t s a n d t h a t b l o w - u p
m e c h a n i s m c a n e x i s t f o r D S ( E - H ) . W e c o n r m t h a t f r o m a n i n i t i a l d a t u m w i t h
'
1
= '
2
= 0 , t h e s o l u t i o n d i s p e r s e s a w a y f o r ( D S I ) . I n a d d i t i o n , w e p r o v e t h a t
C r a n k - N i c o l s o n t y p e s c h e m e s c r e a t e a p e r i o d i c p h a s e e r r o r t h a t c a n b e q u i t e b i g
f o r s o m e v a l u e s o f t . U n f o r t u n a t e l y , w e c a n n o t p r o v e e x i s t e n c e o f s o l u t i o n a n d t h e
c o n v e r g e n c e o f o u r s e m i - d i s c r e t e s c h e m e . H o w e v e r , t h e p r i n c i p l e o f r e l a x a t i o n i s
a p p l i c a b l e t o a w i d e r a n g e o f s y s t e m s , a n d , i n p a r t i c u l a r , o u r s c h e m e i s r e l a t i v e l y
e a s y t o t r a n s p o s e t o o t h e r v e r s i o n s o f D S .
A c k n o w l e d g m e n t
T h e a u t h o r s t h a n k T . C o l i n f o r v a l u a b l e d i s c u s s i o n s a n d f r u i t f u l s u g g e s t i o n s . T h e y
a l s o t h a n k P . F a b r i e f o r h e l p i n g t h e m t o u n d e r s t a n d b e t t e r t h e p h a s e e r r o r c r e a t e d
b y C r a n k - N i c o l s o n s c h e m e . I n a d d i t i o n , t h e r s t a u t h o r i s v e r y g r e a t f u l t o P . D e -
g o n d w h o k i n d l y w e l c o m e h i m a t t h e M a t h e m a t i q u e s p o u r l ' I n d u s t r i e e t l a P h y s i q u e
l a b o r a t o r y i n T o u l o u s e .
1 . D a v e y A . a n d S t e w a r t s o n K . O n t h r e e - d i m e n s i o n a l p a c k e t s o f s u r f a c e w a v e s , P r o c . R .
S o c . L o n d . A , v o l 3 3 8 , p a g e s 1 0 1 { 1 1 0 , 1 9 7 4 .
2 . D j o r d j e v i c V . D . a n d R e d e k o p p L . G . O n t w o - d i m e n s i o n a l p a c k e t s o f c a p i l l a r y - g r a v i t y
w a v e s , J . F l u i d M e c h . , v o l 7 9 , p a g e s 7 0 3 { 7 1 4 , 1 9 7 7 .
3 . G h i d a g l i a J - M . a n d S a u t J - C . O n t h e i n i t i a l v a l u e p r o b l e m f o r t h e D a v e y - S t e w a r t s o n
s y s t e m s , N o n l i n e a r i t y 3 , p a g e s 4 7 5 { 5 0 6 , 1 9 9 0 .
4 . A b l o w i t z M . J . a n d S e g u r H . O n t h e e v o l u t i o n o f p a c k e t s o f w a t e r w a v e s , J . F l u i d M e c h . ,
v o l 9 2 , n
0
( 4 ) , p a g e s 6 9 1 { 7 1 5 , 1 9 7 9 .
5 . H a y a s h i N . a n d H i r a t a H . G l o b a l e x i s t e n c e a n d a s y m p t o t i c b e h a v i o r i n t i m e o f s m a l l
2 6
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s o l u t i o n s t o t h e e l l i p t i c - h y p e r b o l i c D a v e y - S t e w a r t s o n s y s t e m , N o n l i n e a r i t y 9 , p a g e s
1 3 8 7 { 1 4 0 9 , 1 9 9 6 .
6 . H a y a s h i N . L o c a l e x i s t e n c e i n t i m e o f s m a l l s o l u t i o n s t o t h e D a v e y - S t e w a r t s o n s y s t e m s ,
A n n . I n s t . H e n r i P o i n c a r e , v o l 6 5 , n
0
4 , p a g e s 3 1 3 { 3 6 6 , 1 9 9 6 .
7 . B o i t i M . , L e o n J . J . - P . a n d P e m p i n e l l i F . A n e w s p e c t r a l t r a n s f o r m f o r t h e D a v e y -
S t e w a r t s o n I e q u a t i o n , P h y s . L e t t . A , v o l 1 4 1 , n
0
( 3 - 4 ) , p a g e s 1 0 1 { 1 0 7 , 1 9 8 9 .
8 . F o k a s A . S . a n d S a n t i n i P . M . D r o m i o n s a n d a b o u n d a r y v a l u e p r o b l e m f o r t h e D a v e y -
S t e w a r t s o n I e q u a t i o n , P h y s i c a D , v o l 4 4 , p a g e s 9 9 { 1 3 0 , 1 9 9 0 .
9 . F o k a s A . S a n d S u n g L . - Y . O n t h e s o l v a b i l i t y o f t h e n - w a v e , D a v e y - S t e w a r t s o n a n d
K a d o m t s e v - P e t v i a s h v i l i e q u a t i o n s , I n v e r s e P r o b l e m s 8 , v o l 5 , p a g e s 6 7 3 { 7 0 8 , 1 9 9 2 .
1 0 . S a n t i n i P . M . E n e r g y e x c h a n g e o f i n t e r a c t i n g c o h e r e n t s t r u c t u r e s i n m u l t i d i m e n s i o n s ,
P h y s i c a D , v o l 4 1 , p a g e s 2 6 { 5 4 , 1 9 9 0 .
1 1 . W h i t e P . W . a n d W e i d e m a n J . A . C . N u m e r i c a l s i m u l a t i o n o f s o l i t o n s a n d d r o m i o n s i n
t h e D a v e y - S t e w a r t s o n s y s t e m , M a t h e m a t i c s a n d C o m p u t e r s i n S i m u l a t i o n , v o l 3 7 ,
p a g e s 4 6 9 { 4 7 9 , 1 9 9 4 .
1 2 . H i e t a r i n t a J . a n d H i r o t a R . M u l t i d r o m i o n s o l u t i o n s t o t h e D a v e y - S t e w a r t s o n e q u a t i o n ,
P h y s . L e t t . A , v o l 1 4 5 , n
0
( 5 ) , p a g e s 2 3 7 { 2 4 4 , 1 9 9 0 .
1 3 . J a u l e n t M . , M a n n a M . A . a n d M a r t i n e z A l o n s o L . F e r m i o n i c a n a l y s i s o f D a v e y -
S t e w a r t s o n d r o m i o n s , P h y s . L e t t . A , v o l 1 5 1 , n
0
( 6 - 7 ) , p a g e s 3 0 3 { 3 0 7 , 1 9 9 0 .
1 4 . G i l s o n R . a n d N i m m o J . C . A d i r e c t m e t h o d f o r d r o m i o n s o l u t i o n s o f t h e D a v e y -
S t e w a r t s o n e q u a t i o n s a n d t h e i r a s y m p t o t i c p r o p e r t i e s , P r o c . R . S o c . L o n d . A , v o l
4 3 5 , p a g e s 3 3 9 { 3 5 7 , 1 9 9 1 .
1 5 . D e l f o u r M . , F o r t i n M . a n d P a y r e G . F i n i t e - d i e r e n c e s o l u t i o n s o f a n o n - l i n e a r
S c h r o d i n g e r e q u a t i o n , J o u r n a l o f c o m p u t a t i o n a l p h y s i c s , v o l 4 4 , p a g e s 2 7 7 { 2 8 8 , 1 9 8 1 .
1 6 . B i d e g a r a y , B . I n v a r i a n t m e a s u r e s f o r s o m e p a r t i a l d i e r e n t i a l e q u a t i o n s , P h y s i c a D ,
v o l 8 2 , p a g e s 3 4 0 { 3 6 4 , 1 9 9 5 .
1 7 . B r u n e a u C . H . , D i M e n z a L . a n d L e h n e r T . N u m e r i c a l s i m u l a t i o n o f n o n l i n e a r p l a s m a s ,
R a p p o r t I n t e r n e n
0
9 6 0 2 3 , M a t h e m a t i q u e s A p p l i q u e e s d e B o r d e a u x .
1 8 . C o l i n T . a n d F a b r i e P . S e m i d i s c r e t i z a t i o n i n t i m e f o r n o n l i n e a r S c h r o d i n g e r - w a v e s e q u a -
t i o n s , R a p p o r t I n t e r n e n
0
9 6 0 2 8 , M a t h e m a t i q u e s A p p l i q u e e s d e B o r d e a u x .
1 9 . B e s s e C . R e l a x a t i o n s c h e m e s f o r N o n l i n e a r S c h r o d i n g e r E q u a t i o n s a n d D a v e y -
S t e w a r t s o n S y s t e m s , T o b e p u b l i s h e d .
2 0 . G i n i b r e J . a n d V e l o G . O n a c l a s s o f n o n l i n e a r S c h r o d i n g e r e q u a t i o n s p a r t I , I I , J .
F u n c t . A n a l . , v o l 3 2 , p a g e s 1 { 3 2 , 3 3 { 7 1 , 1 9 7 9 .
2 1 . K a t o T . N o n l i n e a r S c h r o d i n g e r e q u a t i o n s , L e c t u r e s N o t e s i n P h y s i c s , 3 4 5 , 1 9 8 8 .
2 2 . G l a s s e y R . T . O n t h e b l o w i n g - u p s o l u t i o n s t o t h e C a u c h y p r o b l e m f o r n o n l i n e a r
S c h r o d i n g e r e q u a t i o n s , J . M a t h . P h y s , v o l 1 8 , n
0
( 9 ) , p a g e s 1 7 9 4 { 1 7 9 7 , 1 9 7 7 .
2 3 . S u l e m P . L . , S u l e m C . a n d P a t e r a A . N u m e r i c a l s i m u l a t i o n o f s i n g u l a r s o l u t i o n s t o t h e
t w o - d i m e n s i o n a l c u b i c S c h r o d i n g e r e q u a t i o n , C o m m . o n P u r e a n d A p p l . M a t h . , v o l
3 7 , p a g e s 7 5 5 { 7 7 8 , 1 9 8 4 .
2 4 . P a p a n i c o l a o u G . C . , S u l e m C . , S u l e m P . - L . a n d W a n g X . P . T h e f o c u s i n g s i n g u l a r i t y
o f t h e D a v e y - S t e w a r t s o n e q u a t i o n s f o r g r a v i t y - c a p i l l a r y s u r f a c e w a v e s , P h y s . D , v o l 7 2 ,
n
0
( 1 - 2 ) , p a g e s 6 1 { 8 6 , 1 9 9 4 .
2 7