The Peakon Limit of the N-Soliton Solution of the Camassa-Holm Equation

0.15 MB·English

by  Yoshimasa Matsuno

The Peakon Limit of the N-Soliton Solution of the Camassa-Holm Equation
Yoshimasa Matsuno∗
Division of Applied Mathematical Science,
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Graduate School of Science and Engineering,
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Yamaguchi University, Ube 755-8611
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n (Received )
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J

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2 WeshowthattheanalyticN-solitonsolutionoftheCamassa-Holm(CH)shallow-water

] model equation converges to the nonanalytic N-peakon solution of the dispersionless CH
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equation when the dispersion parameter tends to zero. To demonstrate this, we develop a
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i novel limiting procedure and apply it to the parametric representation for the N-soliton
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[ solution of the CH equation. In the process, we use Jacobi’s formula for determinants

1 as well as various identities among the Hankel determinants to facilitate the asymptotic
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1 analysis. We also provide a new representation of the N-peakon solution in terms of the
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0 Hankel determinants.
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KEYWORDS: Camassa-Holm equation, soliton, peakon, parametric representation
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∗E-mail address: [email protected]
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1. Introduction
The Camassa-Holm (CH) equation is a model equation describing the unidirectional
propagation of nonlinear shallow-water waves over a flat bottom.1−4) It may be written
in a dimensionless form as
u +2κ2u −u +3uu = 2u u +uu , (1.1)
t x txx x x xx xxx
where u = u(x,t) is the fluid velocity, κ is a positive parameter related to the phase
velocity of the linear dispersive wave and the subscripts t and x appended to u denote
partial differentiation. Like the Korteweg-de Vries equation in shallow-water theory, the
CH equation is a completely integrable nonlinear partial differential equation with a rich
mathematical structure. Because of this fact, a large number of works have been devoted
to the study of the equation from both physical and mathematical points of view. We do
not provide a comprehensive review on various consequences established for the equation.
Instead, we refer to an excellent survey given in ref. 5). Our main concern here is
solutions to the CH equation for both κ 6= 0 and κ = 0. Although equation (1.1) can
be transformed to the corresponding equation with κ = 0 by a Galilean transformation
x′ = x+κ2t,t′ = t,u′ = u+κ2, solutions under the same vanishing boundary condition
at infinity have quite different characteristics. In fact, in the case of κ 6= 0, it typically
exhibits analytic multisoliton solutions (the so-called N-soliton solution with N being an
arbitrary positive integer) like the usual soliton equations.6−15) We also remark that it
has singular cusp soliton solutions as well as solutions consisting of an arbitrary number
of solitons and cusp solitons.16−19) When κ = 0, on the other hand, eq. (1.1) becomes
a dispersionless version of the original CH equation. It admits a new kind of solitary
waves which have a discontinuous slope at their crest.1,2) For this unique feature, they
are now termed peakons. Hence, the peakon has a nonanalytic nature unlike the smooth
soliton. One of its remarkable properties is that the motion of peakons is described by
a finite dimensional completely integrable dynamical system. This fact was used in refs.
20 and 21 to construct the general N-peakon solution which represents the interaction
of N peakons. The careful inspection of the interaction of peakons reveals that it occurs
elastically in pairs and the total effect of the collision is the phase shift whose explicit
2
formula has been given in a closed form.2,9,21) See also a recent paper concerning the
detailed investigation of the dynamics of two peakons.22) An important issue about the
N-peakon solution is how one can reduce it from the analytic N-soliton solution by taking
the singular limit κ → 0. Although this problem has received considerable attention, it
hasbeenresolvedonlypartially. Indeed, inthecaseofN = 1,theconvergence ofthesingle
soliton to the single peakon has been demonstrated using the explicit form of the 1-soliton
solution.2,5) Also, an analysis without recourse to the explicit form of the soliton solution
has been carried out in a general context by employing an abstract theory of dynamical
systems.23,24)) However, the treatment of the general N-soliton case has remained open.
Quite recently, the case N = 2 was completed by two different ways. One will be given in
ref. 25 while the other is included here as a special version of the general N-soliton case.
The purpose of this paper is to demonstrate the convergence of the N-soliton solution
to the N-peakon solution for the general N by using the explicit N-soliton solution of the
CH equation. In §2, we present a parametric representation for the N-soliton solution
of the CH equation which offers a relevant form to the subsequent asymptotic analysis.
In §3, the convergence of the 1-soliton solution to the 1-peakon solution is exemplified to
explain the new idea used in the limiting procedure. In §4, we perform the corresponding
limit for the N-soliton solution and show that it reproduces the N-peakon solution given
by Beals et al.21) We also obtain a new representation of the N-peakon solution in terms
of the Hankel determinants. Section 5 is devoted to the concluding remarks. In Appendix
A, we give a proof of the formula which enables us to rewrite the tau-functions for the N-
soliton solution of the CH equation in terms of the Hankel determinants. In Appendix B,
we establish various identities among the Hankel determinants which are used effectively
to simplify the limiting waveform of the N-soliton solution.
2. The N-Soliton Solution of the CH Equation
The N-soliton solution of the CH equation (1.1) may be represented in a parametric
form11,15)
f
2
u(y,t) = ln , (2.1a)
f
(cid:18) 1(cid:19)t
y f
2
x(y,t) = +ln +d, (2.1b)
κ f
1
3
with
N
f = exp µ (ξ +φ )+ µ µ γ , (2.2a)
1 i i i i j ij
" #
µ=0,1 i=1 1≤i<j≤N
X X X
N
f = exp µ (ξ −φ )+ µ µ γ , (2.2b)
2 i i i i j ij
" #
µ=0,1 i=1 1≤i<j≤N
X X X
where
2κ2
ξ = k (y −κc t−y ), c = , (i = 1,2,...,N), (2.3a)
i i i i0 i 1−κ2k2
i
1−κk
e−φi = i, (0 < κk < 1), (i = 1,2,...,N), (2.3b)
i
1+κk
i
(k −k )2
eγij = i j , (i,j = 1,2,...,N;i 6= j). (2.3c)
(k +k )2
i j
Here, k and y are soliton parameters characterizing the amplitude and phase of the ith
i i0
soliton,respectively, c isthevelocityoftheithsolitoninthe(x,t)coordinatesystemandd
i
is an integration constant. We assume that c > 0 and c 6= c for i 6= j (i,j = 1,2,...,N).
i i j
In (2.2), the notation implies the summation over all possible combination of
µ=0,1
µ = 0,1,µ = 0,1,...,µ = 0,1. The following coordinate transformation (x,t) → (y,t′)
1 2 PN
has been introduced to parametrize the N-soliton solution
dy = rdx−urdt, dt′ = dt, (2.4a)
where the variable r is defined by the relation
r2 = u−u +κ2. (2.4b)
xx
Note in (2.1) that the time variable t′ is identified with the original time variable t by
virtue of (2.4a).
The 1-soliton solution is given by
2κ2c k2
u(y,t) = 1 1 , (2.5a)
1+κ2k2 +(1−κ2k2)coshξ
1 1 1
y (1−κk )eξ1 +1+κk
1 1
x(y,t) = +ln +d. (2.5b)
κ (cid:20)((1+κk1)eξ1 +1−κk1(cid:21)
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Itrepresents asolitarywave travelling totherightwiththeamplitudec −2κ2 andvelocity
1
c . The property of the 1-solitonsolutions has been explored indetail as well as its peakon
1
limit. See, for example ref. 5. For completeness, we write the explicit 1-peakon solution
of equation (1.1) with κ = 0. It reads
u(x,t) = c e−|x−c1t−x10|, (2.6)
1
where x represents the initial position of the peakon. In view of the invariance property
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of the dispersionless version of the CH equation under the transformation x′ = x,t′ =
−t,u′ = −u,italsoadmitsapeakonsolution(2.6)withanegativeamplitude. Itrepresents
a peakon with depression propagating to the left.
3. The Peakon Limit of the 1-Soliton Solution
We first consider the limiting procedure for the 1-soliton solution. This will be helpful
tounderstandthebasicideaindevelopingtheprocedureforthegeneralN-solitonsolution.
We start with the tau-functions for the 1-soliton solution which are the most important
constituents in our analysis. These are given by (2.2) with N = 1. Before proceeding to
the limit, we find it convenient to shift the phase constant y as y → y +φ /k or in
10 10 10 1 1
terms of the phase variable ξ → ξ −φ so that
1 1 1
f = 1+eξ1, (3.1a)
1
f = 1+ν2eξ1, (3.1b)
2 1
where we have put ν = e−φ1. The limit κ → 0 is taken in such a way that the amplitudes
1
of the soliton and peakon given respectively by c − 2κ2 and c coincide and remain
1 1
finite.5) This can be attained by taking the limit κ → 0 with the soliton velocity c in the
1
(x,t) coordinate system being fixed. We see from (2.3a) that the appropriate limit of the
wavenumber k is carried out by taking κk → 1. The limiting procedure described here
1 1
may be called the peakon limit. It now follows from (2.3) that various wave parameters
have the following leading-order asymptotics in the peakon limit
κ2 κ2 λ
κk ∼ 1− , ν ∼ = 1κ2, (3.2a)
1 1
c 2c 4
1 1
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f
eξ1 = eκk1(κy−c1t−x10) ∼ 1ex−c1t−x10, (3.2b)
f
2
where λ = 2/c and x = y /κ. In passing to the last line of (3.2b), we have used (2.1b)
1 1 10 10
to eliminate the y variable and the constant d has been absorbed in the phase constant
x . Substituting (3.2b) into (3.1), the leading terms of f and f are found to be as
10 1 2
f
1
f ∼ 1+ z , (3.3a)
1 1
f
2
f
f ∼ 1+ǫ2λ2 1z , (3.3b)
2 1f 1
2
where we have put
κ2
z = ex−c1t−x10, ǫ = , (3.3c)
1
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for simplicity. If we introduce the new quantity f by f = f /f , we can deduce from (3.3)
2 1
that
f +ǫ2λ2z
f ∼ 1 1. (3.4)
f +z
1
The expression (3.4) yields the following quadratic equation for f:
f2 +(z −1)f −ǫ2λ2z +O(ǫ3) = 0. (3.5)
1 1 1
If we solve this equation, we can express f as a function of x and t. At this instant, it
is crucial to observe that f > 0 since both f and f are positive quantities. A positive
1 2
solution is then substituted into (2.1a) to obtain u. In performing the differentiation,
however, we must replace the t derivative ∂/∂t by ∂/∂t+u∂/∂x in accordance with the
coordinate transformation (2.4a). With this notice in mind, we can rewrite (2.1a) as
∂ ∂
u = +u ln f. (3.6)
∂t ∂x
(cid:18) (cid:19)
Solving (3.6) with respect to u to express it in terms of a single variable f and its deriva-
tives, we have
f
t
u = . (3.7)
f −f
x
The derivatives f and f in (3.7) are obtained simply if one differentiates (3.5) by t and
t x
x, respectively. To be more specific, as ǫ → 0
c z f −ǫ2c λ2z
f ∼ 1 1 1 1 1, (3.8a)
t
2f +z −1
1
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−z f +ǫ2λ2z
f ∼ 1 1 1, (3.8b)
x
2f +z −1
1
where we have used the relations z = −c z and z = z which are derived from (3.3c).
1,t 1 1 1,x 1
After a manipulation using (3.5) and (3.8), we arrive at the expression of u in terms of f:
−c f +c
1 1
u ∼ . (3.9)
f +z
1
The final step of the limiting process is to solve (3.5) under the condition f > 0 and
then take the limit κ → 0 after substituting a positive solution into (3.9). Although the
analytical expression is obtained for f by quadrature, we need only the series solution.
This fact is crucial in developing the peakon limit of the general N-soliton solution where
equation corresponding to (3.5) becomes an algebraic equation of degree N + 1 whose
analytical solution is in general not available.
Now, we expand f in powers of ǫ as
f = f(0) +ǫf(1) +ǫ2f(2) +..., (3.10)
and insert this expression into (3.5). Comparing the coefficients of ǫn(n = 0,1,...), we
obtain a system of algebraic equations for f(n), the first two of which read
f(0)2 +(z −1)f(0) = 0, (3.11a)
1
2f(0)f(1) +(z −1)f(1) = 0, (3.11b)
1
(z −1)f(2) −λ2z = 0. (3.11c)
1 1 1
The above equations can be solved immediately to obtain positive solutions.. Indeed, if
z ≤ 1 ( x−c t−x ≤ 0), then
1 1 10
f ∼ f(0) = 1−z . (3.12a)
1
Substitution of this result into (3.9) yields
u ∼ c z = c ex−c1t−x10. (3.12b)
1 1 1
If, on the other hand, z > 1(or x−c t−x > 0), then
1 1 10
f(0) = f(1) = 0, f ∼ f(2)ǫ2 = λ2z /(z −1)ǫ2, (3.13a)
1 1 1
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u ∼ c z−1 = c e−(x−c1t−x10). (3.13b)
1 1 1
It follows from (3.12) and (3.13) that in the limit κ → 0 (or equivalently ǫ → 0 by (3.3c))
u has a limiting waveform
u = c e−|x−c1t−x10|. (3.14)
1
This expression is just the 1-peakon solution (2.6) of the CH equation with κ = 0.
4. The Peakon Limit of the N-Soliton Solution
4.1 Formulas for determinants
The peakon limit of the general N-soliton solution can be taken along the lines of
the 1-soliton case. However, the calculation involved is quite formidable. To perform the
calculation in an effective manner, we first define the following determinants which are
closely related to the N-peakon solution:
2
1 1 ··· 1
λ λ ··· λ
∆n(i1,i2,...,in) = (cid:12)(cid:12)(cid:12) ...i1 ...i2 ... ...in (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)λn−1 λn−1 ··· λn−1(cid:12)
(cid:12) i1 i2 in (cid:12)
(cid:12) (cid:12)
= (cid:12)(cid:12) (λil −λim)2, (cid:12)(cid:12) (n ≥ 2), (4.1)
1≤l<m≤n
Y
A A ··· A
m m+1 m+n−1
A A ··· A
Dn(m) = (cid:12)(cid:12)(cid:12) m...+1 m...+2 ... m...+n (cid:12)(cid:12)(cid:12). (4.2a)
(cid:12) (cid:12)
(cid:12)Am+n−1 Am+n ··· A2(n−1)+m(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Here
(cid:12) (cid:12)
(cid:12) N (cid:12)
A = λmE , (4.2b)
m i i
i=1
X
y 2
Ei = eλ2it+xi0, xi0 = i0, λi = , (i = 1,2,...,N), (4.2c)
κ c
i
where in (4.2)mis anarbitraryinteger andn is anonnegative integer less thanor equal to
(m) (m) (m)
N. For n greater than N, D = 0. We use the convention ∆ (i ) = 1,D = 1,D =
n 1 1 0 1
(m)
A . The quantity ∆ is the square of the Vandermonde determinant whereas D is the
m n n
determinant of a symmetric matrix. It is a Hankel determinant. We use some properties
of Hankelians in the following analysis.
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(m)
Let D (i ,i ,...,i ;j ,j ,...,j ) (i < i < ... < i , j < j < ... < j , 1 ≤ p,q < N)
n 1 2 p 1 2 q 1 2 p 1 2 q
(m)
be a determinant which is obtained from D by deleting rows i ,i ,...,i and columns
n i 2 p
j ,j ,...,j , respectively. Then, the following Jacobi formula holds which will play a
1 2 q
central role in the present analysis:26)
(m) (m) (m) (m) (m) (m)
D D (1,n+2;1,n+2) = D (1;1)D (n+2;n+2)−D (1;n+2)D (n+2;1).
n+2 n+2 n+2 n+2 n+2 n+2
(4.3)
By virtue of the definition (4.2), we see that
D(m)(1,n+2;1,n+2) = D(m+2), (4.4a)
n+2 n
(m) (m+2)
D (1;1) = D , (4.4b)
n+2 n+1
(m) (m)
D (n+2;n+2) = D , (4.4c)
n+2 n+1
(m) (m) (m+1)
D (1;n+2) = D (n+2;1) = D . (4.4d)
n+2 n+2 n+1
Hence, (4.3) can be rewritten in the form
2
D(m)D(m+2) = D(m+2)D(m) − D(m+1) . (4.5)
n+2 n n+1 n+1 n+1
(cid:16) (cid:17)
The determinant D(m) has an alternative expression in the form of a finite sum21)
n
D(m) = ∆ (i ,i ,...,i )(λ λ ...λ )mE E ...E , (n = 1,2,...,N). (4.6)
n n 1 2 n i1 i2 in i1 i2 in
1≤i1<i2X<...<in≤N
(m)
This formula is very useful in rewriting the tau-functions. Note that D (n = 1,2,...,N)
n
are positive definite since ∆ > 0 and E > 0 (i = 1,2,...,N). We give a simple proof of
n i
(4.6) in Appendix A.
4.2 The peakon limit of the N-soliton solution
We first shift the phase variables as ξ → ξ − φ (i = 1,2,...,N) in (2.2) and take
i i i
the peakon limit κk → 1 with fixed c (i = 1,2,...,N). Using (3.2) and the asymptotic
i i
eγij ∼ (λ −λ )2κ4/16, the leading-order asymptotics of the tau-functions f and f can
i j 1 2
be written in the form
N n
f
f ∼ 1+ ǫn(n−1) 1 ∆ (i ,i ,...,i )z z ...z , (4.7a)
1 f n 1 2 n i1 i2 in
Xn=1 (cid:18) 2(cid:19) 1≤i1<i2X<...<in≤N
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N n
f
f ∼ 1+ ǫn(n+1) 1 (λ λ ...λ )2∆ (i ,i ,...,i )z z ...z , (4.7b)
2 f i1 i2 in n 1 2 n i1 i2 in
Xn=1 (cid:18) 2(cid:19) 1≤i1<i2X<...<in≤N
where
z = ex−cit−xi0 = exE−1, (i = 1,2,...,N). (4.7c)
i i
Furthermore, to compare the limiting form of u resulting from the peakon limit with the
N-peakon solution given by ref. 21), we shift the phase constant appropriately, so that
N λ2 z
z → i=1 i i , (i = 1,2,...,N). (4.8)
i 2 N (λ −λ )2λ2
jQ=1 i j i
(j6=i)
Q
We substitute (4.8) into (4.7) and use (4.6) to modify them into the form
N n
f
f ∼ 1+ ǫn(n−1) 1 d enxD(2) , (4.9a)
1 f n N−n
n=1 (cid:18) 2(cid:19)
X
N n
f
f ∼ 1+2 ǫn(n+1) 1 d enxD(0) , (4.9b)
2 f n+1 N−n
n=1 (cid:18) 2(cid:19)
X
where the positive coefficients d are defined by
n
N 2(n−1)
λ
d = d (t) = i=1 i , (n = 1,2,...,N). (4.9c)
n n 2n∆ N E
Q N i=1 i
Thus, equation corresponding to (3.5) bQecomes an algebraic equation of degree N +1
N+1
ǫn(n−1)h fN−n+1 +O(ǫN(N+1)+1) = 0. (4.10a)
n
n=0
X
Here, as in the 1-soliton case f = f /f and the coefficients h are defined by the relations
2 1 n
h = 1, (4.10b)
0
h = d e(n−1)x exD(2) −2D(0) , (n = 1,2,...,N), (4.10c)
n n N−n N−n+1
(cid:16) (cid:17)
N λ2NeNx
h = − i=1 i . (4.10d)
N+1 2N∆ N E
Q N i=1 i
Q
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