On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation

Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle $\theta$ are constructed and then reduced to the two-component Camassa--Holm model. Only three different independent classes of reductions are encountered corresponding to the angle $\theta$ being 0, $\pi/2$ or taking any value in the interval $0<\theta<\pi/2$. This construction induces B\"{a}cklund transformations between solutions of the two-component Camassa--Holm model associated with different classes of reduction.


Introduction
It is widely known that the standard integrable hierarchies can be supplemented by a set of commuting flows of a negative order in a spectral parameter [1]. A standard example is provided by the modified KdV-hierarchy, which can be embedded in a new extended hierarchy. This extended hierarchy contains in addition to the original modified KdV equation also the differential equation of the sine-Gordon model realized as the first negative flow [2,3,4,5,6,7].
Quite often the negative flows can only be realized in a form of non-local integral differential equations. The cases where the negative flow can be cast in form of local differential equation which has physical application are therefore of special interest. Recently in [11], a negative flow of the extended AKNS hierarchy [8] was identified with a two-component generalization of the Camassa-Holm equation. The standard Camassa-Holm equation [9,10] u t − u txx = −3uu x + 2u x u xx + uu xxx − κu x , κ = const (1.1) enjoys a long history of serving as a model of long waves in shallow water. The two-component extension [11,13] differs from equation (1.1) by presence on the right hand side of a new term ρρ x , with the new variable ρ obeying the continuity equation ρ t + (uρ) x = 0. Such generalization was first encountered in a study of deformations of the bihamiltonian structure of hydrodynamic type [12]. Various multi-component generalizations of the Camassa-Holm model have been subject of intense investigations in recent literature [14,15,16,17,18].
A particular connection between extended AKNS model and a two-component generalization of the Camassa-Holm equation was found in [11] and in [13]. It was pointed out in [19] that the second order spectral equation for a two-component Camassa-Holm model can be cast in form of the first order spectral equation which after appropriate gauge transformations fits into an sl(2) setup of linear spectral problem and associated zero-curvature equations.
The goal of this article is to formulate a general scheme for connecting an extended AKNS model to a two-component Camassa-Holm model which would encompass all known ways of connecting the solution f of the latter model to variables r and q of the former model. Our approach is built on making gauge copies of an extended AKNS model labeled by angle θ belonging to an interval 0 ≤ θ ≤ π/2 and then by elimination of one of two components of the sl(2) wave function reach a second order non-linear partial differential equation which governs the two-component Camassa-Holm model. We found that the construction naturally decomposes into three different classes depending on whether angle θ belongs to an interior of interval 0 ≤ θ ≤ π/2 or is equal to one of two boundary values unifying therefore the results of [11] and [20]. The map between these three cases induces a Bäcklund like transformations between different solutions f of the two-component Camassa-Holm equation.

A simple derivation of a relation between AKNS and two-component Camassa-Holm models
Our starting point is a standard first-order linear spectral problem of the AKNS model: where λ is a spectral parameter, y a space variable and Ψ a two-component object: In addition, the system is augmented by a negative flow defined in terms of a matrix, which is inverse proportional to λ: The compatibility condition arising from equations (2.1) and (2.3): has a general solution: in terms of the zero-grade group element, M 0 , of SL (2). Note that the solution, D (−1) , of the compatibility condition is connected to (1/λ)σ 3 -matrix by a similarity transformation.
The factor 1/4β in (2.5) is a general proportionality factor which implies a determinant formula: for the matrix elements of D (−1) .
From (2.4) we find that When projected on the zero and the first powers of λ the compatibility condition (2.4) yields and A y = qC − rB, B y = −2Aq, C y = 2Ar, (2.9) respectively. Note that the first of equations (2.9) together with equations (2.8) reproduces formula (2.7).
Combining the above equations we find that The spectral equation (2.1) reads in components: Now we eliminate the wave-function component ψ 2 by substituting into the remaining second equation of (2.11). In this way we obtain for ψ 1 ψ 1yy − q y q ψ 1y + λq y q ψ 1 − λ 2 ψ 1 − rqψ 1 = 0.
Eliminating ψ 2 from equation (2.3) yields for ψ the following equation: Compatibility equation ψ yys − ψ syy = 0 yields in total agreement with (2.7). To eliminate r from (2.16) we use that as follows from the first equation from (2.9). Replacing C by 1/(B16β 2 ) − A 2 /B as follows from the determinant relation (2.6) and recalling that B = −q s /2 according to equation (2.8) we obtain after substituting r from (2.17) into (2.16): Note that alternatively we could have eliminated q from equation and obtained an equation for r only. It turns out that the equation for r follows from equation (2.18) by simply substituting r for q.
For brevity we introduce, as in [20], f = ln q. Then expression (2.18) becomes: The above relation can be cast in an equivalent form: which first appeared in [11]. The relation (2.20) is also equivalent to the following condition For a quantity u defined as: with κ being an integration constant, it holds from relation (2.21) that Next, as in [21], we define a quantity m as β 2 f 2 s f y and derive from relations (2.22) and (2.23) that Taking a derivative of m with respect to s yields In terms of quantities u and ρ = f s equations (2.23) and (2.25) take the following form for m given by An inverse reciprocal transformation (y, s) → (x, t) is defined by relations: for an arbitrary function F . Equations (2.26), (2.27) and (2.28) take a form in terms of the (x, t) variables. Equation (2.30) is called the compatibility condition, while equation (2.31) is the two-component Camassa-Holm equation [11], which agrees with standard Camassa-Holm equation (1.1) for ρ = 0.
3 General reduction scheme from AKNS system to the two-component Camassa-Holm equation Next, we perform the transformation on AKNS two-component Ψ function from (2.2). U (θ, f ) stands for an orthogonal matrix: where Ω(θ) is given by and f is a function of y and s, which is going to be determined below for each value of θ. Note that Ω −1 (θ) = Ω(θ) and Ω(0) = σ 3 , Ω(π/2) = σ 1 . Taking a derivative with respect to y and s on both sides of (3.1) one gets Repeating derivation with respect to y one more time yields Next, we will eliminate η in order to obtain an equation for the one-component variable ϕ. This is analogous to the calculation made below equation (2.11), where the first order two-component AKNS spectral problem was reduced to second order equation for the one-component function ψ.
Let us shift a function f by a constant term, ln (tan θ): f −→ f θ = f + ln (tan θ) . (3.10) Then relation (3.9) can be rewritten for 0 < θ < π/2 as which is of the same form as the relation found in reference [11]. It therefore appears that for all values of θ in the the 0 < θ < π/2 relation between function f and AKNS variables q and r remains invariant up to shift of f by a constant. Now, we turn our attention back to equation (3.5) rewritten as For the ϕ component we find: For 0 < θ < π/2 we choose which agrees with the determinant formula A 2 + BC = 1/16β 2 and implies identities: The first of these identities, (3.14), ensures that the last three terms containing η on the right hand side of equation (3.12) cancel. Recall at this point relation (3.6). Simplifying this relation by invoking identity (3.9) and plugging it into equation (3.12) gives From (3.13) we find and therefore Due to the above relation and identity (3.15) equation (3.16) becomes Taking derivative of (3.9) with respect to s we find Thus equation (3.20) becomes We now turn our attention to equation (3.8). The last term containing η vanishes due to the identity (3.9). In addition it holds that as follows from relations (3.19) and (3.21). Also, it holds from relations (3.17)-(3.18) that for 0 < θ < π/2: where Thus, the remaining constant (the ones which do not contain λ) terms on the right hand side of equation (3.8) are equal to Therefore, we can write equation (3.8) as:

The θ = 0 case and Bäcklund transformation between dif ferent solutions
We now consider θ at the boundary of the 0 < θ < π/2 interval. For illustration we take θ = 0, the remaining case θ = π/2 can be analyzed in a similar way. Plugging θ = 0 into relation (3.26) we obtain Comparing with relation (3.24) we get which describes a relation between the product rq for zero and non-zero values of the angle θ, with rq| θ being associated with θ within an interval 0 < θ < π/2. Recall that q = exp(f ) for θ = 0. It follows that A = q sy /4q = (f sy + f s f y )/4 and equation (2.7) is equivalent to On the other hand, it follows from (2.17) and where Obviously P ± (f θ ) = P ± (f ).
We are now ready to show that satisfies the two-component Camassa-Holm equation (2.19) for any f or f θ , which satisfies equation (2.19). For 0 < θ < π/2, it holds that q = exp(f ) and therefore A = q sy /4q = (f sy +f sfy )/4 = (f sy + f s f y )/4 + f s P − y + P − ys + P − s f y 4P − . (4.4) We will now show that Using equation (4.2) one can easily show that equation (4.5) holds if the following relation is true. We note that the above relation can be rewritten as (P 2 − ) s + 2f ys P − + f s P − y + P − sy + P − s f y = 0 . The last equation is fully equivalent to the two-component Camassa-Holm equation (3.28) as can be seen by rewriting Q from relation (3.27) as Q = −(P − + f y /2) 2 − (P − + f y /2) y . This completes the proof for relation (4.5).
It follows from (2.17) and C = 1/(16β 2 B) − A 2 /B that Thus, due to (4.4) and (4.5) we have proved explicitly that is For θ = π/2 we have r = exp(−f ) and comparing with the result for 0 < θ < π/2: we get a Bäcklund transformation Additional Bäcklund transformations can be obtained by comparing expressions for q and r variables in terms of f for the boundary values of θ. We first turn our attention to the case of θ = 0 for which we have q = exp(f ) and Next, for θ = π/2 we have r = exp(−f ) and Comparing expressions for q and r we find find that if f is a solution of the 2-component Camassa-Holm equation (2.19) then so is also f + ln P 2 − + P − f y + P −y .

Conclusions
These notes describe an attempt to construct a general and universal formalism which would realize possible connections between the 2-component Camassa-Holm equation and AKNS hierarchy extended by a negative flow. Construction yields gauge copies of an extended AKNS model connected by a continuous parameter (angle) θ taking values in an interval 0 ≤ θ ≤ π/2. Eliminating one of two components of the sl(2) wave function gives a second order non-linear partial differential equation for a single function f of the two-component Camassa-Holm model. Functions f corresponding to different values of θ in an interior of interval 0 ≤ θ ≤ π/2 differ only by a trivial constant and fall into a class considered in [11]. Two remaining and separate cases correspond to θ equal to 0 and π/2 and agree with a structure described in [20].