PERGAMON
Applied Mathematics Letters
Applied Mathematics Letters 14 (2001) 93-98
www.elsevier.nl/locate/aml
P e a k o n S o l u t i o n s of t h e Shallow W a t e r E q u a t i o n M . S. A L B E R AND C . M I L L E R Department of Mathematics, University of Notre Dame Notre Dame, IN 46556, U.S.A. ©nd, edu
(Received November 1999; accepted December 1999)
Communicated by M. Levi A b s t r a c t - - A new parameterization of the Jacobi inversion problem is used along with the dynamics of the peaks to describe finite time interaction of peakon weak solutions of the shallow water equation. (~) 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - S o l i t o n s , Peakons, Billiards, Shallow water equation, Hamiltonian systems.
1. I N T R O D U C T I O N C a m a s s a and Holm [1] described classes of n-soliton peaked weak solutions, or "peakons", for an integrable (SW) equation Ut + 3UUx = Uxxt + 2U~,U:~x + UU~xx - 2t~Ux ,
(I.I)
arising in the context of shallow water theory. Of particular interest is their description of peakon dynamics in terms of a system of completely integrable Hamiltonian equations for the locations of the "peaks" of the solution, the points at which its spatial derivative changes sign. (Peakons have discontinuities in the x-derivative but both one-sided derivatives exist and differ only by a sign. This makes peakons different from cuspons considered earlier in the literature.) In other words, each peakon solution can be associated with a mechanical system of moving particles. Calogero [2] and Calogero and Francoise [3] further extended the class of mechanical systems of this type. For the K d V equation, the spectral parameter A appears linearly in the potential of the corresponding Schr6dinger equation: V = u - A in the context of the inverse scattering transform (IST) method (see [4]). In contrast, equation (1.1), as well as N - c o m p o n e n t systems in general, were shown to be connected to the energy dependent Schr5dinger operators with potentials with poles in the spectral parameter. Research partially supported by NSF Grant DMS 9626672 and NATO Grant CRG 950897. 0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved, PII: S0893-9659(00)00118-X
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94
M . S . ALBER AND C. MILLER
Alber et al. [5,6] showed that the presence of a pole in the potential is essential in a special limiting procedure that allows for the formation of "billiard solutions". By using algebraic-geometric methods, one finds that these billiard solutions are related to finite-dimensional integrable dynamical systems with reflections. This provides a shortcut to the study of quasiperiodic and solitonic billiard solutions of nonlinear PDEs. This method can be used for a number of equations including the shallow water equation (1.1), the Dym type equation, as well as N-component systems with poles and the equations in their hierarchies [7]. More information on algebraic-geometric methods for integrable systems can be found in [8] and on billiards in [9-11]. In this paper, we consider singular limits of quasiperiodic solutions when the spectral curve becomes singular and its arithmetic genus drops to zero. The solutions are then expressed in terms of purely exponential T-functions and they describe the finite time interaction of two solitary peakons of the shallow water equation (1.1). Namely, we invert the equations obtained by using a new parameterization. First a profile of the two-peakon solution is described by considering different parameterizations for the associated Jacobi inversion problem on three subintervals of the X-axis and by gluing these pieces of the profile together. The dynamics of such solutions is then described by combining these profiles with the dynamics of the peaks of the solution in the form developed earlier in Alber et al. [9,10]. This concludes a derivation in the context of the algebraic geometric approach of the n-peakon ansatz which was used in the initial papers [1,12] for obtaining Hamiltonian systems for peaks. More recently, n-peakon waves were studied in [13,14]. The problem of describing complex traveling wave and quasiperiodic solutions of the equation (1.1) can be reduced to solving finite-dimensional Hamiltonian systems on symmetric products of hyperelliptic curves. Namely, according to Alber et al. [5-7], such solutions can be represeI~ted in the case of two-phase quasiperiodic solutions in the following form (1.2)
U ( x , t) = #1 + #s - M ,
where M is a constant and the evolution of the variables #1 and #2 is given by the equations s k E Pi dpi ~=1 + ~
_
{
dr, dx,
k=l, k=2.
(1.3)
5
Here R ( # ) is a polynomial of degree 6 of the form R ( # ) = # H i = l ( # - mi). The constant from (1.2) takes the form M = 1/2 y~ mi. Notice that (1.3) describes quasiperiodic motion on tori of genus 2. In the limit ml --~ 0, the solution develops peaks. (For details, see [7].) Interaction of Two Peakons In the limit when ms --* m3 ---* al and m4 --+ m5 --* a2, we have two solitary peakons interacting with each other. For this two peakon case, we derive the general form of a profile for a fixed t (t = to, dt = 0) and then see how this profile changes with time knowing how the peaks evolve. Notice that the limit depends on the choice of the branches of the square roots present in (1.3) meaning choosing a particular sign lj in front of each root. The problem of finding the profile, after applying the above limits to (1.3) gives
11
d#l
+ Is
d#s
dX -- a2 - -
= as dY,
(1.4)
ll ]A1 (~tl -- a2) + ls #2 (#2 -- as) -- al --#1#2 = al dY,
(1.5)
/Zl (]A1 -- a l )
d#l
/z2 (122 -- a l )
d#2
/-tl~t2
dX
where Y is a new variable. This is a new parameterization of the Jacobi inversion problem (1.3) which makes the existence of three different branches of the solution obvious. In general, we
Shallow W a t e r Equation
95
c o n s i d e r t h r e e different cases: (/1 = 1,12 = 1), (/1 = 1,/2 = - 1 ) , a n d (/1 = - 1 , [ 2 = - 1 ) . In each case, we i n t e g r a t e a n d invert t h e integrals to c a l c u l a t e t h e s y m m e t r i c p o l y n o m i a l ([21 + P2). A f t e r s u b s t i t u t i n g t h e s e expressions into t h e t r a c e f o r m u l a (1.2) for t h e solution, this results in t h r e e different part s of t h e profile defined on different s u b i n t e r v a l s on t h e real line. T h e union of t h e s e s u b i n t e r v a l s gives t h e whole line. O n t h e last step, these t h r e e p a r t s are glued t o g e t h e r to ()btain a wave profile w i t h two peaks. T h e new p a r a m e t e r i z a t i o n d X = #a[22 d Y plays an i m p o r t a n t role in our a p p r o a c h . In w h a t %llows, each # i ( Y ) will be defined on t h e whole real Y line. However, t h e t r a n s f o r m a t i o n from Y t,nck to X is not s u b j e c t i v e so t h a t p i ( X ) is o n l y defined on a s e g m e n t of t h e real axis. T h i s is w h y different b r a n c h e s are n e e d e d t o c o n s t r u c t a solution on t h e entire real X line. In t h e case (ll = 12 = 1), if we a s s u m e t h a t t h e r e is always one [2 v a r i a b l e b e t w e e n al a n d a2 a n d one b e t w e e n 0 a n d a l a n d t h a t initial c o n d i t i o n s are chosen so t h a t 0 < tt o < a l < po < a2, ~hen we find t h a t : [21 + [2'2 = a l + a'2 - ('ml + n l ) a l a ' 2 e x . T h i s s o l u t i o n is valid on t h e d o m a i n X < - log ( a l n l
+ a2'ml) = X~,
w h e r e n l , ?TZl are c o n s t a n t s d e p e n d i n g on >0, [2o. At t h e p o i n t X~-, ~2 (xT~\I/ -- (t2/}21 @g21Ttl
[2, ( X l ) =- o,
a2?i'~l + al?Zl
N o w we c o n s i d e r (11 = - 1 , 12 = 1). Here we find t h e following e x p r e s s i o n for t h e s y m m e t r i c polynomial: (a`2 - a l ) e - x [21 -}- [22 =- a l -f- a2 --
+ m 2 n ' 2 (a2 - a l ) e x If/,2 q- ?12
which is o n l y defined on t h e interval log
n 2 a z + m2a2 a2 - a l > X > log - X +. m 2 n 2 (a2 - a l ) rtz2al + n2a2
m2, n2 are c o n s t a n t s which m u s t b e chosen so t h a t b o t h #1 a n d #2 are c o n t i n u o u s a t X 1 a n d t h a t t h e e n d s of t h e b r a n c h e s m a t c h up, t h a t is so t h a t X ~ = X +. T h e s e c o n d i t i o n s are satisfied if a2
rn2 = - - (a2 - a l ) m l ,
(1.6)
al al n2 = --
a2
(a2 -- a l ) 7~1.
(1.7)
C o n t i n u i n g in this fashion we arrive at tile final t h r e e b r a n c h e d profile for a fixed t, U = - (aiM
ala2e -x U =
if X < - l o g ( N + M ) ,
+ a2N) e x, +MNe
(1.8)
X (a2 - a l ) 2
a2M + alN
if-log(N+M) < X < l o g u =
+ M N (a2 - a l ) 2'
4M + (a2 - a l ) 2 M N '
+ a?N if X > log (a2 - oq) 2 M N '
where we have m a d e t h e s u b s t i t u t i o n h i = a 2 m l
and N = alnl
(1.9)
(1..10)
a n d used t h e t r a c e f o r m u l a (1.2).
96
M . S . ALBER AND C. I~ILLER
2.5
1.5
0.5 ¸
~
0 -10
1
-8
-6
I
I
I
-4
-2
0
2
4
I
I
6
8
10
F i g u r e 1.1. T h i s is a plot U(x, 0), a profile of t h e s o l u t i o n t o t h e S W e q u a t i o n for t h e two p e a k o n case w h e r e a l = - 1 , a2 -- - 3 , / t O = - . 5 , pt0 = - 1 . 3 . N ot i c e how t h e s o l u t i o n is defined on t h r e e different branches.
Time Evolution So far, only a profile has been derived. Now we will include the time evolution of the peaks to find the general solution for the two peakon case. To do this, we use functions qi(t) for i -- 1, 2 introduced in [9] #~ (x = q i ( t ) , t ) = 0, (1.11) for all t and i = 1, 2 which describe the evolution of the peaks. All peaks belong to a zero level set: Pi = 0. Here the #-coordinates, generalized elliptic coordinates, are used to describe the positions of the peaks. This yields a connection between x and t along trajectories of the peaks resulting in a s y s t e m of equations for the qi(t). T h e solutions of this system are given by ql(t) = GO _ a2t - log 1 - C l e (al-a2)t ~- log (1 - C1),
(1.12)
q2(t) = qO _ a2t + log 1 - C2e ("2-~1)t - log (1 - C2),
(1.13)
where Ci = (q~(0) - al)/(q~(O) - a2). T h e solution defined in (1.8) has the peaks given in t e r m s of the p a r a m e t e r s N and M . So to obtain the solution in t e r m s of b o t h x and t, these p a r a m e t e r s must be considered as functions of time. T h e c o m p l e t e solution now has the form U = - (aiM(t) + a2N(t)) ex, U=
if X < - l o g ( N ( t ) + M ( t ) ) ,
(1.14)
a l a 2 e - x + M ( t ) N ( t ) e x (a2 - a l ) 2 a2M(t) + alN(t) a ~ M ( t ) -F a21N(t)
if - l o g ( N ( t ) + M ( t ) ) < X < log (a2 - - - a ~ U = -e -X
a3M(t) + a3N(t) M ( t ) N ( t ) ( a 2 - a l ) 2'
if X > log
~/--~N-~t) '
a~M(t) + a~N(t)
(a 2 - a l ) 2 M ( t ) N ( t ) '
(1.15) (1.16)
Shallow Water Equation
97
10 B
~U
40
60
Figure 1.2. This is a plot of (ttl + P2), which is what we are seeking in this section. The parameters used axe al = -1, a2 = - 3 , #0 = -.5, ~0 = - t . 3 . Again notice how the solution is defined on three different branches. w h e r e t h e functions M ( t ) , N ( t ) are d e t e r m i n e d b y t h e r e l a t i o n s
N ( t ) + M ( t ) = e -q~(t) = e-'t~ le~t - Cle~'tl 1 - C1
(117)
a ~ M ( t ) + a~N(t) = (a2 - al) 2e q~(t) = (a2 - al) 2 eq~ [e - ~ - C 2 e - ~ t I M(t) N(t) (1 - C2)
(1.18)
w h e r e q l ( t ) , q2(t) a r e t a k e n from (1.12),(1.13). T h i s s y s t e m can b e solved to find t h a t
a~ - a~ + A ( t ) B ( t ) + v/(a~ - a~) 2 - 2 A ( t ) B ( t ) (a T + a~) + A ( t ) 2 B ( t ) 2 M(t) :
,
2B(t)
(1.19)
(1.2o)
N ( t ) = A(t) - M ( t ) ,
where A(t) = e -ql(t) a n d B ( t ) = (a2 - al)2e q2(t). T h e s e functions c o n t a i n four p a r a m e t e r s , b u t m fact t h e s e can be r e d u c e d to two p a r a m e t e r s by using t h e following relations:
ql(0) = - l o g ( M ( 0 ) + N ( 0 ) ) , q2(O) = log
a~M(O)+a~N(O) (a2-al)
2 M(0) N(0)'
q~l(O) =
a2M(O) + alN(O) M(O) + N(O) '
q~(O) = ala2 (a2M(O) + alN(O)) a~M(O) + a~N(O)
(1.21)
(1.22)
S o m e c a r e m u s t be used in choosing t h e sign in (1.19). It is clear t h a t for large n e g a t i v e t, I t l ( q l ( t ) , t ) refers to t h e p a t h of one p e a k o n while for large positive t it refers to t h e other. If t h i s were n o t t h e case, s i m p l e a s y m p t o t i c analysis of (1.12) w o u l d show t h a t t h e p e a k o n s c h a n g e s p e e d which is n o t t h e case. Therefore, ql (t) r e p r e s e n t s t h e p a t h of one of t h e p e a k o n s until s o m e t i m e t* a n d t h e o t h e r one after this time. T h e o p p o s i t e is t r u e for q2(t). A t t h e t i m e t* we say
98
M.S. ALBERAND C. MILLER
that a change of identity has taken place, t* can be found explicitly by using the fact that at this time, the two peaks must have the same height. But the peaks have the same height exactly when a 2 M (t*) = a l N (t*) .
(1.23)
W i t h o u t loss of g e n e r a l i t y we c a n rescale t i m e such t h a t t* = 0. I n t h e case of (1.23), due to t h e original definitions of m l , n l given in t e r m s of #0 # 0 corresponds to a restriction on t h e choice of #0 a n d # 0 n a m e l y - a 2 #0 _ a l _ a2 #0 _ a2 /t o - a2 #0 al
(1.24)
T h i s c o n d i t i o n is satisfied for e x a m p l e w h e n p0 = ( a l a 2 ) / a l + as a n d #o = (al + a2)/2. notice t h a t u n d e r this rescaling, the phase shift is s i m p l y ql(0) - q2(0).
Also
So we now have a procedure to make t h e change of i d e n t i t y occur at t = 0, i.e., P l goes from r e p r e s e n t i n g t h e first peakon to the second one at t = 0.
T h i s change is represented by t h e
change in t h e sign of the p l u s / m i n u s in (1.19). T h a t is, the sign is chosen as positive for t < 0 a n d n e g a t i v e for t > 0. However, M r e m a i n s c o n t i n u o u s despite this sign change since the change of i d e n t i t y occurs precisely w h e n the t e r m u n d e r t h e square root is zero. Therefore, (1.14)-(1.16), a n d (1.19) t o g e t h e r describe the solution U ( X , t) of the S W e q u a t i o n as a f u n c t i o n of x a n d t d e p e n d i n g on two p a r a m e t e r s M ( 0 ) , N ( 0 ) . B y u s i n g t h e a p p r o a c h of this p a p e r weak billiard solutions can be o b t a i n e d for t h e whole class of n - p e a k o n solutions of N - c o m p o n e n t systems.
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