Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I

while in the sequel we study the inverse problem for another equation, discovered by V. Novikov (VN) [7]:

Both these equations admit another, often used, representation

for the CH equation, and

for the VN equation. The solution to the inverse problem presented in this paper is obtained directly on the real axis, instead of transforming the problem to the inhomogeneous string problem as was done in [1] and adapting the method used by T. Stieltjes in [8].

We give a self-contained presentation of the inverse problem for Equation (2.6), emphasizing the role of orthogonal polynomials (Proposition 11) and culminating in the elegant inverse formulas stated in Corollary 14. In the sequel, for the case of the VN equation (1.2), we will show that a different class of polynomials is germane to the inverse problem, namely a class of biorthogonal polynomials (Cauchy biorthogonal polynomials) studied in [2].

The well known Lax pair system for the CH equation can be written as follows (see [3])

We would like to point out that this Lax pair yields a slightly different normalization than in (1.3), here, m = 2u -1 2 u xx . If we set m(x, t) = n j=1 m j (t)δ x j where m j (t) and x j (t) are C ∞ functions, we obtain what is called the n-peakon Lax pair for the CH equation. We assume that for every fixed t ≥ 0 the function ψ(x, t) is continuous on R and x 1 (0) < x 2 (0) < • • • < x n (0). Now the first equation of the system (2.1) implies that on the interval (x j , x j+1 ), we have

It is well known that requiring ψ → 0 as |x| → ∞ is consistent with the system (2.1). Let’s therefore assume that A 0 = 1 and B 0 = 0. Then from the continuity of ψ and the jump conditions imposed by the first equation of (2.1) we have

and

Clearly, Theorem 1. The spectrum of the boundary value problem

is given by roots of A n (z) = 0.

In order to get more information about the spectrum we can rewrite equation (2.6) as an integral equation

where M is the distribution function of the measure m(x) and K(x, y) = 1 2 e -|x-y| is the unique Greens function of 1 -D 2

x vanishing at ±∞. One can easily write the explicit form of this integral equation as:

where, by assumption, all m j > 0. Moreover, evaluating at x i , and introducing the matrix notation

and

we obtain equation (2.7) in the form of the matrix eigenvalue problem:

E is an example of a single-pair matrix (see [5]) which is oscillatory and so is EP.

Oscillatory matrices form a subset of totally nonnegative matrices, which can be characterized as matrices whose all minors of arbitrary size are nonnegative. Oscilatory matrices are special in that they have simple, strictly positive, spectrum [5]. We therefore have the following description of the spectrum of the boundary value problem (2.6).

We need one more fact.

where γ = 1 x dµ(x).

Proof. This theorem follows, after a slight adjustment of notation, from Theorem 4.2 proven in [1] and the observation that W (0) = 0 which is a consequence of T k (0) = I.

The continued fraction expansion ( [8]) turns out to be at the heart of the inverse problem, in a full analogy with the inverse problem for an inhomogeneous string which was studied in the 1950’s by M.G. Krein ([6], [4]).

Theorem 4.

we have

.

(2.12)

This equation is equivalent to

Hence, the proof is complete by induction.

Suppose that P j /Q j (0 ≤ j ≤ 2n) is the jth convergent of the negative of the continued fraction we obtained above so that we are approximating the Weyl function W (z). Then we observe that

and

.

(2.15)

Therefore, it can be verified that

)

(2.17)

These equations guide us to the following theorem Theorem 5.

(2.19)

Proof. By induction.

Corollary 6.

(2.20)

Proof. Use induction and the previous theorem.

Corollary 7.

In preparation for the inverse problem we introduce the following notation. Notation: Given a polynomial p(z) we denote by p[j] the coefficient at the jth power of z. Moreover, n -k is abbreviated to k ′ .

With this notation in place we have Theorem 8.

Proof. By induction.

The Weyl function of the n-peakon problem reads:

Proposition 9.

The inverse problem hinges on the following approximation formulas:

Theorem 10.

(2.22)

Proof. By the previous theorem we have

From these and the definition of the Weyl function we get

Similarly, we see that

Proposition 11. (orthogonality)

x j q 2k+1 (x)dµ(x) = 0, j = 1, . . . , k -1, k = 0, . . . , n -1.

(2.23c)

Proof. By the first approximation formula of Theorem 10 we have

Therefore,

Since W (z) is analytic at z = ∞ so is the remainder and its order there is O( 1 z 2 ). We can therefore use the residue calculus. To this end we choose a circle Γ whose radius is large enough to contain the support of µ defined in Theorem 3. An elementary contour integr

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