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Solitary Wave Solutions of Generalized Korteweg-De Vries Equations
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
105
SOLITARY WAVE SOLUTIONS OF GENERALIZED KORTEWEG-DE VKIES EQUATIONS F. A. Howes
Department of Mathematics U n i v e r s i t y of C a l i f o r n i a a t Davis Davis, CA 95616 U.S.A.
We b e g i n w i t h a d i s c u s s i o n of t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of t h e s c a l a r Korteweg-deVries (KdV) and modified Korteweg-deVries e q u a t i o n s , i n c l u d i n g t h e d e r i v a t i o n of formulas f o r wave amplitude and speed when t h e waves move a g a i n s t a nonzero "background". This i s followed by a c o n s i d e r a t i o n of r e l a t e d q u e s t i o n s f o r system a n a l o g s of (KdV). Our approach i n v o l v e s t h e c o n s i s t e n t u s e of t h e asymptotic t h e o r y of second-order s i n g u l a r l y p e r t u r b e d boundary v a l u e problems, and so i t o f f e r s a convenient a l t e r n a t i v e t o more t r a d i t i o n a l a n a l y t i c a l and numerical methods. SOLITARY WAVE SOLUTIONS OF THE KDV AND MODlFIED KDV EQUATIONS e q u a t i o n i n t h e form
Let us c o n s i d e r f i r s t t h e Korteweg-deVries
- + -2uux + E u = xxx Ut
0,
e
i s a s m a l l p o s i t i v e parameter and s u b s c r i p t s d e n o t e p a r t i a l d i f f e r e n -
tiation.
I n o r d e r t o s t u d y s o l u t i o n s of (KdV) which r e p r e s e n t permanent waves
where
moving t o t h e r i g h t w i t h speed c
-
u = u ( ~ , e )= u ( x , t , g ) .
2
0, we i n t r o d u c e t h e v a r i a b l e s 5
The new v a r i a b l e s a l l o w us t o w r i t e (KdV)
2
x
-
c t and
as t h e ordinary
d i f f e r e n t i a l equation 2
e u/'/ = ( c u ) '
-
2 (u / 2 ) f ,
' =
d/d5,
and a n i n t e g r a t i o n reduces t h i s t h i r d - o r d e r e q u a t i o n t o t h e second-order one
Here Urn i s t h e c o n s t a n t l i m i t i n g s t a t e of l i m u ( ~ ) ( ~ , E= ) 0, f o r k
IF1 (KdV) -4"
equation,
variables
N
5
2
1.
u6,e)
t h a t is,
l i m u(f,e)
151 - b r n
=
Um and
(Note t h a t i f w e had s t a r t e d w i t h t h e unscaled
xt + uux + uxxx = 0,
z g(x-ct),
u,
w e would a l s o o b t a i n (1.1) by d e f i n i n g t h e
= T(x,t,g),
and proceeding a s above.)
Thus t h e s t u d y of permanent wave s o l u t i o n s of t h e Korteweg-deVries duces t o t h e s t u d y of t h e asymptotic b e h a v i o r ( a s s i n g u l a r l y p e r t u r b e d boundary v a l u e problem
g
-+
+
equation re-
0 ) of s o l u t i o n s of t h e
106
F.A. Howes
where
'p
I f cpu(Um) > 0 t h e n Urn is ob-
i s a smooth f u n c t i o n s a t i s f y i n g cp(U m ) = 0.
-
v i o u s l y a maximum p o i n t of t h e p o t e n t i a l energy f u n c t i o n a l @ ( u ) = mathematical t h e o r y of (1.2)
J:: based on phase-plane
(O'Malley (1976)),
cp(s)ds.
t e l l s us t h a t i f t h e r e i s a f i n i t e U-value u* such t h a t @(u+c)= ip(U ) m
’p(u*)
# 0, t h e n t h e problem (1.2) h a s a s o l u t i o n u
= u ( ~ , E as )
E -t
The
m
O+
except t h a t
analysis, =
0 and
satisfying
(1.3) lim
u(O,E)
+
=
u*.
E - t o
This s o l u t i o n c l e a r l y r e p r e s e n t s a s o l i t a r y wave of amplitude )u;k-U,I; wave of e l e v a t i o n ( d e p r e s s i o n ) i f u*
> Urn (u" < U,).
i t is a
We n o t e i n p a s s i n g t h a t i n
addition t o the solution satisfying (1.3), there are solutions u =
of (1.2)
w i t h "spikes" a t r e g u l a r l y spaced p o i n t s i n any f i n i t e i n t e r v a l (-L,L)
C (-m,m),
that is, c
except t h a t
,
n=2,3,...
lim
-+ o+
G(5,s)
l i m +;(Tk,c) s + o
k=l,Z,...,n-l
=
u,,
= u",
5
for
f o r [k
(O'Malley (1976)).
=
i n (-L,L),
L(Zk/n-l),
These f u n c t i o n s correspond t o p e r i -
odic c n o i d a l wave s o l u t i o n s o f e v o l u t i o n e q u a t i o n s l i k e (KdV),
a s discussed, f o r
example, by Karpman (1975). 2 Returning t o t h e s p e c i f i c problem (1.1) we s e e t h a t f o r ~ ( u )= c(u-Urn)- i ( u 2 -Urn),
- u;
cp(Um) = 0 and y U ( u ) = c
c o n s e q u e n t l y , 'pu(U,)
>
0 only i f Urn
<
c.
The poten-
- U ~s)o] ,@(u*) = 0, t i a l energy f u n c t i o n a l @ ( u ) i s e q u a l t o ~ ( U - U ~ ) ~ { U - U ~ - ~ ( C and f o r u* = 3c (1.3),
-
2U,.
This i m p l i e s t h e e x i s t e n c e of a s o l u t i o n of (1.1) s a t i s f y i n g
t h a t i s , a s o l i t a r y wave s o l u t i o n of t h e (KdV)
e l e v a t i o n of h e i g h t u*
-
e q u a t i o n i n t h e form of an
Urn = 3(c-Um), moving t o t h e r i g h t w i t h speed c > U r n .
It
i s i n s t r u c t i v e t o r e w r i t e t h i s amplitude-speed r e l a t i o n a s c = Urn thus, t h e l a r g e r t h e amplitude a d d i t i o n , (1.4) since
c
(1.4)
of t h e wave, t h e g r e a t e r i t s phase speed.
In
i m p l i e s t h a t t h e s o l i t a r y wave of amplitude -3Um i s s t a t i o n a r y ,
i s t h e n zero.
s t a t e (KdV)
A
+ A13 ;
C l e a r l y t h i s s o l i t a r y wave i s a s o l u t i o n of t h e s t e a d y -
equation u u x
+ E 2uxxx
= 0, which e x i s t s provided Urn
<
0.
I f Urn
2
t h e n t h e only time-independent s o l u t i o n of (KdV) s a t i s f y i n g t h e boundary condit i o n s i s t h e uniform s t a t e u
=
Urn.
0
107
Generalised Korteweg-De Vries Equations
A l l of t h e s e r e s u l t s a r e w e l l known; s e e , f o r example, Zabusky (1967) o r Karpman (1975) f o r t h e more t r a d i t i o n a l (but e q u i v a l e n t ) phase-plane a n a l y s i s .
The
novelty of our approach, we t h i n k , l i e s i n t h e e a s e with which r e s u l t s such a s (1.4) a r e obtained from t h e equation (1.1). We c l o s e t h i s s e c t i o n w i t h a d i s c u s s i o n of t h e modified Korteweg-deVries equation i n t h e form
u t
The same v a r i a b l e s
7
-2+ u u
X
f e
2u
XXX
=
o.
(mKd V)
and u = u ( 5 , c ) allow us t o replace (mKdV) with t h e second-
o r d e r equation u" = C(U-Um)
-
4 ( u 3 - Urn)
cp(u),
(1.5)
2 t o which we apply t h e theory f o r (1.2)0 C l e a r l y 'p(Urn) = 0 and s i n c e cp (u) = c-u , Ll 2 we must r e q u i r e Um < c , i n order t h a t 'p (Urn) > 0. The corresponding energy func-
tional i s
1 2 2 @ ( u ) = E ( u - U ~ ) { (u-Um)
2 + 4Um(u-Um)+ 6(Um-c)3
.
(1.6)
Since t h e expression i n braces i s a q u a d r a t i c polynomial with p o s i t i v e discrimina n t , we s e e t h a t (mKdV) has s o l i t a r y wave s o l u t i o n s of e l e v a t i o n and depression 2 The amplitude A of a s o l i t a r y wave of moving t o t h e r i g h t with speed c > U r n . 2 e l e v a t i o n (compressive wave) i s u* Urn, where @(u*) = 0, and so Ac 4UrnAc 2 6(Um-c) = 0 from (1.6). Consequently t h e amplitude-speed r e l a t i o n f o r t h e com-
-
+
p r e s s i v e s o l i t a r y wave i s c = U,
2
1 + ;Ac
+
.
(Ac +&Urn)
On t h e o t h e r hand, t h e amplitude AR of a s o l i t a r y wave of depression ( r a r e f a c t i v e wave) i s Urn
-
u*, where @(u*) = 0, and s o now (1.6) implies t h a t A:
6(U?-c) = 0, t h a t i s ,
2 c = Urn
I n g e n e r a l , t h e amplitudes Ac and A
2
1 + ;AR(% - 4Um)
R
-
+
4UmAR
.
a r e d i f f e r e n t , i f Urn f 0.
W e n o t e a l s o t h a t i f c = Urn then cp (U ) = 0, and t h e theory f o r problem (1.2) u m s t a t e d above does not apply. However, i f Urn < 0, then 'p (U,) > 0, and a modifiuu c a t i o n of t h e theory i n O'Malley (1976) ( s e e a l s o Howes (1978))allows us t o study 1 3 t h e problem (1.5) by a g a i n examining t h e f u n c t i o n a l @ ( u ) , equal t o =(u-Um) 2 { (u-U,) 4Um] f o r c = Urn. Thus t h e modified Korteweg-deVries equation has a
+
s o l i t a r y wave s o l u t i o n of e l e v a t i o n A = -4Um moving t o t h e r i g h t with speed 2 c = U2 = A 116. There is no q u a l i t a t i v e d i f f e r e n c e between t h i s wave and a wave m
f o r which 'pu(Um) > 0; however, t h e l a t t e r type of wave does converge t o Urn ex-
ponentially as
6
+
O',
while t h e former converges only a l g e b r a i c a l l y .
These r e s u l t s f o r (mKdV) a r e a l s o c l a s s i c a l (Zabusky (1967)); nonetheless, t h e i r
F.A. Howes
108
formulation i n terms of t h e s i n g u l a r l y perturbed problem (1.5) i s h e l p f u l , a s i n 2 t h e c a s e c = Urn. F i n a l l y l e t us mention t h a t t h e mathematical theory f o r the general problem (1.2) a l s o t e l l s u s when an e v o l u t i o n equation does n o t have solit a r y wave s o l u t i o n s . For example, consider another modified (KdV) equation 2- -2u u + e uxxx = 0. A s o l i t a r y wave u = u ( 5 , e ) = i ( x , t , ~ )moving t o the
-u
t
r i g h t w i t h speed c > 0 a g a i n s t a zero background (Urn = 0 ) must be a n o n t r i v i a l 2 3 s o l u t i o n of t h e equation 0 u" = cu + u / 3 c cp(u). However, u = 0 i s t h e only r e a l "U
s o l u t i o n of t h e equation
JO
cp(s)ds
=
0, and so ( r e a l ) s o l i t a r y waves do not e x i s t .
SOLITAHY WAVE SOLUTIONS OF SOME VECTOR ANALOGS OF THE KDV EQUATION
Several problems of c u r r e n t i n t e r e s t i n geophysics and p l a n e t a r y physics involve t h e e x i s t e n c e of permanent wave s o l u t i o n s of coupled d i s p e r s i v e equations l i k e
which govern the dynamics of s t r a t i f i e d f l u i d s (Benney (1966), Redekopp and Weidman (1978)).
Here
and
(y
p a r e assumed t o b e p o s i t i v e constants.
In order
t o f i n d s o l u t i o n s o f t h e system (2.1) which r e p r e s e n t permanent waves moving t o t h e r i g h t with speed c
x
5
-
2
0, we proceed a s i n t h e l a s t s e c t i o n and d e f i n e t h e v a r i -
-
-
These u l ( x , t , e ) and u2 = u 2 ( 5 , e ) E u 2 ( x , t , e ) . new v a r i a b l e s enable us t o r e p l a c e (2.1) with t h e system o f ordinary d i f f e r e n t i a l ables
c t , u1 = u1(5,e)
equations 2
-
IN
E u1 = (cu,)'
2
e u; = (cu,)'
for
' =
d/d{,
2
,
2
,
(U1/2+au2Ul)' (u2i2+pu1u2)'
and an i n t e g r a t i o n reduces t h e s e t h i r d - o r d e r equations t o t h e second-
order system e 2 u"1 = cul e 2u" = cu2
-
(u,.2 /2+W2U1) 2
(U2/2+pU1U2)
I
.
(2.2)
The constants of i n t e g r a t i o n a r e taken t o be zero, and s o we seek s o l u t i o n s of (2.2)
whose constant l i m i t i n g s t a t e i s (O,O),
(0,O)
for k
2
that is,
l i m (up),uF))(5,e)
151+m
0.
=
The problem (2.2) i s a s p e c i a l c a s e of t h e more g e n e r a l problem
where
u
N
=
2=
(u,,u,)
((p
,(p ) i s a smooth vector-valued f u n c t i o n of t h e v e c t o r - v a r i a b l e 1 2 I n a n attempt t o mimic t h e s c a l a r theory f o r satisfying =
z(2) 2.
109
Generalised KortewegDe Vries Equations problem (1.2), we now ask (following the theory of O'Donnell (1983)) t h a t , i n a d d i t i o n , t h e zero equilibrium s t a t e s a t i s f y t h e equation cp(u) = -
Y
2 in
t h e compon-
entwise sense t h a t CP 1( 0 , ~ ~ = )0
v2(u1,0) = 0,
and
u2 i n t h e region of i n t e r e s t .
f o r a l l ul,
(2.4)
The analog of t h e c o n d i t i o n t h a t
'P (0) > 0 i s then t h e condition t h a t
( 0 , ~ ~ > )0
,U1
f o r a l l such u1,u2.
cp
and
2,u2
(ul,O)
>
(2.5)
0,
These two f a i r l y r e s t r i c t i v e c o n d i t i o n s e s s e n t i a l l y allow
US
t o decouple t h e system (2.3) and apply t h e s c a l a r theory f o r problem ( 1 . 2 ) componentwise.
In o r d e r t o determine t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of
( 2 . 3 ) and t h e i r amplitudes, we d e f i n e t h e two p o t e n t i a l energy f u n c t i o n a l s @l(u1,u2)
=
-r
u1 (P1(s.u 2 Ids "0
and
u2
-Jo
m2(u1,u2) It follows from (2.4) and (2.5)
that u = N
(P2(U1,S)ds-
2 is
a maximum point of @
on t h e n a t u r e of t h i s maximum, w e d i s t i n g u i s h two c a s e s . CASE I
i'
and depending
S o l i t a r y Waves i n Each Component Centered a t D i f f e r e n t Points.
In t h i s
c a s e t h e s o l i t a r y waves i n each component evolve independently of each o t h e r (asymptotically, a s E @l(uJ,O) = @,(O,O)
-t
+
0 ).
I f t h e r e a r e f i n i t e values up, u f ~such t h a t 2 = 0 and cp 1(u-X,O) 1 f 0 , cp2(o,u;) f 0,
= 0, @ 2 (O,u*) 2 = @2( 0 , O )
then t h e theory (Howes (1983)) t e l l s u s t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,e) a s
6
+ O+ s a t i s f y i n g
except t h a t
t1 # t2’
a t points
ti
CASE I1
S o l i t a r y Waves i n Each Component Centered a t t h e Same Point.
in
(-,a),
In t h i s
case we must t a k e i n t o account t h e coupling between components, s i n c e t h e waves " i n t e r a c t " through a term l i k e u1u2 i n equation (2.2).
uy, uy such (pl(u);*,uy)
#
I f there a r e f i n i t e values
t h a t @ (u**,u**) = @ ( 0 , O ) = 0, @ 2 ( u T * , u y ) = @,(O,O) = 0 and 1 1 2 1 0 , (p2(u;*,uy) # 0 , then t h e theory (Howes (1983),(1984)) t e l l s us
t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,g) a s lim
s+o+ except t h a t
u.(<,~= ) 0,
=
--
<5 <
my
g
+
0'
satisfying
110
F.A. Howes
We note t h a t , i n g e n e r a l , t h e wave amplitudes u* and i t h e coupling of t h e c m p o n e n t s i n t h e c a s e of u’?.
uy
a r e d i f f e r e n t , owing t o
Let u s i l l u s t r a t e t h e s e r e s u l t s by looking f o r s o l i t a r y wave s o l u t i o n s of t h e s y s tem (2.1)
moving t o t h e r i g h t w i t h speed c = 1, t h a t i s , we c o n s i d e r t h e problem
I n any s u b i n t e r v a l of (-L,L) where respectively u l ) i s a s y m p t o t i c a l l y z e r o , t h e 2 ul-equation ( r e s p e c t i v e l y u2-equation) reduces t o t h e uncoupled e q u a t i o n E w“ = 2 w - w / 2 , which i s a s p e c i a l c a s e of e q u a t i o n (1.1). Consequently, t h e r e a r e s o l u t i o n s of ( 2 . 6 ) a s
6
+ ’0
r e p r e s e n t i n g two-component s o l i t a r y waves of ampli-
t u d e 3, moving t o t h e r i g h t w i t h u n i t speed, and l o c a l i z e d a t d i f f e r e n t <-values.
I n o t h e r words, we have s o l u t i o n s such t h a t
and lim
t1 # 5 2 0
for locations
+
u2(t2,c) = 3,
e + o On t h e o t h e r hand, t h e r e a l s o e x i s t s o l i t a r y wave s o l u -
t i o n s , moving t o t h e r i g h t w i t h u n i t speed, which a r e l o c a l i z e d a t t h e same {-value,
say
5
= 0, w i t h t h e amplitude of t h e f i r s t component equal t o 3(301-1) t
(9018-1) and t h a t of t h e second component equal t o 3 ( 3 p - l ) / ( 9 ~ p - l ) , provided @
--6
f 1/9. ‘ 4 3 -ul
To s e e t h i s , n o t e t h a t t h e p a i r of a l g e b r a i c e q u a t i o n s @1(u1,u2) =
- 3m21
= 0,
rn2(Ul,U2) =
-6‘-u;[3
-
-u2 3pu ] = 0 h a s t h e n o n t r i v i a l 1
5,
5
p f and Olp # s o l u t i o n u** = 3(301-1)/(9&-1), u’;” = 3 ( 3 @ - 1 ) / ( 9 & - 1 ) , i f ci # 1 I n p a r t i c u l a r , f o r t h e symmetric c a s e cy = p (f t h e r e i s a s o l u t i o n of ( 2 . 6 )
5)
s q t i s f y i n g f o r i =1 , 2
and
Thus t h e coupling term m1u2 acts t o d e c r e a s e t h e amplitude of t h e wave i n each component from 3 t o 3/(3&1).
1 T.
111
Generalised Korteweg-De Vries Equations CONCLUDING REMARKS
The reasoning employed i n t h e l a s t s e c t i o n a p p l i e s m u t a t i s mutandis t o t h e g e n e r a l p e r t u r b e d second-order boundary v a l u e problem (3.1)
where (3.1)
u,
N
v,
N
and
U
N
a r e N-vectors, a s w e l l a s t o nonautonomous v e r s i o n s of
(Howes (1983), O'Donnell (1983)).
Korteweg-deVries
Thus we a r e a b l e t o t r e a t a g e n e r a l i z e d
system l i k e
and even a system w i t h s o u r c e o r r e a c t i o n terms such a s
ACKNaWLEDGMENTS The a u t h o r thanks U.C.
Davis f o r p r o v i d i n g a t r a v e l g r a n t which enabled h i s p a r t i -
c i p a t i o n i n t h i s t i m e l y and i n t e r e s t i n g meeting.
He a l s o thanks I d a Mae Zalac f o r
doing a s p l e n d i d j o b of typing. REFERENCES Benney, D. J., Long n o n - l i n e a r waves i n f l u i d f l o w s , J. Math. and Phys. 45 (1966) 52-62. Howes, F. A., B o u n d a r y - i n t e r i o r l a y e r i n t e r a c t i o n s i n n o n l i n e a r s i n g u l a r p e r t u r b a t i o n t h e o r y , Memoirs Amer. Math. SOC. no. 203 (1978). Howes, F. A , , Nonlinear d i s p e r s i v e systems: t h e o r y and examples, S t u d i e s i n Appl. Math. 69 (1983) 75-97. Howes, F. A., Asymptotic s t r u c t u r e s i n multicomponent f i e l d s , SIAM J. Math. Anal. (1984). Karpman, V. 1975).
I., Non-Linear Waves i n D i s p e r s i v e Media (Pergamon P r e s s , Oxford,
O'Donnell, M. A., Boundary and c o r n e r l a y e r b e h a v i o r i n s i n g u l a r l y p e r t u r b e d s e m i l i n e a r systems of boundary v a l u e problems, SIAM J. Math. Anal. (1983). O'Malley, R. E., Jr., Phase-plane s o l u t i o n s t o some s i n g u l a r p e r t u r b a t i o n problems, J. Math. Anal. Appl. 5 4 (1976) 449-466. Redekopp, L. G. and Weidman, P. D., S o l i t a r y Rossby waves i n z o n a l s h e a r flows and t h e i r i n t e r a c t i o n s , J. Atmospheric S c i . 35 (1978) 790-804. Zabusky, N. J., A s y n e r g e t i c approach t o problems of n o n l i n e a r d i s p e r s i v e wave p r o p a g a t i o n , i n : Ames, W. F. (ed.), Nonlinear P a r t i a l D i f f e r e n t i a l Equat i o n s (Academic P r e s s , New York, 1967).
105
SOLITARY WAVE SOLUTIONS OF GENERALIZED KORTEWEG-DE VKIES EQUATIONS F. A. Howes
Department of Mathematics U n i v e r s i t y of C a l i f o r n i a a t Davis Davis, CA 95616 U.S.A.
We b e g i n w i t h a d i s c u s s i o n of t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of t h e s c a l a r Korteweg-deVries (KdV) and modified Korteweg-deVries e q u a t i o n s , i n c l u d i n g t h e d e r i v a t i o n of formulas f o r wave amplitude and speed when t h e waves move a g a i n s t a nonzero "background". This i s followed by a c o n s i d e r a t i o n of r e l a t e d q u e s t i o n s f o r system a n a l o g s of (KdV). Our approach i n v o l v e s t h e c o n s i s t e n t u s e of t h e asymptotic t h e o r y of second-order s i n g u l a r l y p e r t u r b e d boundary v a l u e problems, and so i t o f f e r s a convenient a l t e r n a t i v e t o more t r a d i t i o n a l a n a l y t i c a l and numerical methods. SOLITARY WAVE SOLUTIONS OF THE KDV AND MODlFIED KDV EQUATIONS e q u a t i o n i n t h e form
Let us c o n s i d e r f i r s t t h e Korteweg-deVries
- + -2uux + E u = xxx Ut
0,
e
i s a s m a l l p o s i t i v e parameter and s u b s c r i p t s d e n o t e p a r t i a l d i f f e r e n -
tiation.
I n o r d e r t o s t u d y s o l u t i o n s of (KdV) which r e p r e s e n t permanent waves
where
moving t o t h e r i g h t w i t h speed c
-
u = u ( ~ , e )= u ( x , t , g ) .
2
0, we i n t r o d u c e t h e v a r i a b l e s 5
The new v a r i a b l e s a l l o w us t o w r i t e (KdV)
2
x
-
c t and
as t h e ordinary
d i f f e r e n t i a l equation 2
e u/'/ = ( c u ) '
-
2 (u / 2 ) f ,
' =
d/d5,
and a n i n t e g r a t i o n reduces t h i s t h i r d - o r d e r e q u a t i o n t o t h e second-order one
Here Urn i s t h e c o n s t a n t l i m i t i n g s t a t e of l i m u ( ~ ) ( ~ , E= ) 0, f o r k
IF1 (KdV) -4"
equation,
variables
N
5
2
1.
u6,e)
t h a t is,
l i m u(f,e)
151 - b r n
=
Um and
(Note t h a t i f w e had s t a r t e d w i t h t h e unscaled
xt + uux + uxxx = 0,
z g(x-ct),
u,
w e would a l s o o b t a i n (1.1) by d e f i n i n g t h e
= T(x,t,g),
and proceeding a s above.)
Thus t h e s t u d y of permanent wave s o l u t i o n s of t h e Korteweg-deVries duces t o t h e s t u d y of t h e asymptotic b e h a v i o r ( a s s i n g u l a r l y p e r t u r b e d boundary v a l u e problem
g
-+
+
equation re-
0 ) of s o l u t i o n s of t h e
106
F.A. Howes
where
'p
I f cpu(Um) > 0 t h e n Urn is ob-
i s a smooth f u n c t i o n s a t i s f y i n g cp(U m ) = 0.
-
v i o u s l y a maximum p o i n t of t h e p o t e n t i a l energy f u n c t i o n a l @ ( u ) = mathematical t h e o r y of (1.2)
J:: based on phase-plane
(O'Malley (1976)),
cp(s)ds.
t e l l s us t h a t i f t h e r e i s a f i n i t e U-value u* such t h a t @(u+c)= ip(U ) m
’p(u*)
# 0, t h e n t h e problem (1.2) h a s a s o l u t i o n u
= u ( ~ , E as )
E -t
The
m
O+
except t h a t
analysis, =
0 and
satisfying
(1.3) lim
u(O,E)
+
=
u*.
E - t o
This s o l u t i o n c l e a r l y r e p r e s e n t s a s o l i t a r y wave of amplitude )u;k-U,I; wave of e l e v a t i o n ( d e p r e s s i o n ) i f u*
> Urn (u" < U,).
i t is a
We n o t e i n p a s s i n g t h a t i n
addition t o the solution satisfying (1.3), there are solutions u =
of (1.2)
w i t h "spikes" a t r e g u l a r l y spaced p o i n t s i n any f i n i t e i n t e r v a l (-L,L)
C (-m,m),
that is, c
except t h a t
,
n=2,3,...
lim
-+ o+
G(5,s)
l i m +;(Tk,c) s + o
k=l,Z,...,n-l
=
u,,
= u",
5
for
f o r [k
(O'Malley (1976)).
=
i n (-L,L),
L(Zk/n-l),
These f u n c t i o n s correspond t o p e r i -
odic c n o i d a l wave s o l u t i o n s o f e v o l u t i o n e q u a t i o n s l i k e (KdV),
a s discussed, f o r
example, by Karpman (1975). 2 Returning t o t h e s p e c i f i c problem (1.1) we s e e t h a t f o r ~ ( u )= c(u-Urn)- i ( u 2 -Urn),
- u;
cp(Um) = 0 and y U ( u ) = c
c o n s e q u e n t l y , 'pu(U,)
>
0 only i f Urn
<
c.
The poten-
- U ~s)o] ,@(u*) = 0, t i a l energy f u n c t i o n a l @ ( u ) i s e q u a l t o ~ ( U - U ~ ) ~ { U - U ~ - ~ ( C and f o r u* = 3c (1.3),
-
2U,.
This i m p l i e s t h e e x i s t e n c e of a s o l u t i o n of (1.1) s a t i s f y i n g
t h a t i s , a s o l i t a r y wave s o l u t i o n of t h e (KdV)
e l e v a t i o n of h e i g h t u*
-
e q u a t i o n i n t h e form of an
Urn = 3(c-Um), moving t o t h e r i g h t w i t h speed c > U r n .
It
i s i n s t r u c t i v e t o r e w r i t e t h i s amplitude-speed r e l a t i o n a s c = Urn thus, t h e l a r g e r t h e amplitude a d d i t i o n , (1.4) since
c
(1.4)
of t h e wave, t h e g r e a t e r i t s phase speed.
In
i m p l i e s t h a t t h e s o l i t a r y wave of amplitude -3Um i s s t a t i o n a r y ,
i s t h e n zero.
s t a t e (KdV)
A
+ A13 ;
C l e a r l y t h i s s o l i t a r y wave i s a s o l u t i o n of t h e s t e a d y -
equation u u x
+ E 2uxxx
= 0, which e x i s t s provided Urn
<
0.
I f Urn
2
t h e n t h e only time-independent s o l u t i o n of (KdV) s a t i s f y i n g t h e boundary condit i o n s i s t h e uniform s t a t e u
=
Urn.
0
107
Generalised Korteweg-De Vries Equations
A l l of t h e s e r e s u l t s a r e w e l l known; s e e , f o r example, Zabusky (1967) o r Karpman (1975) f o r t h e more t r a d i t i o n a l (but e q u i v a l e n t ) phase-plane a n a l y s i s .
The
novelty of our approach, we t h i n k , l i e s i n t h e e a s e with which r e s u l t s such a s (1.4) a r e obtained from t h e equation (1.1). We c l o s e t h i s s e c t i o n w i t h a d i s c u s s i o n of t h e modified Korteweg-deVries equation i n t h e form
u t
The same v a r i a b l e s
7
-2+ u u
X
f e
2u
XXX
=
o.
(mKd V)
and u = u ( 5 , c ) allow us t o replace (mKdV) with t h e second-
o r d e r equation u" = C(U-Um)
-
4 ( u 3 - Urn)
cp(u),
(1.5)
2 t o which we apply t h e theory f o r (1.2)0 C l e a r l y 'p(Urn) = 0 and s i n c e cp (u) = c-u , Ll 2 we must r e q u i r e Um < c , i n order t h a t 'p (Urn) > 0. The corresponding energy func-
tional i s
1 2 2 @ ( u ) = E ( u - U ~ ) { (u-Um)
2 + 4Um(u-Um)+ 6(Um-c)3
.
(1.6)
Since t h e expression i n braces i s a q u a d r a t i c polynomial with p o s i t i v e discrimina n t , we s e e t h a t (mKdV) has s o l i t a r y wave s o l u t i o n s of e l e v a t i o n and depression 2 The amplitude A of a s o l i t a r y wave of moving t o t h e r i g h t with speed c > U r n . 2 e l e v a t i o n (compressive wave) i s u* Urn, where @(u*) = 0, and so Ac 4UrnAc 2 6(Um-c) = 0 from (1.6). Consequently t h e amplitude-speed r e l a t i o n f o r t h e com-
-
+
p r e s s i v e s o l i t a r y wave i s c = U,
2
1 + ;Ac
+
.
(Ac +&Urn)
On t h e o t h e r hand, t h e amplitude AR of a s o l i t a r y wave of depression ( r a r e f a c t i v e wave) i s Urn
-
u*, where @(u*) = 0, and s o now (1.6) implies t h a t A:
6(U?-c) = 0, t h a t i s ,
2 c = Urn
I n g e n e r a l , t h e amplitudes Ac and A
2
1 + ;AR(% - 4Um)
R
-
+
4UmAR
.
a r e d i f f e r e n t , i f Urn f 0.
W e n o t e a l s o t h a t i f c = Urn then cp (U ) = 0, and t h e theory f o r problem (1.2) u m s t a t e d above does not apply. However, i f Urn < 0, then 'p (U,) > 0, and a modifiuu c a t i o n of t h e theory i n O'Malley (1976) ( s e e a l s o Howes (1978))allows us t o study 1 3 t h e problem (1.5) by a g a i n examining t h e f u n c t i o n a l @ ( u ) , equal t o =(u-Um) 2 { (u-U,) 4Um] f o r c = Urn. Thus t h e modified Korteweg-deVries equation has a
+
s o l i t a r y wave s o l u t i o n of e l e v a t i o n A = -4Um moving t o t h e r i g h t with speed 2 c = U2 = A 116. There is no q u a l i t a t i v e d i f f e r e n c e between t h i s wave and a wave m
f o r which 'pu(Um) > 0; however, t h e l a t t e r type of wave does converge t o Urn ex-
ponentially as
6
+
O',
while t h e former converges only a l g e b r a i c a l l y .
These r e s u l t s f o r (mKdV) a r e a l s o c l a s s i c a l (Zabusky (1967)); nonetheless, t h e i r
F.A. Howes
108
formulation i n terms of t h e s i n g u l a r l y perturbed problem (1.5) i s h e l p f u l , a s i n 2 t h e c a s e c = Urn. F i n a l l y l e t us mention t h a t t h e mathematical theory f o r the general problem (1.2) a l s o t e l l s u s when an e v o l u t i o n equation does n o t have solit a r y wave s o l u t i o n s . For example, consider another modified (KdV) equation 2- -2u u + e uxxx = 0. A s o l i t a r y wave u = u ( 5 , e ) = i ( x , t , ~ )moving t o the
-u
t
r i g h t w i t h speed c > 0 a g a i n s t a zero background (Urn = 0 ) must be a n o n t r i v i a l 2 3 s o l u t i o n of t h e equation 0 u" = cu + u / 3 c cp(u). However, u = 0 i s t h e only r e a l "U
s o l u t i o n of t h e equation
JO
cp(s)ds
=
0, and so ( r e a l ) s o l i t a r y waves do not e x i s t .
SOLITAHY WAVE SOLUTIONS OF SOME VECTOR ANALOGS OF THE KDV EQUATION
Several problems of c u r r e n t i n t e r e s t i n geophysics and p l a n e t a r y physics involve t h e e x i s t e n c e of permanent wave s o l u t i o n s of coupled d i s p e r s i v e equations l i k e
which govern the dynamics of s t r a t i f i e d f l u i d s (Benney (1966), Redekopp and Weidman (1978)).
Here
and
(y
p a r e assumed t o b e p o s i t i v e constants.
In order
t o f i n d s o l u t i o n s o f t h e system (2.1) which r e p r e s e n t permanent waves moving t o t h e r i g h t with speed c
x
5
-
2
0, we proceed a s i n t h e l a s t s e c t i o n and d e f i n e t h e v a r i -
-
-
These u l ( x , t , e ) and u2 = u 2 ( 5 , e ) E u 2 ( x , t , e ) . new v a r i a b l e s enable us t o r e p l a c e (2.1) with t h e system o f ordinary d i f f e r e n t i a l ables
c t , u1 = u1(5,e)
equations 2
-
IN
E u1 = (cu,)'
2
e u; = (cu,)'
for
' =
d/d{,
2
,
2
,
(U1/2+au2Ul)' (u2i2+pu1u2)'
and an i n t e g r a t i o n reduces t h e s e t h i r d - o r d e r equations t o t h e second-
order system e 2 u"1 = cul e 2u" = cu2
-
(u,.2 /2+W2U1) 2
(U2/2+pU1U2)
I
.
(2.2)
The constants of i n t e g r a t i o n a r e taken t o be zero, and s o we seek s o l u t i o n s of (2.2)
whose constant l i m i t i n g s t a t e i s (O,O),
(0,O)
for k
2
that is,
l i m (up),uF))(5,e)
151+m
0.
=
The problem (2.2) i s a s p e c i a l c a s e of t h e more g e n e r a l problem
where
u
N
=
2=
(u,,u,)
((p
,(p ) i s a smooth vector-valued f u n c t i o n of t h e v e c t o r - v a r i a b l e 1 2 I n a n attempt t o mimic t h e s c a l a r theory f o r satisfying =
z(2) 2.
109
Generalised KortewegDe Vries Equations problem (1.2), we now ask (following the theory of O'Donnell (1983)) t h a t , i n a d d i t i o n , t h e zero equilibrium s t a t e s a t i s f y t h e equation cp(u) = -
Y
2 in
t h e compon-
entwise sense t h a t CP 1( 0 , ~ ~ = )0
v2(u1,0) = 0,
and
u2 i n t h e region of i n t e r e s t .
f o r a l l ul,
(2.4)
The analog of t h e c o n d i t i o n t h a t
'P (0) > 0 i s then t h e condition t h a t
( 0 , ~ ~ > )0
,U1
f o r a l l such u1,u2.
cp
and
2,u2
(ul,O)
>
(2.5)
0,
These two f a i r l y r e s t r i c t i v e c o n d i t i o n s e s s e n t i a l l y allow
US
t o decouple t h e system (2.3) and apply t h e s c a l a r theory f o r problem ( 1 . 2 ) componentwise.
In o r d e r t o determine t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of
( 2 . 3 ) and t h e i r amplitudes, we d e f i n e t h e two p o t e n t i a l energy f u n c t i o n a l s @l(u1,u2)
=
-r
u1 (P1(s.u 2 Ids "0
and
u2
-Jo
m2(u1,u2) It follows from (2.4) and (2.5)
that u = N
(P2(U1,S)ds-
2 is
a maximum point of @
on t h e n a t u r e of t h i s maximum, w e d i s t i n g u i s h two c a s e s . CASE I
i'
and depending
S o l i t a r y Waves i n Each Component Centered a t D i f f e r e n t Points.
In t h i s
c a s e t h e s o l i t a r y waves i n each component evolve independently of each o t h e r (asymptotically, a s E @l(uJ,O) = @,(O,O)
-t
+
0 ).
I f t h e r e a r e f i n i t e values up, u f ~such t h a t 2 = 0 and cp 1(u-X,O) 1 f 0 , cp2(o,u;) f 0,
= 0, @ 2 (O,u*) 2 = @2( 0 , O )
then t h e theory (Howes (1983)) t e l l s u s t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,e) a s
6
+ O+ s a t i s f y i n g
except t h a t
t1 # t2’
a t points
ti
CASE I1
S o l i t a r y Waves i n Each Component Centered a t t h e Same Point.
in
(-,a),
In t h i s
case we must t a k e i n t o account t h e coupling between components, s i n c e t h e waves " i n t e r a c t " through a term l i k e u1u2 i n equation (2.2).
uy, uy such (pl(u);*,uy)
#
I f there a r e f i n i t e values
t h a t @ (u**,u**) = @ ( 0 , O ) = 0, @ 2 ( u T * , u y ) = @,(O,O) = 0 and 1 1 2 1 0 , (p2(u;*,uy) # 0 , then t h e theory (Howes (1983),(1984)) t e l l s us
t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,g) a s lim
s+o+ except t h a t
u.(<,~= ) 0,
=
--
<5 <
my
g
+
0'
satisfying
110
F.A. Howes
We note t h a t , i n g e n e r a l , t h e wave amplitudes u* and i t h e coupling of t h e c m p o n e n t s i n t h e c a s e of u’?.
uy
a r e d i f f e r e n t , owing t o
Let u s i l l u s t r a t e t h e s e r e s u l t s by looking f o r s o l i t a r y wave s o l u t i o n s of t h e s y s tem (2.1)
moving t o t h e r i g h t w i t h speed c = 1, t h a t i s , we c o n s i d e r t h e problem
I n any s u b i n t e r v a l of (-L,L) where respectively u l ) i s a s y m p t o t i c a l l y z e r o , t h e 2 ul-equation ( r e s p e c t i v e l y u2-equation) reduces t o t h e uncoupled e q u a t i o n E w“ = 2 w - w / 2 , which i s a s p e c i a l c a s e of e q u a t i o n (1.1). Consequently, t h e r e a r e s o l u t i o n s of ( 2 . 6 ) a s
6
+ ’0
r e p r e s e n t i n g two-component s o l i t a r y waves of ampli-
t u d e 3, moving t o t h e r i g h t w i t h u n i t speed, and l o c a l i z e d a t d i f f e r e n t <-values.
I n o t h e r words, we have s o l u t i o n s such t h a t
and lim
t1 # 5 2 0
for locations
+
u2(t2,c) = 3,
e + o On t h e o t h e r hand, t h e r e a l s o e x i s t s o l i t a r y wave s o l u -
t i o n s , moving t o t h e r i g h t w i t h u n i t speed, which a r e l o c a l i z e d a t t h e same {-value,
say
5
= 0, w i t h t h e amplitude of t h e f i r s t component equal t o 3(301-1) t
(9018-1) and t h a t of t h e second component equal t o 3 ( 3 p - l ) / ( 9 ~ p - l ) , provided @
--6
f 1/9. ‘ 4 3 -ul
To s e e t h i s , n o t e t h a t t h e p a i r of a l g e b r a i c e q u a t i o n s @1(u1,u2) =
- 3m21
= 0,
rn2(Ul,U2) =
-6‘-u;[3
-
-u2 3pu ] = 0 h a s t h e n o n t r i v i a l 1
5,
5
p f and Olp # s o l u t i o n u** = 3(301-1)/(9&-1), u’;” = 3 ( 3 @ - 1 ) / ( 9 & - 1 ) , i f ci # 1 I n p a r t i c u l a r , f o r t h e symmetric c a s e cy = p (f t h e r e i s a s o l u t i o n of ( 2 . 6 )
5)
s q t i s f y i n g f o r i =1 , 2
and
Thus t h e coupling term m1u2 acts t o d e c r e a s e t h e amplitude of t h e wave i n each component from 3 t o 3/(3&1).
1 T.
111
Generalised Korteweg-De Vries Equations CONCLUDING REMARKS
The reasoning employed i n t h e l a s t s e c t i o n a p p l i e s m u t a t i s mutandis t o t h e g e n e r a l p e r t u r b e d second-order boundary v a l u e problem (3.1)
where (3.1)
u,
N
v,
N
and
U
N
a r e N-vectors, a s w e l l a s t o nonautonomous v e r s i o n s of
(Howes (1983), O'Donnell (1983)).
Korteweg-deVries
Thus we a r e a b l e t o t r e a t a g e n e r a l i z e d
system l i k e
and even a system w i t h s o u r c e o r r e a c t i o n terms such a s
ACKNaWLEDGMENTS The a u t h o r thanks U.C.
Davis f o r p r o v i d i n g a t r a v e l g r a n t which enabled h i s p a r t i -
c i p a t i o n i n t h i s t i m e l y and i n t e r e s t i n g meeting.
He a l s o thanks I d a Mae Zalac f o r
doing a s p l e n d i d j o b of typing. REFERENCES Benney, D. J., Long n o n - l i n e a r waves i n f l u i d f l o w s , J. Math. and Phys. 45 (1966) 52-62. Howes, F. A., B o u n d a r y - i n t e r i o r l a y e r i n t e r a c t i o n s i n n o n l i n e a r s i n g u l a r p e r t u r b a t i o n t h e o r y , Memoirs Amer. Math. SOC. no. 203 (1978). Howes, F. A , , Nonlinear d i s p e r s i v e systems: t h e o r y and examples, S t u d i e s i n Appl. Math. 69 (1983) 75-97. Howes, F. A., Asymptotic s t r u c t u r e s i n multicomponent f i e l d s , SIAM J. Math. Anal. (1984). Karpman, V. 1975).
I., Non-Linear Waves i n D i s p e r s i v e Media (Pergamon P r e s s , Oxford,
O'Donnell, M. A., Boundary and c o r n e r l a y e r b e h a v i o r i n s i n g u l a r l y p e r t u r b e d s e m i l i n e a r systems of boundary v a l u e problems, SIAM J. Math. Anal. (1983). O'Malley, R. E., Jr., Phase-plane s o l u t i o n s t o some s i n g u l a r p e r t u r b a t i o n problems, J. Math. Anal. Appl. 5 4 (1976) 449-466. Redekopp, L. G. and Weidman, P. D., S o l i t a r y Rossby waves i n z o n a l s h e a r flows and t h e i r i n t e r a c t i o n s , J. Atmospheric S c i . 35 (1978) 790-804. Zabusky, N. J., A s y n e r g e t i c approach t o problems of n o n l i n e a r d i s p e r s i v e wave p r o p a g a t i o n , i n : Ames, W. F. (ed.), Nonlinear P a r t i a l D i f f e r e n t i a l Equat i o n s (Academic P r e s s , New York, 1967).