Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

Fengfeng Dong; Lingjun Zhou

doi:10.1080/14029251.2018.1452674

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Volume 25, Issue 2, March 2018, Pages 290 - 308

Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

Authors

Fengfeng Dong

School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China,1433494@tongji.edu.cn

Lingjun Zhou

School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China,zhoulj@tongji.edu.cn

Received 19 November 2017, Accepted 17 December 2017, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1452674 How to use a DOI?
Keywords: peakon; two-component Camassa-Holm system; continued fractions; inverse spectral approach
Abstract: In this paper, we are concerned with the explicit construction of peakon solutions of the integrable twocomponent system with cubic non-linearity. We establish the spectral and inverse spectral problem associated to the Lax pairs of the system. The inverse problem is solved by the classical results of Stieltjes continued fractions, which also contributes a lot to the spectral problem. The explicit formulas are obtained from solutions of the inverse problem. The positivity of the spectral measures is implied by J. Moser's work on the Jacobi spectral problem.
Copyright: © 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

In this paper, we consider the following two-component system

{mt=12[m(uv-uxvx)]x-12m(uvx-uxv),nt=12[n(uv-uxvx)]x+12n(uvx-uxv),m=u-uxx, n=v-vxx. (1.1)

The completely integrable system (1.1) is proposed by Xia and Qiao in [45], it admits bi-Hamiltonian structure and infinity many conservation laws and also integrable in the sense of geometry for describing pseudo-spherical surfaces. The blow-up scenario, persistence properties and unique continuation have been discussed in [23,46].

When v = −2, (1.1) is reduced to the Camassa-Holm (CH) equation

mt+mxu+2mux=0, m=u-uxx, (1.2)

which models the unidirectional propagation of shallow water waves over a flat bottom [5,6,14,26] and can also be derived as the tri-Hamiltonian duality of the Korteweg-de Vries (KdV) equation [36]. Compared with the KdV equation, the CH equation has the advantage of admitting peaked solitons (peakons) of the form c e^−|^x⁻^ct^|, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations for water waves in irrotational flow [11–13], and modeling wave breaking [4,6,7].

Since the discovery of the CH equation and its peakon solutions, partial differential equations admitting peakons have attracted much attention in the past two decades. Several equations, such as Degasperi-Procesi (DP) equation [18], V. Novikov equation [24,35], modified CH equation [37] and the Geng-Xue (G-X) equation [19], have similar properties; see [20,38,40,42–45] for other multicomponent versions, and [1,17,21,39] for non-integrable peakon equations. These equations are interesting for introducing many challenging problems, including existence, uniqueness, stability and breakdown of solutions, etc. We recall that the CH (periodic) peakons [8,31] and DP peakons [32] are stable in the sense that their shape is stable under small perturbations, which shows the peakons are detectable. Besides, the peakon interactions were a key ingredient in the development of the theory of continuation after blow-up of global weak solutions of the CH equation [2,22].

It was shown in [3,4,9,10] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation. Particularly, peakon dynamic system of (1.2) was explicitly solved in terms of the classical theory of Stieltjes continued fractions [3] and their interactions were investigated via orthogonal polynomials [4]. Because of the Lax integrability, multipeakons to other peakon equations had also been explicitly formulated via the inverse spectral method (for details, see [15,16,25,27–29]). All of these work shows that the mathematics of peakons are closely connected with significant topics and theories in classical analysis, such as sting problem, Stieltjes continued fractions, orthogonal polynomials, Padé approximation, oscillatory kernels, etc.

Back to (1.1), we remark that it is not equivalent to the two-component modified CH equation [40] for (1.2) can be reduced to (1.1) and the latter cannot. We now review the work on peakons of (1.1) in [45]. The two-component Camassa-Holm system (1.1) admits two-component peakons:

u(x,t)=∑k=1Nmk(t)e-|x-xk(t)|, v(x,t)=∑k=1Nnk(t)e-|x-yk(t)|, (1.3)

where mknk=0,mk2+nk2≠0 . In this case, m and n will be discrete measures with disjoint supports:

m=u-Dx2u=2∑i=1Nmiδxi, n=v-Dx2v=2∑j=1Nnjδyj. (1.4)

The peakon ansatz (1.3) satisfies (1.1) if and only if x_k(t), y_k(t), m_k(t) and n_k(t) satisfy the following peakon ODEs:

{x˙k=-12∑i,j=1Nminj(1-sgn(xi-xk)sgn(yj-xk))e-|xk-xi|-|xk-yj|,y˙k=-12∑i,j=1Nminj(1-sgn(xi-yk)sgn(yj-yk)e-|yk-xi|-|yk-yj|,m˙k=-12mk∑i,j=1Nminj(sgn(yj-xk)-sgn(xi-xk))e-|xk-xi|-|xk-yj|,n˙k=12nk∑i,j=1Nminj(sgn(yj-yk)-sgn(xi-yk))e-|yk-xi|-|yk-yj|, (1.5)

for k = 1,2,…,N. The 1 + 1 peakons can be integrated directly, while the peakon ODEs (1.5) with N > 1 becomes considerably complicated, which makes it difficult to obtain the explicit solution formulas.

This paper is devoted to deriving the explicit formulas for the solutions to (1.5) using the inverse spectral method. We will only deal with the interlacing case where

u(x,t)=∑k=1Km2k-1(t)e-|x-x2k-1(t)|, v(x,t)=∑k=1Kn2k(t)e-|x-x2k(t)|, (1.6)

with

m2k-1,n2k>0(1≤k≤K), (1.7a)

x2k-1<x2k(1≤k≤K), x2k<x2k+1(1≤k≤K-1). (1.7b)

The work pertinent to this paper is [29,30]. In [29], Lundmark and Szmigielski studied a third order non-selfadjoint boundary value problem coming from the Lax pair(s) of the G-X equation

mt+(3mux+mxu)v=0, m=u-uxx,nt+(3nvx+nxv)u=0, n=v-vxx, (1.8)

and solved the inverse spectral problem associated with two positive interlacing discrete measures m and n:

m=2∑k=1Km2k-1δx2k-1, n=2∑k=1Kn2kδx2k. (1.9)

The peakons of the form (1.6) were obtained explicitly. The dynamics of interlacing peakons were analyzed in [30].

The rest of our paper is organized as follows. In section 2, we give Lax pairs of (1.1), and set up our approach to deriving explicit formulas for solutions of the peakon ODEs (1.5) in the interlacing case. In Section 3, we study two forward spectral problems associated with positive interlacing discrete measures. In Section 4, we solve the inverse spectral problem via Stieltjes continued fractions, and give two constants of motion. In Section 5, the solutions of the inverse problem construct explicitly the interlacing peakons of (1.1), involving Hankel determinants associated to two different spectral measures.

2. Preliminaries

In this section, we recall the Lax integrability of (1.1), state the inverse spectral method applied in our paper.

Equation (1.1) can be derived as the compatibility condition of the linear spectral problem [45]:

(ϕ1ϕ2)x=12(-1λm-λn1)(ϕ1ϕ2), (2.1a)

(ϕ1ϕ2)t=-12(2λ-2+12Q+12P-λ-1(u-ux)-12λmQλ-1(v+vx)+12λnQ-12Q-12P)(ϕ1ϕ2), (2.1b)

where λ is a spectral parameter, m = u − u_xx, n = v − v_xx, P = uv_x − u_xv and Q = uv − u_xv_x, but also as the compatibility condition of a different linear spectral problem obtained by interlacing u and v:

(ϕ1ϕ2)x=12(-1λn-λm1)(ϕ1ϕ2), (2.2a)

(φ1φ2)t=-12(2λ-2+12Q-12P-λ-1(v-vx)-12λnQλ-1(u+ux)+12λmQ-12Q+12P)(φ1φ2). (2.2b)

We call (2.2) the twin Lax pair with respect to the Lax pair (2.1) of (1.1).

Under the gauge transformation

ψ1=φ1ex2, ψ2=φ2e-x2, (2.3)

(2.1a) and (2.2a) become the following simpler counterparts, respectively,

(ψ1ψ2)x=(0λg-λh0)(ψ1ψ2), m=2ge-x, n=2hex, (2.4)

(ψ1ψ2)x=(0λh^-λg^0)(ψ1ψ2), m=2g^ex, n=2h^e-x. (2.5)

We will recover m and n from spectral data obtained by imposing boundary condition

ψ1(-∞)=ψ2(+∞)=0 (2.6)

on equations (2.4) and (2.5) when m and n are positive discrete meatures given by (1.9). In this case,

hk=n2ke-x2k, gk=m2k-1ex2k-1, h^k=n2kex2k, g^k=m2k-1e-x2k-1. (2.7)

For later convenience, we make convention that x₀ = − ∞, x₂_K _{+ 1} = + ∞.

To describe our approach specifically, we define the spectral maps as follows:

Definition 2.1

(Forward and inverse spectral maps). Let P⊂𝕉4K be the set of tuples

p=(h1,g1,h^1,g^1,…,hK,gK,h^K,g^K)

satisfying

hk,gk,h^k,g^k>0, k=1,…,K.

Let R⊂𝕉4K be the set of tuples

r=(λ12,⋯,λK2;λ^12,⋯,λ^K-12;a1,…,aK;b1,…,bK-1;γ,γ˜)

satisfying

0<λ12<⋯<λK2, 0<λ^12<⋯<λ^K-12, γ,γ˜>0,ai>0,i=1,…,K, bj>0,j=1,…,K-1. (2.8)

We denote the forward spectral map by

S:P→R, (2.9)

which will be given in Section 3, and the the inverse spectral map

T:R→P (2.10)

will be defined in Section 4.

According to the Theorems 3.1, 3.4, 4.4, 4.8 and 4.7, the definition above is suitable. Using the relation (2.7), we have the following relations:

x2k=12ln(h^khk), x2k-1=12ln(gkg^k), (2.11a)

n2k=(h^khk)12, m2k-1=(g^kgk)12. (2.11b)

By (2.11a), (2.11b), we can recover the positive interlacing discrete measures (1.9), and give the interlacing peakons (1.6) locally by Theorem 5.1.

Finally, we introduce two types of half-strictly interlacing relation [29], which will be used in Section 3 to describe the solutions of spectral problems.

Definition 2.2.

Let ([1,k]l) be the set of l-element subsets of the integer interval [1, k] = {1,…,k}, I,J∈(([1,k]l) or I∈([1,k]l),J∈([1,k]l-1) , then

I≼J⇔i1≤j1<i2≤j2<⋯<il≤jl or i1≤j1<i2≤j2<⋯<il,I≼J⇔i1<j1≤i2<j2≤⋯≤il<jl or i1<j1≤i2<j2≤⋯≤il,

depending on whether |I| − |J| = 0 or 1.

3. Forward Spectral Problem

In this section, our goal is to define the forward spectral map (2.9). Suppose that hk,gk,h^k,g^k>0(1≤k≤K) , we obtain the corresponding spectral data, a 4K-tuple r ∈ R. To simplify the structure of the paper, γ˜ will be given separately in Section 4.

3.1. Spectrum

Imposing boundary condition (2.6) on (2.4) and (2.5), respectively, we obtain two spectral problems, solving them in the discrete case will give all the 2 K − 1 pairs of eigenvalues.

Definition 3.1

(Eigenvalue). The eigenvalues of the interlacing peakon spectral problem are those values of λ for which (2.4) or (2.5), in the case when m and n are positive discrete meatures with interlacing supports, has nontrivial solutions satisfying (2.6).

Spectral problem I.

(ψ1ψ2)x=(0λg-λh0)(ψ1ψ2), ψ1(-∞)=ψ2(+∞)=0, (3.1)

where

g=∑k=1Kgkδx2k-1, h=∑k=1Khkδx2k.

Since g and h are interlacing discrete measures, we have the following facts:

•

For any 0 ≤ k ≤ K, ψ₁ is constant in the interval x₂_k _{− 1} < x<x₂_k _{+ 1} while ψ₂ remains constant in x₂_k < x< x₂_k _{+ 2}.
•

For any 1 ≤ k ≤ K, ψ₁ has a jump λ g_k ψ₂(x₂_k _{− 1}) at the point x₂_k _{− 1} while ψ₂ has a jump −λh_kψ₁(x₂_k) at x₂_k.

Let q_k = ψ₁(x₂ _k _{+ 1} −), p_k = ψ₂(x₂ _k −), then the problem (3.1) becomes

{qk-qk-1=λgkpk,1≤k≤K,pk+1-pk=-λhkqk,1≤k≤K,q0=pK+1=0, (3.2)

where q_K = lim_x _{→ +} _∞ψ₁(x), p_K _{+ 1} = lim_x _{→ +} _∞ψ₂(x). When lim_x _→−∞ψ₂(x) = 1, consider the finite difference problem

{qk-qk-1=λgkpk,1≤k≤K,pk+1-pk=-λhkqk,1≤k≤K,q0=0, p1=1. (3.3)

Proposition 3.1.

The problem (3.3) has a unique solution, which can be given by

qk=∑l=1k(-1)l+1λ2l-1∑I∈([1,k]l),J∈([1,k]l-1)I≼JgIhJ,pk+1=1+∑l=1k(-1)lλ2l∑I,J∈([1,k]l)I≼JgIhJ,

where gI=∏i∈Igi, hJ=∏j∈Jhj .

Proof. According to (3.3),

(qkpk+1)=Tk(λ)⋯T1(λ)(01), Tk(λ)=(1λgk-λhk1-λ2gkhk).

Using induction, we obtain the conclusion.

Spectral problem II.

(ψ^1ψ^2)x=(0λh^-λg^0)(ψ^1ψ^2), ψ^1(-∞)=ψ^2(+∞)=0, (3.4)

where

h^=∑k=1Kh^kδx2k, g^=∑k=1Kg^kδx2k-1.

Let q^k=ψ^1(x2k-2+),p^k=ψ^2(x2k-1+) , then the problem (3.4) becomes

{q^k+1-q^k=λh^kp^k,1≤k≤K,p^k-p^k-1=-λg^kq^k,1≤k≤K,q^1=p^K=0. (3.5)

When lim_x _→−∞ψ₂(x) = 1, the finite difference problem

{q^k+1-q^k=λh^kp^k,1≤k≤K,p^k-p^k-1=-λg^kq^k,1≤k≤K,p^0=1, q^1=0, (3.6)

has a unique solution, which can be written as

(p^Kq^K+1)=T^K(λ)⋯T^1(λ)(10), T^k(λ)=(1-λg^kλh^k1-λ2g^kh^k).

By induction, we have the following proposition as an analogue of Proposition 3.1.

Proposition 3.2.

Let g^I=∏i∈Ig^i, h^J=∏j∈Jh^j , then

p^k=1+∑l=1k-1(-1)lλ2l∑I∈([2,k]l),J∈([1,k]l)J≼Ig^Ih^J,q^k+1=∑l=1k(-1)lλ2l-1∑I∈([2,k]l-1),J∈([1,k]l)J≼Ig^Ih^J.

By Definition 3.1, the spectra of the interlacing peakon spectral problem are zeros of p_K _{+ 1} and p^K .

Theorem 3.1.

If all g_k, h_k, g^k and h^k are positive, then the spectra of (3.1) and (3.4) are non-zero, real, simple and appear in pairs with opposite signs. The eigenvalues can be denoted by

0<λ12<⋯<λK2, 0<λ^12<⋯<λ^K-12.

The Theorem 3.1 follows from the following algebraic lemma on the Jacobi matrix:

𝔍=(0d10⋯00d10d2⋯000d20⋯00⋮⋮⋮⋱⋮⋮000⋯0dN-1000⋯dN-10), d1,…,dN-1>0. (3.7)

Lemma 3.1.

Let Δ_k(λ) denote the k-th order principal minor of λI-𝔍 , with the proviso that Δ₀ = 1, then the roots of Δ_N(λ) = 0 are real and simple.

Proof. It is easy to obtain the recurrence formula

Δk+1(λ)=λΔk(λ)-dk2Δk-1(λ), (k≥1),

Therefore, none of adjacent polynomials in the following sequence

ΔN,ΔN-1,…,Δ1,Δ0

vanishes simultaneously for Δ₀ ≡ 1. For any eigenvalue λ_j, Δ_N(λ_j) = 0, then Δ_N₋₁(λ_j) ≠ 0. Thus, rank (λjI-𝔍)=N-1 , so that the geometric multiplicity of λ_j is 1. Since 𝔍 is real and symmetric, the multiplicity of λ_j equals 1. Therefore, the eigenvalues of 𝔍 are real and simple, which completes the proof.

Remark 3.1.

Let K = diag (1,−1,1,−1,…), then K ^{− 1} JK = − J. Thus

ΔN(-λ)=det(-λI-𝔍)=det(K-1(-λI-𝔍)K)=det(-λI+𝔍)=(-1)NΔN(λ),

taking determinants, we have Δ_N(λ) = (−1)^N Δ_N(−λ). Therefore, when N is even, the roots of Δ_N(λ) = 0 are non-zero, real, simple and appear in pairs with opposite signs.

The proof of Theorem 3.1. According to Lemma 3.1 and its remark, we only need to show that (3.2) and (3.5) are equivalent to some Jacobi matrix eigenvalue problems, respectively.

For the first one, we can rewrite (3.2) as J¯ω¯=λω¯ , where

ω¯=(-g112p1,-h112q1,g212p2,h212q2,…,(-1)KgK12pK,(-1)KhK12qK)t,

J¯ has the form of (3.7) with N = 2K and

d¯2k-1=(gkhk)-12, d¯2k=(gk+1hk)-12, k=1,2…,K.

For the second one, we would like to make some modifications first. Note that (3.5) has nontrivial solutions if and only if the following spectral problem has nontrivial solutions:

{q^k+1-q^k=λh^kp^k,1≤k≤K-1,p^k-p^k-1=-λg^kq^k,2≤k≤K,q^1=p^K=0. (3.8)

Therefore, (3.5) is equivalent to (3.8), which can be written as

J˜ω˜=λω˜, (3.9)

where

ω˜=(h^112p^1,g^212q^2,-h^212p^2,-g^312q^3,…,(-1)Kh^K-112p^K-1,(-1)Kg^K12q^K)t,

J˜ has the form of (3.7) with N = 2 K − 2 and

d˜2k-1=(g^k+1h^k)-12, d˜2k=(g^k+1h^k+1)-12, k=1,2…,K-1.

This completes the proof of Theorem 3.1.

Remark 3.2.

Since pK+1(0)=p^K(0)=1 , we have

pK+1=∏j=1K(1-λ2λj2), p^K=∏j=1K-1(1-λ2λ^j2).

3.2. Weyl functions

We now define Weyl functions associated to the boundary value problems (3.1) and (3.4) as the following two rational functions:

W(λ)=ψ1(∞,λ)λψ2(∞,λ)=qK(λ)λpK+1(λ), (3.10)

W^(λ)=ψ^1(∞,λ)λψ^2(∞,λ)=q^K+1(λ)λp^K(λ). (3.11)

By Propositions 3.1, 3.2 and 3.2 and Theorem 3.1, we have the following theorem.

Theorem 3.2.

If all g_k, h_k, g^k and h^k are positive, then Weyl functions W(λ), W^(λ) have partial fraction decompositions:

W(λ)=∑i=1Kaiλi2-λ2,

W^(λ)=γ+∑j=1K-1bjλ^j2-λ2. (3.12)

It is elementary to verify that the compatibility conditions of both the Lax pair (2.1) and the twin Lax pair (2.2) with m and n given by (1.4), are equivalent to the non-overlapping peakon ODEs (1.5). Therefore, the interlacing case of (1.5) can induce the evolution of spectral data defined above.

Theorem 3.3

(Evolution). Let m and n be positive measures given by (1.9), x_{2k − 1}, m_{2k − 1}, x_2k and n_2k satisfy (1.5) in the interlacing case, then

daidt=-aiλi2, dbjdt=-bjλ^j2, dλidt=0, dλ^jdt=0, dγdt=0. (3.13)

Hence, λ_i, λ^j and γ are constants of motion.

Proof. Easy to check that both sides of (2.1b) and (2.2b) vanish when ψ₁(− ∞) = 0, ψ₂(− ∞) = 1, i.e. the boundary condition imposed when defining the spectral data is consistent with the time evolution.

For x > x₂_K, substitute ψ₁(x, λ) = q_K(λ), ψ₂(x, λ) = p_K _{+ 1} into (2.1b), we obtain

dqKdt=-qKλ2+LλpK+1, dpK+1dt=0, L=∑k=1Kgk. (3.14)

Similarly, substituting ψ^1(x,λ)=q^K+1,ψ^2(x,λ)=p^K into (2.2b) leads to

dq^K+1dt=-q^K+1λ2+L^λp^K, dp^Kdt=0, L^=∑k=1Kh^k. (3.15)

Thus, the eigenvalues λ_i, λ^j are time invariant, and

dWdt=-Wλ2+Lλ2. (3.16)

dW^dt=-W^λ2+L^λ2, (3.17)

Taking residues of both sides of (3.16) and (3.17) at λ_i, λ^j , respectively, we obtain the evolution of a_i, b_j. Computing limλ→∞dW^dt leads to dγdt=0 , i.e. γ is a constant of motion.

Corollary 3.1.

The quantities

Mk=∑I,J∈([1,k]k)I≼JgIhJ,k=1,…K

and

M^k=∑I∈([2,k]k),J∈([1,k]k)J≼Ig^Ih^J,k=1,…K-1

form 2 K − 1 constants of motion for the peakon ODEs (1.5) in the interlacing case.

Proof. From (3.14) and (3.15), the coefficients of p_K _{+ 1} and p^K are time invariant, therefore, the conclusion follows Propositions 3.1 and 3.2.

Theorem 3.4.

If all g_k, h_k, g^k and h^k are positive, then a_i, b_j in Theorem 3.2 are positive.

The proof of Theorem 3.4 is closely related to the following results of J. Moser on Jacobi spectral problem [33,34].

Lemma 3.2

(J. Moser). Let f(λ) be the (N, N) entry of (λI-𝔍)-1 , i.e.

f(λ)=ΔN-1(λ)/ΔN(λ),

then f(λ) has partial fraction decomposition

f(λ)=∑j=1l2λrjλ2-λj2+κNrl+1λ, rj>0, (3.18)

where l = [N / 2], κ_N = 1 when N is odd and κ_N = 0 when N is even.

The proof of Theorem 3.4. Due to the equivalence established in the proof of Theorem 3.1, we can obtain the relations between Weyl functions of interlacing peakon problems (3.1) and (3.4) and some rational functions admitting the properties above.

There is a bit of difficulty in proving that b_j > 0, we deal with it first. Let e˜=(0,…,0,1) be a row vector of dimension 2K − 2, then (3.8) without p^K=0 can be written as

e˜t(-1)K+1p^K/g^K12=(λI-J˜)ω˜,

Denote the (2K − 2,2K − 2) entry of (λI-J˜)-1 by W˜(λ) , then -g^Kq^K/p^K=W˜(λ) . Hence,

W˜(λ)=-λg^KW^(λ)+λh^Kg^K. (3.19)

By Lemma 3.2, W˜(λ)=∑j=1K-12λr˜jλ2-λ^j2(r˜j>0) , taking residue of (3.19) at λ^j leads to 2r˜j=g^Kbj , therefore, bj>0

The proof of a_i > 0 is analogous. Denote the (2K, 2K) entry of (λI-J¯)-1 by W¯(λ) , then W¯(λ)=-λhKW(λ) , which leads to a_i > 0.

Remark 3.3.

Theorems 3.1 and 3.4 can also be obtained in the theory of orthogonal polynomials.

Remark 3.4.

γ˜ will be given as certain duality of γ and shown to be time invariant at the end of Section 4, and γ,γ˜>0 will follow from Theorem 4.3.

We have now completed the spectral characterization of the boundary value problems (3.1) and (3.4), and the construction of the forward spectral map (2.9).

4. Inverse Spectral Problem

In this section, with the help of Stieltjes continued fractions, we will solve the inverse spectral problem of recovering two pairs of discrete interlacing measures g, h and g^,h^ from the spectral data {λi2,λ^j2,ai,bj,γ,γ˜} that they give rise to.

We first introduce two theorems to characterize the properties of Weyl functions W(λ) and W^(λ) .

Theorem 4.1.

Let z = −λ², μ_i = λ_i², vj=λ^j2 , define two spectral measures as follows:

dα(μ)=∑i=1Kaiδμi, dβ(v)=∑j=1K-1bjδvj,

then W(λ) and W^(λ) admit the following integral representations

W(λ)=∫dα(μ)z+μ, W^(λ)=γ+∫dβ(v)z+v.

Theorem 4.2.

Weyl functions W(λ) and W^(λ) have continued fraction expansions:

W(λ)=1-λ2hK+1gK+⋯+1-λ2h1+1g1, (4.1)

W^(λ)=h^K+1-λ2g^K+1h^K-1+⋯+1-λ2g^2+1h^1. (4.2)

Proof. Using the recursive relations (3.6), we can obtain (4.2) by induction. In fact,

q^k+1λp^k=q^k+λh^kp^kλp^k=h^k+1λ(p^k-1-λg^kq^k)q^k=h^k+1-λ2g^k+1q^kλp^k-1.

Taking account of q^1=0 , we have the continued fraction expansion (4.2). The proof of (4.1) is analogues.

The following results of T. Stieltjes [41] will recover h_k, g_k(k = 1,…,K), h^1,h^k,g^k(k=2,…,K-1) , g^K by elementary functions of spectral data, and verify that h^K and g^1 are constant.

Theorem 4.3

(Stieltjes). Suppose F(z) is a rational function admitting the following integral representation:

F(z)=c+∫dμ(x)x+z, (4.3)

here, dμ(x) is the Stieltjes measure corresponding to the increasing piecewise constant function μ(x) with finite jumps on 𝕉 , then F(z) can be developed in a finite Stieltjes continued fraction:

F(z)=c+1za1+1a2+1za3+⋯, ai>0, (4.4)

conversely, any rational function with this type of continued fraction admits integral representation of the form (4.3). Particularly, when dμ=∑j=1Nbjδλj , set

Hkl=det(Ai+j+l)i,j=0k-1(l≥0), Ak=∫-∞+∞xkdμ(x)

with the proviso that H0l=H00=1 , if Hi0,Hj1>0(i,j≤N) , then

a2k=(Hk0)2Hk1Hk-11, a2k+1=(Hk1)2Hk0Hk+10.

By Theorem 4.3, γ=h^K. Taking account of Theorem 3.3 and the relation (2.7), we have the following theorem.

Theorem 4.4.

γ>0,h^K is a constant of motion, and

h^K=n2K(0)exp(x2K(0))>0. (4.5)

To recover h_k, g_k(k = 1,…,K) and h^1,g^2,…,h^K-1,g^K , we need to ensure the positivity of some Hankel determinants. The following lemma follows from Heine’s formula on k × k Hankel determinant of moments.

Lemma 4.1.

Hkl=∑J∈([1,N]k)bJλJlΔJ2, 0≤k≤N, (4.6)

where

bJ=∏j∈Jbj, λJl=∏j∈Jλjl, ΔJ2=∏i,j∈J,i<j(λj-λi)2.

Remark 4.1.

Lemma 4.1 together with the properties of spectral data given by Theorems 3.1 and 3.4 shows that Hil>0(0≤i≤K), H^jl>0(0≤j≤K-1) .

Applying Theorem 4.3 to W(λ) recovers h_k, g_k(k = 1,…,K).

Theorem 4.5.

gK+1-k=(Hk0)2Hk1Hk-11, hK+1-k=(Hk-11)2Hk0Hk-10, k=1,…,K, (4.7)

where Hkl=det(Ai+j+l)i,j=0k-1(l≥0),Ak=∫μkdα(μ)=∑j=1Kλj2kaj .

Apply Theorem 4.3 to W^(λ)-γ( or W^(λ)) , we have h^1,g^2,…,h^K-1 and g^K .

Theorem 4.6.

h^K-k=(H^k0)2H^k1H^k-11, g^K+1-k=(H^k-11)2H^k0H^k-10, k=1,…,K-1, (4.8)

where H^kl=det(A^i+j+l)i,j=0k-1(l≥0),A^k=∫vkdβ(v)=∑j=1K-1λ^j2kbj .

The following theorem follows from Remark 4.1 and the two theorems above.

Theorem 4.7.

g1,h1,…,gK,hK,h^1,g^2,…,h^K-1,g^K given by (4.7) and (4.8) are always positive.

We will finish this section by defining γ˜>0 and recovering g^1 .

Let us consider the spectral problem (3.1) again, and define another Weyl function W^*(λ) . To avoid confusion, we rewrite (3.1) as follows:

(Ψ˜1ψ˜2)x=(0λh^-λg^0)(Ψ˜1ψ˜2), ψ˜1(-∞)=ψ˜2(+∞)=0. (4.9)

Let q˜k=ψ˜1(x2k-),p˜k=ψ˜2(x2k+1-) and ψ˜1(+∞)=1 , consider the finite difference equation

{q˜k+1-q˜k=λh^kp˜k,1≤k≤K,p˜k-p˜k-1=-λg^kq˜k,1≤k≤K,p˜K=0, q˜K+1=1. (4.10)

Using (4.10), we have

(p˜0q˜1)=S^(λ)-1(01), S^(λ)≔T^K(λ)⋯T^1(λ).

Note that S^(λ)-1=(S^22-S^12-S^21S^11) , thus, q˜1=limx→-∞ψ˜1(x)=0 if and only if S^11=0 . Hence, (4.9) really has the same spectra as (3.4).

Define Weyl function of spectral problem (4.9) by the second column of S^(λ)-1 ,

W^*(λ)=-S^12(λ)λS^11(λ).

Recall that

W^(λ)=q^K+1(λ)λp^K(λ)=S^21(λ)λS^11(λ),

and

degS^(λ)=(2K-22K-12K-12K).

Therefore, the partial fraction decompositions of Weyl functions W^(λ) and W^*(λ) have the same form, and we can set

W^*(λ)=γ˜+∑j=1K-1bj*λ^j2-λ2. (4.11)

The proof of bj*>0 . We show bj*>0 by using the positivity of bj .

Obviously, detS^=detT^K⋯detT^1=1 , since ±λ^j are zeros of S^11 , we obtain

S^12(λ^j)S^21(λ^j)=S^12(-λ^j)S^21(-λ^j)=-1.

Easy to know that

bj=-2S^21(±λ^j)/S^11′(±λ^j), bj*=2S^12(±λ^j)/S^11′(±λ^j),

therefore,

bjbj*=(2S^11′(±λ^j))2.

By Theorem 3.1, S^11′(±λ^j)≠0 , which leads to bj*>0 .

Using the positivity of bj* , we can write W^* as the Stieltjes transformation of a positive discrete measure

W^*(λ)=γ˜+∫dβ*(v)z+v, β*(v)=∑j=1K-1bj*δvj.

Due to the fact that (1.1) can also be derived as the compatibility condition of the following linear spectral problem:

(φ1φ2)x=12(-1λn-λm1)(φ1φ2), (4.12a)

(φ1φ2)t=-12(12Q-12P-λ-1(v-vx)-12λnQλ-1(u+ux)+12λmQ-2λ-2-12Q+12P)(φ1φ2), (4.12b)

we have the following theorem.

Theorem 4.8.

γ˜>0,g^1 is a constant of motion and

g^1=m1(0)exp(-x1(0)). (4.13)

Proof. Using (4.10), W^*(λ) can be written as the following continued fraction

W^*(λ)=g^1+1-λ2h^1+1g^2+⋯+1-λ2h^K-1+1g^K. (4.14)

Thus, by Theorem 4.3, γ˜=g^1>0

Easy to check that (1.5) is equivalent to the compatibility condition of (4.12) with m, n given by (1.4), and both sides of (4.12b) vanish when ψ₁(+ ∞) = 1, ψ₂(+ ∞) = 0. Hence, for x < x₁, substituting ψ˜1(x,λ)=q˜1,ψ˜2(x,λ)=p˜0 into (4.12b), we obtain

dp˜0dt=p˜0λ2-L˜λq˜1, dq˜1dt=0, L˜=∑k=1Kg^k. (4.15)

Therefore, γ˜ in (4.11) is a constant of motion, which completes the proof.

Remark 4.2.

In the proof above, we obtain the evolution of bj* :

dbj*dt=bj*λ^j2.

Apply Theorem 4.3 to W^*(λ) , we can obtain the following explicit formulas for h^1,h^k,g^k(k=2,…,K-1) and g^K :

h^k=(H˜k-11)2H˜k-10H˜k0, g^k+1=(H˜k0)2H˜k1H˜k-11, k=1,…,K-1,

where H˜kl=det(A˜i+j+l)i,j=0k-1(l≥0), A˜k=∫vkdβ*(v)=∑j=1K-1λ^j2kbj* .

5. Interlacing Peakons

By Remark 4.1, together with the relations (2.11a) and (2.11b), we have the explicit formulas for positive interlacing peakons.

Theorem 5.1.

The new two-component Camassa-Holm system (1.1) admits positive interlacing peakons:

u(x,t)=∑k=1Km2k-1(t)e-|x-x2k-1(t)|, v(x,t)=∑k=1Kn2k(t)e-|x-x2k(t)|,

where

x1=12ln(ex1(0)m1(0)(HK0)2HK1HK-11), m1=(m1(0)e-x1(0)(HK0)2HK1HK-11)12,x2K=12ln(n2K(0)ex2K(0)A0), n2K=(n2K(0)ex2K(0)/A0)12,x2K-2k+1=12ln((Hk0)2Hk1Hk-11H^k0H^k-10(H^k-11)2),m2K-2k+1=Hk0H^k-11(Hk1Hk-11H^k-10H^k0)-12, 1≤k≤K-1x2K-2k+2=12ln((H^k-10)2H^k-11H^k-21Hk0Hk-10(Hk-11)2),n2K-2k+2=Hk-11H^k-10(H^k-11H^k-21Hk0Hk-10)-12, 2≤k≤K

with

Hkl=det(Ai+j+l)i,j=0k-1(l≥0),Ak=∫μkdα(μ)=∑i=1Kλi2kai,H^kl=det(A^i+j+l)i,j=0k-1(l≥0), A^k=∫vkdβ(v)=∑j=1K-1λ^j2kbj,

and

ai=ai(0)exp(-tλi2), bj=bj(0)exp(-tλ^j2), ai(0),bj(0)>0.

Remark 5.1.

The formulas in Theorem 5.1 give positive interlacing peakons globally if and only if

x2k-1<x2k(1≤k≤K), x2k<x2k+1(1≤k≤K-1). (5.1)

Due to the relations (2.11a) and (2.11b), (5.1) is equivalent to

gkhk<g^kh^k(1≤k≤K), g^k+1h^k<gk+1hk(1≤k≤K-1). (5.2)

For K = 1, since

m1(0)n2(0)ex2(0)>ex1(0)/λ12, λ12=(g1(0)h1(0))-1,

(5.1) holds for all t if x₁(0) < x₂(0). When K > 1, (5.1) holds only for the spectral data from certain subset of R, which will be studied in the future.

6. Conclusion and Discussions

In this paper, we have formulated and solved the peakon problem arising from the explicit construction of interlacing peakons for the new two-component Camassa-Holm system derived by Xia and Qiao, and obtained the explicit formulas for positive interlacing peakons locally. As a generalization of the CH equation and modified CH equation (a special case of (1.1) when v = − 2u), there are some new features and complexity of the solutions. Further studies on the global existence and other properties of positive interlacing peakons to (1.1) will be developed.

In our paper, the classical result of T. Stieltjes on continued fractions allow us to solve the inverse problem associated to the positive interlacing peakons in the case N = 2K, how about N = 2K + 1, where the number of peakons in u and v is not equal, for example, the measure m is supported on the sites x₁, x₃,…, x₂_K _{+ 1}? In [29,30], Lundmark and Szmigielski use the inverse spectral method to derive explicit formulas for the interlacing K + K peakons, and explore their dynamical properties. In their work, a sort of symmetry in the case N = 2K is key to the inverse problem, for the interlacing case N = 2K + 1 or more general one, certain limiting procedures is needed. However, in our case, the interlacing peakons for N = 2K + 1 can also be explicitly constructed, the operation is similar to the case N = 2K, and somewhat easier.

Acknowledgments

This work was supported by the National Natural Science Foundation of China [NSFC11301398] and the Fundamental Research Funds for the Central Universities.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 2
Pages: 290 - 308
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
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TY  - JOUR
AU  - Fengfeng Dong
AU  - Lingjun Zhou
PY  - 2021
DA  - 2021/01/06
TI  - Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System
JO  - Journal of Nonlinear Mathematical Physics
SP  - 290
EP  - 308
VL  - 25
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1452674
DO  - 10.1080/14029251.2018.1452674
ID  - Dong2021
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