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Peakons and coshoidal waves: Traveling wave solutions of the Camassa-Holm equation
li N O g r H . HOIIAND
P e a k o n s and Coshoidal Waves: Traveling W a v e Solutions of t h e C a m a s s a - H o l m E q u a t i o n J o h n P. B o y d
Department of Atmospheric, Oceanic ~4 Space Sciences University of Michigan 2455 Hayward Avenue Ann Arbor, Michigan 48109
ABSTRACT Camassa and Holm [1] have recently derived a new integrable wave equation. For the special case K = 0, they showed it has solitary waves of the form c e x p ( - ] x - ctl) which they named "peakons". In this work, we derive a perturbation series for general K which converges even at the peakon limit. We also give three analytical representations for the spatially periodic generalization of the peakon, the "coshoidal wave". The three representations are (i) a closed form, analytical solution, (ii) a Fourier series with coefficients that are explicit rational functions~ and (iii) an imbricate Fourier series, which is the superposition of an infinite number of peakons, each separated from its neighbors by distance P where P is the spatial period. Lastly, we have numerically tested the soliton superposition principle. Although the Camassa-Holm equation is integrable for general K, it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases K = O, K/c = 1/2. However, the imbricate~soliton series is a very good approximate solution for general K, even when the spatial period is small and the solution resembles a sine wave more than a solitary wave. © Elsevier Science Inc., 1997
1.
INTRODUCTION
C a m a s s a a n d Holm [1] derived a c o m p l e t e l y i n t e g r a b l e wave equation for w a t e r waves b y r e t a i n i n g two t e r m s t h a t are u s u a l l y neglected in t h e small a m p l i t u d e , shallow w a t e r limit (which gives t h e K o r t e w e g - d e V r i e s equation).
APPLIED MATHEMATICS AND COMPUTATION 81:173-187 (1997) 0096-3003/97/$17.00
© Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010
SSDI 0096-3003(95)00326-6
174
J . P . BOYD
The result is what we shall dub the " C a m a s s a - H o l m " or " C H " equation u t + 2K%
-
uxz t + 3uu z = 2uzU,x
+ u u x z ~.
(1.1)
For a wave that travels at a steady speed c, i.e., u(x, t) -- v ( X ) where X = x - ct, (1.1) reduces to the ordinary differential equation (2K -
c ) v x + CVxx x + 3 v v x = 2 V x V x x
+ VVxx x.
(1.2)
The CH equation differs from the familiar Benjamin-Bona-Mahony or " B B M " equation [2], also known as the Regularized Long Wave equation of Peregrine, only through the two terms on the right-hand side of (1.1). These terms are small in the small amplitude, shallow water regime where the BBM equation, the Korteweg-deVries (KdV), and the CH equation are all good approximations to the full inviscid water wave equations. Nevertheless, these extra terms profoundly alter the solitary waves. The KdV and BBM solitons are both proportional to the square of the hyperbolic secant function and are smooth, infinitely differentiable functions of x for all real x. Camassa and Holm show that the solitary waves of (1.1) for the special case K = 0 are u( x, t) = c exp(
-I x -
ctl),
(1.3)
where the phase speed c is arbitrary. Because these have a sharp peak with a discontinuous first derivative, they dubbed these solutions " p e a k o n s ' . (See Figure 1). Camassa, Holm, and H y m a n [3] present numerical solutions of the time-dependent form and a discussion of the CH equation as a Hamiltonian system. Cooper and Shepard [4] derive variational approximations to the solitary waves of (1.2) for general K. The CH solitons which (i) decay to 0 as I XI ~ oo and (ii) are largest at X = 0 in the moving coordinate system are a two-parameter family where the parameters are (c, K). However, dilating the amplitude scales both parameters. The solution for general c and K = K c is c times the solution for unit phase speed and K - - K. Consequently, the solitons are only a one-parameter family of shapes where the parameter is K = K/c, the "relative kappa". In the rest of the article, we shall compute solutions only for c = l a n d K--- K. The K -- 0 shape is Camassa and Holm's peakon (1.3); K = 1 / 2 has the sech 2 profile of a solution to the KdV equation. There are no solitary waves for K > 1/2; sufficiently far from the peak of the solitary wave that the
Imbrication of Camassa-Holm Peakons
175
°--AU__ 0.~ 0.! OJ 0.4 O~ O~
0.1 •4
-3
-2
-1
0
!
2
3
4
S
(X FIC. 1. The C a m a s s a - H o l m soliton ( c = 1) for three different values of K. T h e profiles have been " c o m p o s i t e d " b y p l o t t i n g v/v(O) as functions of 8X. (The solutions for E < 1 (¢* K > 0) are in r e a l i t y m u c h wider t h a n the a n a l y t i c a l p e a k o n (K = 0 ¢* s = 1), which is the i n n e r m o s t curve. T h e m i d d l e g r a p h (dashed) is ~2 ___ 1 / 2 (¢* K = 1 / 4 ) . T h e o u t e r m o s t curve is
sech2(s X/2), which is the analytical solution to the Benjamin-Bona-Mahony equation.
nonlinear terms can be neglected, the a s y m p t o t i c behavior is
v ( X ; K ) ~ [constant] e x p ( - ~/1 - 2 K X ) ,
] X ] - ~ 0%
(1.4)
t h a t is, the solitary waves are oscillatory, rather t h a n exponentially decaying, where K > 1 / 2 . In the next section, we derive a p e r t u r b a t i o n series for the solitary wave. This is rapidly convergent for K ~ 1 / 2 but converges, albeit slowly, even for the peakon ( K -- 0). T h e spatially periodic solutions to (1.2), which we shall d u b "coshoidal waves", are a four-parameter family. T h e p a r a m e t e r s are (c, K, P, ~ ) where P is the spatial period and ~/ is an internal degree of freedom t h a t can be interpreted as the constant of integration in the second order, once-integrated form of (1.2) -
c)v+
CVxx + ( 3 / 2 ) v
2 = vvxx + 12
+
(1.5)
T h e solitary waves contain two fewer p a r a m e t e r s because (i) the solitons are the special case P = ¢c and (ii) T = 0 in order t h a t the soliton decay to zero
176
J . P . BOYD
The same amplitude scaling law applies to coshoidal waves as to the solitons, however, so it is sufficient to specialize to c = 1: the periodic, steadily translating waves are a three-parameter family of shapes. In Section 3, we explicitly construct the two-parameter subset of these shapes (for unit phase speed) which have slope discontinuities like the peakon. 2.
P E R T U R B A T I O N SERIES F O R T H E SOLITON
The CH equation (1.5) differs from the BBM equation only by the two terms on the right-hand side. If the solitary wave varies slowly with X, then these two extra terms will be small and the soliton is given to lowest order by the solution of
( 2 , , - c)v + cvxx + (3/2)v 2 = o,
(2.1)
vl( X; s ) ~ c6 2 s e c h 2 ( s X / 2 ) ,
(2.2)
which is
where 6 2 ~ 1-
2K
(2.3)
and where K - K/c. The sech-squared approximation can be extended by writing j
v( X; 6) = ~_, 6 2j E aj,nsech2m(6X/2), j=l
(2.4)
m=l
matching powers of 6, and using the identities tanh2(z) = 1 - sech2(z) for any z, d sech( z)/dz = - sech( z)tanh(z) and d tanh( z)/dz = sech2(z). As noted in the introduction, it is sufficient to specialize to c = 1, because the solution for c = Z is related to that for unit speed by
v( x ; c = , , 6 ) = , v ( x ; c = 1, 6 ) .
(2.5)
)te that K and the perturbation parameter 8 are invariant under such caling of the phase speed. The solution at 0 ( 6 2 0 is a polynomial of Tee 2 j in sech(6X/2).
Imbrication of Camassa-Holm Peakons
177
The first six terms are v( X; e) ~ ~2 sech2(6X/2) + e4(_sech2(eX/2) + sech4(sX/2)) + oo6( 1 sech2(~'X/2) - 5 seeh4(eX/2) + 4 sech6(3X/2))
+ o,,8 ( l s e c h 2 ( ~ . X / 2 ) + ~6 sech4(eX/2) 16
31 ) sech6(sX/2) + ~ sechS(sx/2)
1
11 373 ' sech2(eX/2) - --21sech4(eX/2) + --105sech6(eX/2) 123 2062 sech8(eX/2) + sechl°(6X/2) 315 35
+ el 0 ~
16 37 / 1 sech2(¢X/2) + sech4(~X/2) sech6(~X/2 ) + el2 315 105 15 668 3103 sechS(ex/2) _ 1459 sechl°(eX/2) + __sech12(eX/2 ) + 315 105 105 (2.6)
The MATLAB program is shown in Appendix 1. The only complication is that the polynomials in sech(e X / 2 ) become badly conditioned at high order (above e40), but this could be fixed by computing in Maple or Mathematica, which allow exact rational arithmetic. Series of powers of sech(c X) are usually asymptotic but divergent, as for the Fifth-Order Korteweg-deVries (FKdV) equation [5, 6]. Here, however, the perturbation does not contain a derivative higher than that of the unperturbed (BBM) equation. Even at 8 = 1, where the soliton is singular (v x discontinuous at X = 0), the series appears to converge to the peakon with the m a x i m u m of the j-th perturbation order decreasing roughly as O(1/j2); the total error decreases as O ( 1 / N ) where N is the truncation of the perturbation series. Figure 2 compares the peakon ( e = 1) with the N = 20 partial sum of the perturbation series. If the radius of convergence is unity, as conjectured, then the series will converge geometrically for ~ < 1 as O(exp[ N log(e)]/N).
178
J . P . BOYD
I
'°kkil FIG. 2. Comparisonof the analytical peakon, exp(-I X]), (solid curve) versus the first 20 terms (up to and including ~40) of the perturbation series for e = 1 (Dashed). The circles show the error, which has a maximum pointwise value of 0.048.
3.
PERIODIC GENERALIZATIONS OF SOLITARY WAVES
T h e periodic travelling waves have two extra degrees of freedom compared to the soliton, including (i) the spatial period P and (ii) the constant of integration ~/ in the second order form of the CH equation, (1.5). T h e requirement t h a t the solitary wave decay to zero demands t h a t 3' = 0 because every other term in (1.5) tends to zero as v ( X ) ~ 0 as I XI ~ 0. For periodic solutions, however, there is no longer any reason w h y ~/should be zero. The periodic waves are a four-parameter family, but the shapes depend on only the three parameters ( P , ~/, and K = K / c ) as explained in the Introduction. As shown in Section 2, the peakons, t h a t is, solitary waves with slope discontinuities, are the special case K = 0. T h e periodic waves with slope discontinuities are likely to be a two-parameter family with parameters ( P , 3~). Our construction of a two-parameter set of explicit periodic solutions has three parts. T h e first is to demonstrate a constraint on the solution. In the sense of distributions [8], the first derivative of a step function is the Dirac delta function. Because the first derivative of a peakon is proportional to a step function at X = 0, it follows t h a t the second derivative of a soliton or
179
Imbrication of Camassa-Hohn Peakons
periodic solution with a similar singularity will be proportional to the &function. If we integrate the second order form of the CH equation (1.5) over the infinitesimal interval [-IZ,/z], we obtain from the usual properties of the delta function, lim f ~
1 2 _ 3 -~v2 + ( c - 2 K ) v d Z ( v - c) vx~ + ~v~
+ ~/= A(v(0)
c)
/z--* 0 J - / z
(3.1) for some proportionality constant A. Because the integrand is zero for any solution of (1.5), it follows that v(0) = c.
(3.2)
This constraint applies to any solitary wave or periodic solution with a slope discontinuity at X = 0. The second step is to write down a one-parameter family of solutions with arbitrary period P. (The other parameters are c = 1, K = 0, and y ( P ) which can be determined by substituting the solution into (1.5).) The closed form expression may be demonstrated by direct substitution into (1.5) for the special case K = 0. Any multiple of cosh(X - P / 2 ) also satisfies the differential equation everywhere except the singular point X = 0, but only the expression shown satisfies (3.2) for c = 1. The equivalent Fourier series is obtained by analytical evaluation of the usual Fourier coefficient integrals. The so-called "imbricate" series is derived from the Fourier series by Poisson summation [9]. (I) Closed-form Expression cosh( X v( X; 1, 0, P) =
P/2)
cosh(P/2) v(X+ P;P)
'
= v(X; P ) ,
X ~ [0, P]
(3.3)
X ~ [0, P].
(II) Fourier series v( X; 1, 0, P) = -~ tanh
1 1 + 2 =1 (1 + 4~rZn2/P 2)
[ 21rn
°st-P-
X)
(3.4)
.
J. P. BOYD
180 (III) Imbricate series c~
v( X; 1,0, P ) = t a n h ( P / 2 )
exp( -I X - mPI).
~., Tn~
(3.5)
--or
The imbricate series is built upon a " p a t t e r n " function, which is repeated with even spacing at X = 0, ± P, + 2 P, etc. In this case, the pattern is the peakon multiplied by a constant. This one-parameter family of solutions can be generalized by using the following rescaling theorem which can be verified by direct substitution into the differential equation: if v( X; c, K, P) is a solution, then
w( X; c, K, P, B) - - B c + (I + B)v( X; c, K, P)
(3.6)
is also a solution for K ~ K ' in the equation, all other parameters unchanged, where K'=
B + (1 + B) K
(3.7)
with B as a new parameter, an arbitrary constant. For the special case K = 0, c = 1, this implies the statement that for any member of the one-parameter family defined by (3.3), (3.4), or (3.5), the modified function
q(X; B, P) =- - B +
(3.8)
(1 + B) v ( X ; 1,0, P )
solves the reduced CH equation with K / c = B. This rescaling law gives a two-parameter family of solutions where (P, B) are the parameters. Because these all can be expressed in terms of cosh(X), we shall dub these the "coshoidal" waves. We shall not attempt to prove that (3.3) with (3.6) is the most general set of periodic solutions with slope discontinuities but conjecture that it is. Because B is a degree of freedom that merely adds a constant while simultaneously rescaling the amplitude, the dependence of the coshoidal wave on B is not very interesting. For graphical purposes, it is convenient to fix B by imposing a condition such as the zero-mean constraint ta~h(P/2)
"lPv(X) dX = 0 ~ B = P/2 "o
tanh( P/2)
.
(3.9)
181
Imbrication of Camassa-Holm Peakons
S~d~ Pmod F~ ~ 1
ii!
0.5
'~ -0.5
j/
,
,,
~2
-1 ,I
X Spatial Pedod I ~ 2 i~
~
0.
Q.
! o.
°0.~
i
i
,
~
4
,
i
i
-2
0 X
2
, 4
i
,
8
8
10
F](~. 3. The zero-mean coshoidal solution for unit phase speed for various spatial periods P. (The solution for arbitrary phase speed c is obtained by multiplying v( X; 1, B, P) by c.) The solid curve is the coshoidal wave. The dashed curves show the peakons which are superimposed in the imbricate series; these asymptote to - B . (a) P (b) P (c) P the sense
= ~r ( B = 1.4031) = 2~r ( B ~ 0.4644) = 47r ( B = 0.1893) (For P so large, there is little difference from the limit P ~ ~ in that v( X; 1, B, P) is graphically indistinguishable from the nearest peakon.)
182
J. P. BOYD Spl~ml Pw'iod I:~ 4 pi
~- o.4 ~• 0.2
-~
-15
-10
-5
FIG. 3.
0 X
5
10
15
20
Continued.
The zero-mean coshoidal wave is illustrated in Figure 3 for three different spatial periods P. Note that K = K / c does not equal zero except in the limit P ~ ~, in which case the coshoidal wave becomes a peakon. Related work on imbricate series includes Parker (10), Chow (11), Boyd and Haupt (12) and older articles cited therein. Imbricate series are useful for non-soliton problems, too [Parker (13) and Boyd (14)]. Figure 3 illustrates coshoidal waves for several spatial periods along with the peakons that are imbricated to make them.
4.
S O L I T O N - I M B R I C A T I O N F O R SOLITONS W I T H O U T DERIVATIVE DISCONTINUITIES
Because solitons decay exponentially with I XI, it is obvious that an imbricate series of solitons will be a good approximate solution if the period P is large. Remarkably, Toda showed that for the KdV equation, solitonimbrication generates the exact solution for all periods. It is now known [9] that this soliton-imbrication principle can be extended to most solitary waves that can be expressed in terms of elliptic functions or rational functions. Here, we have added the imbrication of peakons to the list of exact solutions that are the superposition of solitary waves.
183
Imbrication of Camassa-Holm Peakons
This inspired us to numerically test soliton-imbrication for the full three-parameter family of shapes of periodic solutions to the CH equation. (We showed this is true for the two-parameter family of imbricated peakons in the previous section.) Because no closed-form analytical solution is known for the smooth solitary waves, free of slope discontinuities, the question must be answered numerically: Is the superposition of solitary waves an exact, nonlinear periodic solution? The imbrication of the solitary wave is defined by oo
V ( X ; 1 , K, P) =
~_~ V s o l ( X - mP;1, K ) . m
~
(4.1)
-¢~
If this is an exact solution, then the function 6'(X) -
2 K Vx + 3 V Vx - 2 Vx V x x v ~ - vxx~
VVxx x
(4.2)
should be equal to a constant, the phase speed. (Note that the phase speed of the imbricated soliton is usually different from that of the constituent solitons, so C need not equal unity.) Given that the phase speed is altered by the imbrication of solitons, perhaps the parameter K is, too. We can force C(0) to equal 6-I(P/2) by choosing
(c(o; ~o,) - c( K = K~o I +
n/2; ,,~o,)
( r(P/2) - r(0))
(4.3)
where C( X; K~ol) denotes (4.2) evaluated with K = K~oZ, the parameter of the solitons in the imbrication, and where
2yx r(x)
= ( vx - y ~ . )
"
(4.4)
By numerically solving (1.2) by means of rational Chebyshev functions [9], one can evaluate 6"(X) to within an error that decreases exponentially fast with the truncation of the spectral series. A representative graph of C ( X ) minus its spatial mean is shown as Figure 4. The difference should be zero if the soliton-imbrication has generated an exact periodic solution. This difference is not zero. However, the difference
184
J. P. BOYD 0~!
0~
-0.000~
-0.001
0.5
1
x/2
X FlC. 4. C ( X ) ( m i n u s its m e a n value) is p l o t t e d for P = Ir a n d t h e i m b r i c a t i o n of s o l i t a r y w a v e s w i t h K = 1 / 1 0 ( 8 2 = 4 / 5 ) a n d u n i t p h a s e speed. T h e m e a n v a l u e of C ( X ) is 1.1417 a n d v a l u e of K t h a t m i n i m i z e s t h e v a r i a t i o n s of C ( X ) is K = 0.1164.
is smaller than the height of the constituent solitary saves by a factor of 1000! In the limit K ~ 0, we have already seen that the superposition is exact; Toda proved that for sech 2 solitons (here, K = 1/2), the superposition is exact, too. For intermediate K, the imbrication of the soliton is a very good approximation but not exact. Although we have not explored the (K, P) parameter space in full, we not have found any cases where C~ X) varied by more than a small fraction of the amplitude of the solitons.
5.
CONCLUSIONS
The Camassa-Holm equation is of interest for two reasons. First, it is a model for small amplitude, shallow water waves--just as consistent in this limit as the KdV equation, which has been intensively studied for a century. Second, the equation is of great intrinsic interest in the study of solitary waves because (i) it is exactly integrable (ii) its solutions, including multiple solitons given in Camassa and Holm [1] are very simple, and (iii) it is novel in that its solitary waves have a discontinuous first derivative, in contrast to the great smoothness of most previously known species of solitary waves. Boyd [9] conjectured that the imbrication of the soliton was an exact solution only when the solitary wave was either an elliptic function or a rational function. (He showed that soliton imbrication fails, except as an approximation, for the quartically nonlinear KdV equation, whose solitons are hyperelliptic functions.) The coshoidal wave shows that this conjecture is false and too restrictive; the peakon is neither elliptic nor rational, but its imbrication is an exact periodic solution. There are probably other cases, as
Imbrication of Carnassa-Holm Peakons
185
y e t undiscovered, where t h e s o l i t o n - i m b r i c a t i o n is exact; a general principle to connect all cases now seems elusive. A P P E N D I X 1. A M A T L A B P R O G R A M PERTURBATION SERIES
TO COMPUTE
THE
m a x j = 6; a(1,1) = 1; a ( 2 , 1 ) = - 1 ; a ( 2 , 2 ) = 1; a v x x ( 2 , 1 ) = 1; a v x x ( 2 , 2 ) = - 1 . 5 ; avx(1,1) = - 1; for j = 3:maxj % Beginning of loop over p e r t u r b a t i o n order j % F i r s t a n d Second d e r i v a t i v e c o e f f i c i e n t s . . , used for larger j % v x x = sum[j = 2:infin] eps^(2j) sum[m = l:j + 1] avxx(j,m) sech(epsx) % ^(2m) % vx = t a n h ( e p s x ) sum[j = 2:infin] eps^(2j + 1) s u m [ m = l:j] avx(j,m) % sech(eps x)^(2m) form= l:j- 1 avxx(j,m) -- a v x x ( j , m ) + m* m*a(j - 1,m); a v x x ( j , m + 1) = - 0 . 5 * m * ( 2 * m + 1)*a(j - 1,m); avx(j - 1,m) = - m* a(j - 1,m); end % of m loop % Nonlinear c o n t r i b u t i o n to Rjk from - 1.5 v* v for m = 2:(j - 1) for p = l : m for q = l:(j + 1 - m) k=p+q; R(j,k) = R(j,k) - 1.5*a(m,p)*a(j + 1 - m,q); end % of q loop end % of p loop end % of m loop % Nonlinear c o n t r i b u t i o n to Rjk from v* vxx for m = l:(j - 1) for p = l : m for q = l:(j + 1 - m) k=p+q; R(j,k) = R(j,k) + a ( m , p ) * a v x x ( j + 1 - m,q); end % of q loop end % of p loop end % of m loop % Nonlinear c o n t r i b u t i o n of R j k from 0.5* vx* v x for m = l:(j - 1) for p = l : m for q = l:(j - p)
186
J . P . BOYD
k=p+q; glug = 0.5* avx(m,p)* avx(j - m,p); R(j,k) = R(j,k) + glug; R(j,k + 1) = R(j,k + 1) - glug; end % of q loop end % of p loop end % of m loop % Now compute the coefficients at j-th order. a(j,j) = R(j,j + 1)/3 - 0.5*j*(2*j + 1)); form=j:1:2 a(j,m- 1)= (R(j,m)- (m*m- 1)*a(j,m))/(30.25*(2* m - 2)*(2* m - 1)); end of loop in m % The leading order term in the residual is R(j,j + 1), which is proportional % to O(eps**(2*j + 2))* sech(eps x)**(2j + 2). end; % End of loop over perturbation order j
APPENDIX 2 LIST OF SYMBOLS B c C
scaling parameter phase speed ratio
K K/c m n P q r t u v w x X 8 K A /~ ~r
sum index sum index spatial period rescaled solution denominator of residual time time-dependent solution steadily-translating solution rescaled solution spatial coordinate coordinate in moving reference frame perturbation parameter parameter of CH equation scaling parameter limits of infinitesimal interval 3.14159
Imbrication of Camassa-Holm Peakons
187
This work was supported by the N S F under Grant 0CE9119459 and the Department of Energy through KC070101. I thank Roberto Camassa for reprints which inspired this note. REFERENCES 1 R. Camassa and D. D. Holm, An Integrable Shallow Water Equation with Peaked Solitons, Phys. Rev. Lett., 71:1661-1664 (1993). 2 T. B. Benjamin, J. L. Bona, and J. Mahoney, Phil. Trans. Royal Soc. Lond. A227, 47 (1972). 3 R. Camassa, D. D. Holm, and J. M. Hyman, A new integrable shallow water equation, in Advances in Applied Mechanics 31 (T.-Y. Wu and J. W. Hutchinson, Eds.), Academic, pp. 1-33, 1994. 4 F. Cooper and H. Shepard, Solitons in the Camassa-Holm Shallow Water Equation. To be published. 5 J.P. Boyd, Weakly Nonlocal Solitons for Capillary-Gravity Waves: Fifth-degree Korteweg-deVries Equation, Physica D48, 129-146 (1991). 6 J . P . Boyd, Weakly Nonlocal Solitary Waves and Other Exponentially Small Phenomena, Cambridge University Press, New York (1995). 7 J . P . Boyd, Solitons From Sine Waves: Analytical and Numerical Methods for Nonintegrable Solitary and Cnoidal Waves, Physica D21, 227-246 (1986). 8 M . J . Lighthill, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, New York. 9 J.P. Boyd, New directions in solitons and nonlinear periodic waves: polycnoidal waves, imbricated solitons, weakly non-local solitary waves and numerical boundary value algorithms, in Advances in Applied Mechanics, 27 (T.-Y. Wu and J. W. Hutchinson, Eds.), Academic, New York, pp. 1-82, 1989. 10 A. Parker, Periodic solutions of the intermediate long-wave equation: A nonlinear superposition principle. J. Math. Phys. A25:2005-2032 (1992). 11 K. W. Chow, Theta functions and dispersion relations of periodic waves. J. Phys. Soc. Japan 62:2007-2011 (1993). 12 J . P . Boyd and S. E. Haupt, Polycnoidal waves: Spatially periodic generalizations of multiple solitary waves. In Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX (A. R. Osborne, Ed.) North-Holland, Amsterdam, 1991, pp. 827-856. 13 A. Parker, On the periodic solution of the Burgers equation: A unified approach. Proc. R. Soc..London A438:113-132 (1992). 14 J . P . Boyd, The arctan/tan and Kepler-Burger mappings for periodic solutions with a shock, front, or internal boundary layer. J. Comp. Phys., 98:181-193 (1992).
P e a k o n s and Coshoidal Waves: Traveling W a v e Solutions of t h e C a m a s s a - H o l m E q u a t i o n J o h n P. B o y d
Department of Atmospheric, Oceanic ~4 Space Sciences University of Michigan 2455 Hayward Avenue Ann Arbor, Michigan 48109
ABSTRACT Camassa and Holm [1] have recently derived a new integrable wave equation. For the special case K = 0, they showed it has solitary waves of the form c e x p ( - ] x - ctl) which they named "peakons". In this work, we derive a perturbation series for general K which converges even at the peakon limit. We also give three analytical representations for the spatially periodic generalization of the peakon, the "coshoidal wave". The three representations are (i) a closed form, analytical solution, (ii) a Fourier series with coefficients that are explicit rational functions~ and (iii) an imbricate Fourier series, which is the superposition of an infinite number of peakons, each separated from its neighbors by distance P where P is the spatial period. Lastly, we have numerically tested the soliton superposition principle. Although the Camassa-Holm equation is integrable for general K, it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases K = O, K/c = 1/2. However, the imbricate~soliton series is a very good approximate solution for general K, even when the spatial period is small and the solution resembles a sine wave more than a solitary wave. © Elsevier Science Inc., 1997
1.
INTRODUCTION
C a m a s s a a n d Holm [1] derived a c o m p l e t e l y i n t e g r a b l e wave equation for w a t e r waves b y r e t a i n i n g two t e r m s t h a t are u s u a l l y neglected in t h e small a m p l i t u d e , shallow w a t e r limit (which gives t h e K o r t e w e g - d e V r i e s equation).
APPLIED MATHEMATICS AND COMPUTATION 81:173-187 (1997) 0096-3003/97/$17.00
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174
J . P . BOYD
The result is what we shall dub the " C a m a s s a - H o l m " or " C H " equation u t + 2K%
-
uxz t + 3uu z = 2uzU,x
+ u u x z ~.
(1.1)
For a wave that travels at a steady speed c, i.e., u(x, t) -- v ( X ) where X = x - ct, (1.1) reduces to the ordinary differential equation (2K -
c ) v x + CVxx x + 3 v v x = 2 V x V x x
+ VVxx x.
(1.2)
The CH equation differs from the familiar Benjamin-Bona-Mahony or " B B M " equation [2], also known as the Regularized Long Wave equation of Peregrine, only through the two terms on the right-hand side of (1.1). These terms are small in the small amplitude, shallow water regime where the BBM equation, the Korteweg-deVries (KdV), and the CH equation are all good approximations to the full inviscid water wave equations. Nevertheless, these extra terms profoundly alter the solitary waves. The KdV and BBM solitons are both proportional to the square of the hyperbolic secant function and are smooth, infinitely differentiable functions of x for all real x. Camassa and Holm show that the solitary waves of (1.1) for the special case K = 0 are u( x, t) = c exp(
-I x -
ctl),
(1.3)
where the phase speed c is arbitrary. Because these have a sharp peak with a discontinuous first derivative, they dubbed these solutions " p e a k o n s ' . (See Figure 1). Camassa, Holm, and H y m a n [3] present numerical solutions of the time-dependent form and a discussion of the CH equation as a Hamiltonian system. Cooper and Shepard [4] derive variational approximations to the solitary waves of (1.2) for general K. The CH solitons which (i) decay to 0 as I XI ~ oo and (ii) are largest at X = 0 in the moving coordinate system are a two-parameter family where the parameters are (c, K). However, dilating the amplitude scales both parameters. The solution for general c and K = K c is c times the solution for unit phase speed and K - - K. Consequently, the solitons are only a one-parameter family of shapes where the parameter is K = K/c, the "relative kappa". In the rest of the article, we shall compute solutions only for c = l a n d K--- K. The K -- 0 shape is Camassa and Holm's peakon (1.3); K = 1 / 2 has the sech 2 profile of a solution to the KdV equation. There are no solitary waves for K > 1/2; sufficiently far from the peak of the solitary wave that the
Imbrication of Camassa-Holm Peakons
175
°--AU__ 0.~ 0.! OJ 0.4 O~ O~
0.1 •4
-3
-2
-1
0
!
2
3
4
S
(X FIC. 1. The C a m a s s a - H o l m soliton ( c = 1) for three different values of K. T h e profiles have been " c o m p o s i t e d " b y p l o t t i n g v/v(O) as functions of 8X. (The solutions for E < 1 (¢* K > 0) are in r e a l i t y m u c h wider t h a n the a n a l y t i c a l p e a k o n (K = 0 ¢* s = 1), which is the i n n e r m o s t curve. T h e m i d d l e g r a p h (dashed) is ~2 ___ 1 / 2 (¢* K = 1 / 4 ) . T h e o u t e r m o s t curve is
sech2(s X/2), which is the analytical solution to the Benjamin-Bona-Mahony equation.
nonlinear terms can be neglected, the a s y m p t o t i c behavior is
v ( X ; K ) ~ [constant] e x p ( - ~/1 - 2 K X ) ,
] X ] - ~ 0%
(1.4)
t h a t is, the solitary waves are oscillatory, rather t h a n exponentially decaying, where K > 1 / 2 . In the next section, we derive a p e r t u r b a t i o n series for the solitary wave. This is rapidly convergent for K ~ 1 / 2 but converges, albeit slowly, even for the peakon ( K -- 0). T h e spatially periodic solutions to (1.2), which we shall d u b "coshoidal waves", are a four-parameter family. T h e p a r a m e t e r s are (c, K, P, ~ ) where P is the spatial period and ~/ is an internal degree of freedom t h a t can be interpreted as the constant of integration in the second order, once-integrated form of (1.2) -
c)v+
CVxx + ( 3 / 2 ) v
2 = vvxx + 12
+
(1.5)
T h e solitary waves contain two fewer p a r a m e t e r s because (i) the solitons are the special case P = ¢c and (ii) T = 0 in order t h a t the soliton decay to zero
176
J . P . BOYD
The same amplitude scaling law applies to coshoidal waves as to the solitons, however, so it is sufficient to specialize to c = 1: the periodic, steadily translating waves are a three-parameter family of shapes. In Section 3, we explicitly construct the two-parameter subset of these shapes (for unit phase speed) which have slope discontinuities like the peakon. 2.
P E R T U R B A T I O N SERIES F O R T H E SOLITON
The CH equation (1.5) differs from the BBM equation only by the two terms on the right-hand side. If the solitary wave varies slowly with X, then these two extra terms will be small and the soliton is given to lowest order by the solution of
( 2 , , - c)v + cvxx + (3/2)v 2 = o,
(2.1)
vl( X; s ) ~ c6 2 s e c h 2 ( s X / 2 ) ,
(2.2)
which is
where 6 2 ~ 1-
2K
(2.3)
and where K - K/c. The sech-squared approximation can be extended by writing j
v( X; 6) = ~_, 6 2j E aj,nsech2m(6X/2), j=l
(2.4)
m=l
matching powers of 6, and using the identities tanh2(z) = 1 - sech2(z) for any z, d sech( z)/dz = - sech( z)tanh(z) and d tanh( z)/dz = sech2(z). As noted in the introduction, it is sufficient to specialize to c = 1, because the solution for c = Z is related to that for unit speed by
v( x ; c = , , 6 ) = , v ( x ; c = 1, 6 ) .
(2.5)
)te that K and the perturbation parameter 8 are invariant under such caling of the phase speed. The solution at 0 ( 6 2 0 is a polynomial of Tee 2 j in sech(6X/2).
Imbrication of Camassa-Holm Peakons
177
The first six terms are v( X; e) ~ ~2 sech2(6X/2) + e4(_sech2(eX/2) + sech4(sX/2)) + oo6( 1 sech2(~'X/2) - 5 seeh4(eX/2) + 4 sech6(3X/2))
+ o,,8 ( l s e c h 2 ( ~ . X / 2 ) + ~6 sech4(eX/2) 16
31 ) sech6(sX/2) + ~ sechS(sx/2)
1
11 373 ' sech2(eX/2) - --21sech4(eX/2) + --105sech6(eX/2) 123 2062 sech8(eX/2) + sechl°(6X/2) 315 35
+ el 0 ~
16 37 / 1 sech2(¢X/2) + sech4(~X/2) sech6(~X/2 ) + el2 315 105 15 668 3103 sechS(ex/2) _ 1459 sechl°(eX/2) + __sech12(eX/2 ) + 315 105 105 (2.6)
The MATLAB program is shown in Appendix 1. The only complication is that the polynomials in sech(e X / 2 ) become badly conditioned at high order (above e40), but this could be fixed by computing in Maple or Mathematica, which allow exact rational arithmetic. Series of powers of sech(c X) are usually asymptotic but divergent, as for the Fifth-Order Korteweg-deVries (FKdV) equation [5, 6]. Here, however, the perturbation does not contain a derivative higher than that of the unperturbed (BBM) equation. Even at 8 = 1, where the soliton is singular (v x discontinuous at X = 0), the series appears to converge to the peakon with the m a x i m u m of the j-th perturbation order decreasing roughly as O(1/j2); the total error decreases as O ( 1 / N ) where N is the truncation of the perturbation series. Figure 2 compares the peakon ( e = 1) with the N = 20 partial sum of the perturbation series. If the radius of convergence is unity, as conjectured, then the series will converge geometrically for ~ < 1 as O(exp[ N log(e)]/N).
178
J . P . BOYD
I
'°kkil FIG. 2. Comparisonof the analytical peakon, exp(-I X]), (solid curve) versus the first 20 terms (up to and including ~40) of the perturbation series for e = 1 (Dashed). The circles show the error, which has a maximum pointwise value of 0.048.
3.
PERIODIC GENERALIZATIONS OF SOLITARY WAVES
T h e periodic travelling waves have two extra degrees of freedom compared to the soliton, including (i) the spatial period P and (ii) the constant of integration ~/ in the second order form of the CH equation, (1.5). T h e requirement t h a t the solitary wave decay to zero demands t h a t 3' = 0 because every other term in (1.5) tends to zero as v ( X ) ~ 0 as I XI ~ 0. For periodic solutions, however, there is no longer any reason w h y ~/should be zero. The periodic waves are a four-parameter family, but the shapes depend on only the three parameters ( P , ~/, and K = K / c ) as explained in the Introduction. As shown in Section 2, the peakons, t h a t is, solitary waves with slope discontinuities, are the special case K = 0. T h e periodic waves with slope discontinuities are likely to be a two-parameter family with parameters ( P , 3~). Our construction of a two-parameter set of explicit periodic solutions has three parts. T h e first is to demonstrate a constraint on the solution. In the sense of distributions [8], the first derivative of a step function is the Dirac delta function. Because the first derivative of a peakon is proportional to a step function at X = 0, it follows t h a t the second derivative of a soliton or
179
Imbrication of Camassa-Hohn Peakons
periodic solution with a similar singularity will be proportional to the &function. If we integrate the second order form of the CH equation (1.5) over the infinitesimal interval [-IZ,/z], we obtain from the usual properties of the delta function, lim f ~
1 2 _ 3 -~v2 + ( c - 2 K ) v d Z ( v - c) vx~ + ~v~
+ ~/= A(v(0)
c)
/z--* 0 J - / z
(3.1) for some proportionality constant A. Because the integrand is zero for any solution of (1.5), it follows that v(0) = c.
(3.2)
This constraint applies to any solitary wave or periodic solution with a slope discontinuity at X = 0. The second step is to write down a one-parameter family of solutions with arbitrary period P. (The other parameters are c = 1, K = 0, and y ( P ) which can be determined by substituting the solution into (1.5).) The closed form expression may be demonstrated by direct substitution into (1.5) for the special case K = 0. Any multiple of cosh(X - P / 2 ) also satisfies the differential equation everywhere except the singular point X = 0, but only the expression shown satisfies (3.2) for c = 1. The equivalent Fourier series is obtained by analytical evaluation of the usual Fourier coefficient integrals. The so-called "imbricate" series is derived from the Fourier series by Poisson summation [9]. (I) Closed-form Expression cosh( X v( X; 1, 0, P) =
P/2)
cosh(P/2) v(X+ P;P)
'
= v(X; P ) ,
X ~ [0, P]
(3.3)
X ~ [0, P].
(II) Fourier series v( X; 1, 0, P) = -~ tanh
1 1 + 2 =1 (1 + 4~rZn2/P 2)
[ 21rn
°st-P-
X)
(3.4)
.
J. P. BOYD
180 (III) Imbricate series c~
v( X; 1,0, P ) = t a n h ( P / 2 )
exp( -I X - mPI).
~., Tn~
(3.5)
--or
The imbricate series is built upon a " p a t t e r n " function, which is repeated with even spacing at X = 0, ± P, + 2 P, etc. In this case, the pattern is the peakon multiplied by a constant. This one-parameter family of solutions can be generalized by using the following rescaling theorem which can be verified by direct substitution into the differential equation: if v( X; c, K, P) is a solution, then
w( X; c, K, P, B) - - B c + (I + B)v( X; c, K, P)
(3.6)
is also a solution for K ~ K ' in the equation, all other parameters unchanged, where K'=
B + (1 + B) K
(3.7)
with B as a new parameter, an arbitrary constant. For the special case K = 0, c = 1, this implies the statement that for any member of the one-parameter family defined by (3.3), (3.4), or (3.5), the modified function
q(X; B, P) =- - B +
(3.8)
(1 + B) v ( X ; 1,0, P )
solves the reduced CH equation with K / c = B. This rescaling law gives a two-parameter family of solutions where (P, B) are the parameters. Because these all can be expressed in terms of cosh(X), we shall dub these the "coshoidal" waves. We shall not attempt to prove that (3.3) with (3.6) is the most general set of periodic solutions with slope discontinuities but conjecture that it is. Because B is a degree of freedom that merely adds a constant while simultaneously rescaling the amplitude, the dependence of the coshoidal wave on B is not very interesting. For graphical purposes, it is convenient to fix B by imposing a condition such as the zero-mean constraint ta~h(P/2)
"lPv(X) dX = 0 ~ B = P/2 "o
tanh( P/2)
.
(3.9)
181
Imbrication of Camassa-Holm Peakons
S~d~ Pmod F~ ~ 1
ii!
0.5
'~ -0.5
j/
,
,,
~2
-1 ,I
X Spatial Pedod I ~ 2 i~
~
0.
Q.
! o.
°0.~
i
i
,
~
4
,
i
i
-2
0 X
2
, 4
i
,
8
8
10
F](~. 3. The zero-mean coshoidal solution for unit phase speed for various spatial periods P. (The solution for arbitrary phase speed c is obtained by multiplying v( X; 1, B, P) by c.) The solid curve is the coshoidal wave. The dashed curves show the peakons which are superimposed in the imbricate series; these asymptote to - B . (a) P (b) P (c) P the sense
= ~r ( B = 1.4031) = 2~r ( B ~ 0.4644) = 47r ( B = 0.1893) (For P so large, there is little difference from the limit P ~ ~ in that v( X; 1, B, P) is graphically indistinguishable from the nearest peakon.)
182
J. P. BOYD Spl~ml Pw'iod I:~ 4 pi
~- o.4 ~• 0.2
-~
-15
-10
-5
FIG. 3.
0 X
5
10
15
20
Continued.
The zero-mean coshoidal wave is illustrated in Figure 3 for three different spatial periods P. Note that K = K / c does not equal zero except in the limit P ~ ~, in which case the coshoidal wave becomes a peakon. Related work on imbricate series includes Parker (10), Chow (11), Boyd and Haupt (12) and older articles cited therein. Imbricate series are useful for non-soliton problems, too [Parker (13) and Boyd (14)]. Figure 3 illustrates coshoidal waves for several spatial periods along with the peakons that are imbricated to make them.
4.
S O L I T O N - I M B R I C A T I O N F O R SOLITONS W I T H O U T DERIVATIVE DISCONTINUITIES
Because solitons decay exponentially with I XI, it is obvious that an imbricate series of solitons will be a good approximate solution if the period P is large. Remarkably, Toda showed that for the KdV equation, solitonimbrication generates the exact solution for all periods. It is now known [9] that this soliton-imbrication principle can be extended to most solitary waves that can be expressed in terms of elliptic functions or rational functions. Here, we have added the imbrication of peakons to the list of exact solutions that are the superposition of solitary waves.
183
Imbrication of Camassa-Holm Peakons
This inspired us to numerically test soliton-imbrication for the full three-parameter family of shapes of periodic solutions to the CH equation. (We showed this is true for the two-parameter family of imbricated peakons in the previous section.) Because no closed-form analytical solution is known for the smooth solitary waves, free of slope discontinuities, the question must be answered numerically: Is the superposition of solitary waves an exact, nonlinear periodic solution? The imbrication of the solitary wave is defined by oo
V ( X ; 1 , K, P) =
~_~ V s o l ( X - mP;1, K ) . m
~
(4.1)
-¢~
If this is an exact solution, then the function 6'(X) -
2 K Vx + 3 V Vx - 2 Vx V x x v ~ - vxx~
VVxx x
(4.2)
should be equal to a constant, the phase speed. (Note that the phase speed of the imbricated soliton is usually different from that of the constituent solitons, so C need not equal unity.) Given that the phase speed is altered by the imbrication of solitons, perhaps the parameter K is, too. We can force C(0) to equal 6-I(P/2) by choosing
(c(o; ~o,) - c( K = K~o I +
n/2; ,,~o,)
( r(P/2) - r(0))
(4.3)
where C( X; K~ol) denotes (4.2) evaluated with K = K~oZ, the parameter of the solitons in the imbrication, and where
2yx r(x)
= ( vx - y ~ . )
"
(4.4)
By numerically solving (1.2) by means of rational Chebyshev functions [9], one can evaluate 6"(X) to within an error that decreases exponentially fast with the truncation of the spectral series. A representative graph of C ( X ) minus its spatial mean is shown as Figure 4. The difference should be zero if the soliton-imbrication has generated an exact periodic solution. This difference is not zero. However, the difference
184
J. P. BOYD 0~!
0~
-0.000~
-0.001
0.5
1
x/2
X FlC. 4. C ( X ) ( m i n u s its m e a n value) is p l o t t e d for P = Ir a n d t h e i m b r i c a t i o n of s o l i t a r y w a v e s w i t h K = 1 / 1 0 ( 8 2 = 4 / 5 ) a n d u n i t p h a s e speed. T h e m e a n v a l u e of C ( X ) is 1.1417 a n d v a l u e of K t h a t m i n i m i z e s t h e v a r i a t i o n s of C ( X ) is K = 0.1164.
is smaller than the height of the constituent solitary saves by a factor of 1000! In the limit K ~ 0, we have already seen that the superposition is exact; Toda proved that for sech 2 solitons (here, K = 1/2), the superposition is exact, too. For intermediate K, the imbrication of the soliton is a very good approximation but not exact. Although we have not explored the (K, P) parameter space in full, we not have found any cases where C~ X) varied by more than a small fraction of the amplitude of the solitons.
5.
CONCLUSIONS
The Camassa-Holm equation is of interest for two reasons. First, it is a model for small amplitude, shallow water waves--just as consistent in this limit as the KdV equation, which has been intensively studied for a century. Second, the equation is of great intrinsic interest in the study of solitary waves because (i) it is exactly integrable (ii) its solutions, including multiple solitons given in Camassa and Holm [1] are very simple, and (iii) it is novel in that its solitary waves have a discontinuous first derivative, in contrast to the great smoothness of most previously known species of solitary waves. Boyd [9] conjectured that the imbrication of the soliton was an exact solution only when the solitary wave was either an elliptic function or a rational function. (He showed that soliton imbrication fails, except as an approximation, for the quartically nonlinear KdV equation, whose solitons are hyperelliptic functions.) The coshoidal wave shows that this conjecture is false and too restrictive; the peakon is neither elliptic nor rational, but its imbrication is an exact periodic solution. There are probably other cases, as
Imbrication of Carnassa-Holm Peakons
185
y e t undiscovered, where t h e s o l i t o n - i m b r i c a t i o n is exact; a general principle to connect all cases now seems elusive. A P P E N D I X 1. A M A T L A B P R O G R A M PERTURBATION SERIES
TO COMPUTE
THE
m a x j = 6; a(1,1) = 1; a ( 2 , 1 ) = - 1 ; a ( 2 , 2 ) = 1; a v x x ( 2 , 1 ) = 1; a v x x ( 2 , 2 ) = - 1 . 5 ; avx(1,1) = - 1; for j = 3:maxj % Beginning of loop over p e r t u r b a t i o n order j % F i r s t a n d Second d e r i v a t i v e c o e f f i c i e n t s . . , used for larger j % v x x = sum[j = 2:infin] eps^(2j) sum[m = l:j + 1] avxx(j,m) sech(epsx) % ^(2m) % vx = t a n h ( e p s x ) sum[j = 2:infin] eps^(2j + 1) s u m [ m = l:j] avx(j,m) % sech(eps x)^(2m) form= l:j- 1 avxx(j,m) -- a v x x ( j , m ) + m* m*a(j - 1,m); a v x x ( j , m + 1) = - 0 . 5 * m * ( 2 * m + 1)*a(j - 1,m); avx(j - 1,m) = - m* a(j - 1,m); end % of m loop % Nonlinear c o n t r i b u t i o n to Rjk from - 1.5 v* v for m = 2:(j - 1) for p = l : m for q = l:(j + 1 - m) k=p+q; R(j,k) = R(j,k) - 1.5*a(m,p)*a(j + 1 - m,q); end % of q loop end % of p loop end % of m loop % Nonlinear c o n t r i b u t i o n to Rjk from v* vxx for m = l:(j - 1) for p = l : m for q = l:(j + 1 - m) k=p+q; R(j,k) = R(j,k) + a ( m , p ) * a v x x ( j + 1 - m,q); end % of q loop end % of p loop end % of m loop % Nonlinear c o n t r i b u t i o n of R j k from 0.5* vx* v x for m = l:(j - 1) for p = l : m for q = l:(j - p)
186
J . P . BOYD
k=p+q; glug = 0.5* avx(m,p)* avx(j - m,p); R(j,k) = R(j,k) + glug; R(j,k + 1) = R(j,k + 1) - glug; end % of q loop end % of p loop end % of m loop % Now compute the coefficients at j-th order. a(j,j) = R(j,j + 1)/3 - 0.5*j*(2*j + 1)); form=j:1:2 a(j,m- 1)= (R(j,m)- (m*m- 1)*a(j,m))/(30.25*(2* m - 2)*(2* m - 1)); end of loop in m % The leading order term in the residual is R(j,j + 1), which is proportional % to O(eps**(2*j + 2))* sech(eps x)**(2j + 2). end; % End of loop over perturbation order j
APPENDIX 2 LIST OF SYMBOLS B c C
scaling parameter phase speed ratio
K K/c m n P q r t u v w x X 8 K A /~ ~r
sum index sum index spatial period rescaled solution denominator of residual time time-dependent solution steadily-translating solution rescaled solution spatial coordinate coordinate in moving reference frame perturbation parameter parameter of CH equation scaling parameter limits of infinitesimal interval 3.14159
Imbrication of Camassa-Holm Peakons
187
This work was supported by the N S F under Grant 0CE9119459 and the Department of Energy through KC070101. I thank Roberto Camassa for reprints which inspired this note. REFERENCES 1 R. Camassa and D. D. Holm, An Integrable Shallow Water Equation with Peaked Solitons, Phys. Rev. Lett., 71:1661-1664 (1993). 2 T. B. Benjamin, J. L. Bona, and J. Mahoney, Phil. Trans. Royal Soc. Lond. A227, 47 (1972). 3 R. Camassa, D. D. Holm, and J. M. Hyman, A new integrable shallow water equation, in Advances in Applied Mechanics 31 (T.-Y. Wu and J. W. Hutchinson, Eds.), Academic, pp. 1-33, 1994. 4 F. Cooper and H. Shepard, Solitons in the Camassa-Holm Shallow Water Equation. To be published. 5 J.P. Boyd, Weakly Nonlocal Solitons for Capillary-Gravity Waves: Fifth-degree Korteweg-deVries Equation, Physica D48, 129-146 (1991). 6 J . P . Boyd, Weakly Nonlocal Solitary Waves and Other Exponentially Small Phenomena, Cambridge University Press, New York (1995). 7 J . P . Boyd, Solitons From Sine Waves: Analytical and Numerical Methods for Nonintegrable Solitary and Cnoidal Waves, Physica D21, 227-246 (1986). 8 M . J . Lighthill, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, New York. 9 J.P. Boyd, New directions in solitons and nonlinear periodic waves: polycnoidal waves, imbricated solitons, weakly non-local solitary waves and numerical boundary value algorithms, in Advances in Applied Mechanics, 27 (T.-Y. Wu and J. W. Hutchinson, Eds.), Academic, New York, pp. 1-82, 1989. 10 A. Parker, Periodic solutions of the intermediate long-wave equation: A nonlinear superposition principle. J. Math. Phys. A25:2005-2032 (1992). 11 K. W. Chow, Theta functions and dispersion relations of periodic waves. J. Phys. Soc. Japan 62:2007-2011 (1993). 12 J . P . Boyd and S. E. Haupt, Polycnoidal waves: Spatially periodic generalizations of multiple solitary waves. In Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX (A. R. Osborne, Ed.) North-Holland, Amsterdam, 1991, pp. 827-856. 13 A. Parker, On the periodic solution of the Burgers equation: A unified approach. Proc. R. Soc..London A438:113-132 (1992). 14 J . P . Boyd, The arctan/tan and Kepler-Burger mappings for periodic solutions with a shock, front, or internal boundary layer. J. Comp. Phys., 98:181-193 (1992).