Two-component generalizations of the Camassa-Holm equation

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.


Introduction
In recent years there has been a growing interest in integrable non-evolutionary partial differential equations of the form where F is some function of u and its derivatives with respect to x. The most celebrated example of this type of equation is the Camassa-Holm equation [1]: Other examples of integrable equations of the form (1) include the Degasperis-Procesi equation (see [5,6]) as well as equations with cubic nonlinearity, such as (see [13,19] and [9,24], respectively). All the of the latter equations of Camassa-Holm type are integrable by the inverse scattering transform. They possess infinite hierarchies of local conservation laws and (quasi-)local higher symmetries, bi-Hamiltonian structures and other remarkable attributes of integrable systems. Part of the fascination with these sorts of equations is due to the fact that as well as having traditional (smooth) multi-soliton solutions, they admit weak solutions of peakon (peaked soliton) type, and also display interesting blowup and wavebreaking phenomena [14]. The complete classification of integrable equations of the form (1) was carried out in [19] using the perturbative symmetry approach introduced in [17]. Various approaches to generating multicomponent systems of Camassa-Holm type have been proposed recently, based on energy-dependent spectral problems [11], or Novikov algebras [23].
In this paper we study integrable two-component systems of the form where F, G are polynomials over C in u, v and their x-derivatives. An example of an integrable system of the form (3) is x . (4) The above system is related to a system which (up to sending t → −t and renaming variables) was given as by Chen, Liu and Zhang [3], and related to an alternative system of the form (3) presented by Falqui [7], namely (again, up to renaming variables, and fixing the value of a parameter). To be precise, under the transformation which is of Miura type, solutions of the system (4) are mapped to solutions of (5), while is a Miura map from (6) to (5).
The rest of the paper is concerned with classifying integrable systems of the form (3). In the next section we outline the perturbative symmetry approach in the context of non-evolutionary systems with two dependent variables, and explain how it leads to an integrability test for such systems. Section 3 contains the result of applying this integrability test, in the form of a list of systems with quadratic, cubic and mixed quadratic/cubic nonlinear terms; there are six systems in total, presented in Theorems 2,3 and 4 below. The fourth section is concerned with a different problem, namely that of classifying pairs of compatible Hamiltonian operators in two dependent variables. However, this turns out to be highly relevant to the preceding considerations, since it provides a bi-Hamiltonian structure for (almost) every system in the aforementioned list. In the fifth section we consider changes of independent variables, specifically reciprocal transformations (sending conservation laws to conservation laws); these are helpful for the construction of Lax pairs and exact solutions, which we illustrate in some cases. The paper ends with conclusions and suggestions for future work.

Integrability test: perturbative symmetries
In this section we briefly recall the basic definitions and notations of the perturbative symmetry approach (for details see [17,18]). We also present the integrability test which we will subsequently apply to isolate integrable generalizations of the Camassa-Holm equation.

Quasi-local polynomials and definition of symmetries
Let u, v be functions in x, t. Polynomials in u, v and their x-derivatives over C form a differential ring R with an x-derivation where u k , v k denote k-th derivatives of u, v with respect to x. In particular, u 0 and v 0 denote the functions u and v themselves. We often omit the zero index of u 0 and v 0 and simply write u and v.
We will assume that 1 ∈ R. Elements of the ring R are finite sums of monomials in u, v and their x-derivatives with complex coefficients. The degree of a monomial is defined as a total power, i.e. the sum of all powers of variables that contribute to the monomial. Let R n denote the set of polynomials of degree n in u, v and their x-derivatives. Then ring R has a gradation Elements of R 1 are linear functions of the u, v and their derivatives, elements of R 2 are quadratic, etc. It is convenient to define a "little-oh" order symbol o(R n ). We say that f = o(R n ) if f ∈ k>n R k , i.e. the degree of every monomial of f is bigger than n.
Since 1 ∈ R, the kernel of the linear map D x : R → Im D x ⊂ R is empty and therefore D −1 x is defined uniquely on Im D x .
To an element g ∈ R we associate differential operators g * ,u and g * ,v called Fréchet derivatives with respect to u and v and defined as Now we need to introduce a concept of quasi-local differential polynomials and the corresponding extension of the ring R. The idea of this extension is similar to that in [15,26,17].
To rewrite the Camassa-Holm type system (3) in evolutionary form, we introduce a pair of pseudo-differential operators System (3) then can be rewritten as Clearly, if F, G ∈ R then the right hand side of the system (9) no longer consists of differential polynomials and we need an extension of the original differential ring R.
Consider the following sequence of ring extensions: where the set ∆ ± (R n ) = {∆ ± (a) : a ∈ R n } and the horizontal line denotes the ring closure. The index n indicates the "nesting depth" of operators ∆ ± . We then define quasi-local differential polynomials as follows.
Definition 1. An element f is called a quasi-local differential polynomial if f ∈ R n for sufficiently large n.
The right-hand side of equations in (9) lies in R 1 . Its symmetries and densities of conservation are also generally speaking all quasi-local and belong to R k for some k ≥ 0 .
We now recall the definition of a symmetry.

Definition 2.
A pair of quasi-local differential polynomials P and Q is called a symmetry of an evolutionary system u t = f, v t = g, where f, g are quasi-local polynomials, if the system If a = (f, g) T and b = (P, Q) T then the above definition is equivalent to the vanishing of the Lie bracket We finally define a notion of formal pseudodifferential series (or just formal series) as an object of the form with coefficients being quasi-local differential polynomials or constants. The order of the formal series (11) is N (we assume that the leading coefficient a N = 0). The formal series form a ring: the sum of formal series is defined in the obvious way, while multiplication (composition) is defined by For positive n the sum (12) is finite since the binomial coefficients vanish for k > n, and for negative n the composition is well-defined in the sense of formal series.
In the symmetry approach we admit the following definition of integrability: (9) is integrable if it possesses an infinite hierarchy of symmetries.
In the following subsections we present the necessary conditions for existence of a hierarchy of symmetries. For this it is convenient to introduce the symbolic representation of the ring of quasi-local polynomials and derive the necessary conditions in the symbolic representation.
Using the addition and multiplication operations where necessary, we thus construct the symbolic representation ofR k , k = 1, 2, . . ..
Finally, we define the symbolic representation for pseudo-differential formal series. For any two terms f D p x , gD q x of formal series (p, q ∈ Z and f, g ∈ R k ) with symbols f →û nvm a(ξ 1 , . . . , ξ n , ζ 1 , . . . , ζ m ), g →û svr b(ξ 1 , . . . , ξ s , ζ 1 , . . . , ζ r ) the composition rule in the symbolic representation reads where the symmetrisation is taken with respect to permutations of arguments ξ and arguments ζ, but not the argument η.
The set of formal series (18) has the structure of an associative noncommutative ringR ∆ (η). It inherits the natural gradation from R, namelŷ whereR n ∆ (η) with n = 0, 1, 2, 3, . . . are constant (i.e. independent of u, v ), linear in u or v, quadratic, cubic, etc. We say that a formal series

Formal recursion operator and necessary conditions for integrability
In this subsection we formulate the necessary conditions for integrability of a system of the form Letf ,ĝ be the symbolic representations of the differential polynomials f, g, so that Define F = f * ,uf * ,v g * ,uĝ * ,v and let where L (i) , i = 1, . . . , 4 are formal series, Definition 5. A formal series Λ (22) is called a formal recursion operator for system (20) if all the coefficients φ (i) jk are quasi-local and it satisfies the equation In the above definition Λ t stands for a formal series obtained from Λ by differentiating all the coefficients φ jk by t and replacing u t and v t according to the system (20). Theorem 1. Assume that the system (20) is such that ω 2 (ζ) = 0 = ω 3 (ξ) = 0 (24) and ω 1 (ξ) = c 1 ξ, ω 4 (ζ) = c 4 ζ (for constants c 1 , c 4 ). Suppose that the system (20) possesses an infinite hierarchy of quasi-local higher symmetries. Then the system possesses a formal recursion operator (22) The assumption (24) implies that the linear part of the system (20) is diagonal; in principle, this condition may be removed. (Note that Falqui's system (6) is excluded by this assumption.) In the diagonal case the proof of the theorem is essentially the same as the proof of the analogous Theorem 2 from [17] and therefore we omit it here.
To classify integrable systems of the Camassa-Holm type (see the next section) we only need to verify quasi-locality of φ (i) jk , i = 1, . . . , 4 with j + k ≤ 3.

Classification theorems
In this section we present the classification of integrable Camassa-Holm type systems of the form where f, g are polynomials containing terms of degree two or above in u, v, u 1 , v 1 , u 2 , v 2 . We will also assume that λ 2 = −λ 1 and µ 2 = µ 1 as otherwise the linear part of each equation of the system will be λ 1 (1 − D x )u 1 and µ 1 (1 + D x )v 1 , and individually these terms are removable by a Galilean transformation.
We will restrict the classification to non-linearisable systems and therefore require the existence of non-trivial conservation laws. This allows us to further restrict the admissible linear terms in (25).
Using the above proposition with a combination of a Galilean transformation and rescaling t, we thus can consider only systems of the form where we assume that f, g are polynomials in u, v, u 1 , v 1 , u 2 , v 2 with only quadratic terms, cubic terms, or both, and we consider these different cases separately.
Applying the integrability test as described above leads to a total of six different systems, which are listed below according to the type of nonlinearity.
Systems with quadratic nonlinearity: Theorem 2. If the system (27) with f, g being quadratic polynomials in u, v, u 1 , v 1 , u 2 , v 2 possesses an infinite hierarchy of higher symmetries then modulo the scaling transformations u → αu, v → βv, x → γx, t → δt it is one of the list Systems with cubic nonlinearity: Theorem 3. If the system (27) with f, g being cubic polynomials in u, v, u 1 , v 1 , u 2 , v 2 possesses an infinite hierarchy of higher symmetries then modulo the scaling transformations Mixed nonlinearity: Theorem 4. If the system (27) with f, g being mixed quadratic and cubic polynomials in u, v, u 1 , v 1 , u 2 , v 2 possesses an infinite hierarchy of higher symmetries then modulo the scaling In the next section we consider compatible pairs of Hamiltonian operators, which will lead to bi-Hamiltonian structures for each of the six systems listed above.

Compatible Hamiltonian Operators
In this section, we attempt to classify integrable two-component Camassa-Holm equations based on bi-Hamiltonian structures. This is similar to the approach of [23], based on Novikov algebras; however, in due course we will obtain systems with cubic nonlinearity, which do not appear in the latter approach.
We use the multivector method, as described in the standard reference [20], to investigate the conditions such that the specified types of antisymmetric operators H with entries H ij , i, j = 1, 2, depending on a pair of fields m, n, form Hamiltonian pairs with a nondegenerate constant-coefficient differential Hamiltonian operator given by For the purpose of deriving coupled two-component Camassa-Holm equations, we are going to study three cases: Moreover, we also use elimination requirements to get rid of non-coupled (triangular) or non-Camassa-Holm type equations by removing pairs satisfying one or more of the conditions • The determinant of J is a multiple of D x ; • (H 11 ) ,n = (H 12 ) ,n = 0 and H 11 J 12 − H 12 J 11 = 0; Otherwise, we refer to the Hamiltonian pairs H and J as non-trivial CH Hamiltonian pairs.

Compatible linear Hamiltonian operators
We consider linear antisymmetric differential operators in dependent variables m, n, of the form where a i , i = 1, . . . , 8 are constants.
Theorem 5. Let the operators H and J be given by (36) and (34), respectively.
• Suppose that c 4 = 1. There are three non-trivial CH Hamiltonian pairs: • Suppose that c 4 = 0, c 2 = 1 and the parameter c 3 is arbitrary.There are two non-trivial CH Hamiltonian pairs: x , a 1 a 6 − a 2 2 c 6 + a 2 a 8 − a 2 6 = 0 and c 6 = 0; • Suppose that c 4 = c 2 = 0, c 6 = 1 and the parameter c 3 is arbitrary. There is only one non-trivial CH Hamiltonian pair: Proof: For the operators (34) and (36), acting on the univector ξ, we have where we used the notation D i x θ = θ i and D i x η = η i . We define the bivector associated to the operator H by The operator H is Hamiltonian if and only if it satisfies the Jacobi identity, which is equivalent [20] to the vanishing of the trivector where P and Q are given by (38). We substitute them into it and simplify the expression, which leads to an algebraic system for the constants a j , j = 1, . . . , 8, that is The system (40) provides necessary and sufficient conditions for H to be Hamiltonian. For the purposes of this theorem, we solve it together with the conditions for H to be compatible with the constant Hamiltonian operator J , that is, the trivector Pr J (ξ) (Θ H ) vanishes [20]. We carry out this calculation in the same way as for (39) and obtain an algebraic system for the constants c i , a j , for i = 1, . . .
Upon solving the latter system together with (40), when c 4 = 1 we get the following three solutions after applying our elimination requirements: these correspond to the three nontrivial CH Hamiltonian pairs (i)-(iii) in the statement. Similarly, we treat the other two cases with the help of the Maple package Gröebner and obtained the listed pairs (iv)-(vi). This completes the proof.
Any compatible Hamiltonian pair H and J which does not depend explicitly on the independent variables x and t, with J nondegenerate, leads to an integrable equation for the vector of dependent variables m, that is In fact, for scalar m, the Camassa-Holm equation (2) was first constructed in this way in [8] (although the correct form of the equation itself did not appear until [1]). In the case at hand, with the vector m = (m, n) T , we apply this construction to the compatible Hamiltonian pairs listed in Theorem 5. Since the pairs of operators J (i) , H (i) for i = 1, . . . , 6 in the six cases above depend linearly on arbitrary constant parameters, in each case we have a lot of freedom to obtain different compatible pairs, by fixing the constants in the operator J (i) to get J , and taking linear combinations of J (i) and H (i) with different constants to get H.
From case (i), we get the integrable equation it follows that In the same way, from cases (ii) and (iii) we get two pairs of integrable equations, given by respectively. Notice that system (45) is the same as (44), upon swapping dependent variables u ↔ v and sending x → −x; they do not belong in the list in the previous section since they include third derivatives, putting them outside the family (25). In fact, the system (44) can be seen to be a reduction of Example 2 on p.97 of [23] by setting the parameters h = 0, f = 1 and performing a Galilean transformation. It is also worth pointing out that it is possible to relax our elimination conditions slightly and still obtain interesting bi-Hamiltonian systems; for instance, setting c 3 = c 5 = c 6 = 0 in (45) or a 1 = c 3 = c 5 = 0 in (46), with a 8 = c 1 = 1 in both cases, gives Falqui's system (6), which is almost triangular (it would be with c 1 = 0).
For the system (46), if we take c 1 = c 5 = 0 and c 3 = 1 and rescale u and v, we get the system (29) in Theorem 2. Thus we arrive at the following result: Corollary 1. Define m and n as in (43). System (29) is bi-Hamiltonian, having the form where the compatible Hamiltonian operators H 1 and H 2 are given by and the corresponding Hamiltonian functions are given by the densities ρ 1 = un and ρ 2 = uv(2m + 2n + 1).
where the star denotes the Fréchet derivative, and the dagger denotes the adjoint operator.
For case (iv), if we let then we obtain the integrable equation in the explicit form    m t = a 1 (2mu x + m x u) + a 2 (2nu x + n x u + 2mv x + m x v) + a 6 c 6 (2nv x + n x v) + c 1 u x n t = a 2 c 6 (2mu x + m x u)+a 6 (2nu x + n x u + 2mv x + m x v)+ a 8 c 6 (2nv x + n x v) + c 1 v x a 1 a 6 − a 2 2 c 6 + a 2 a 8 − a 2 6 = 0 and c 6 = 0 In particular, if we take either a 6 = 0 or a 2 = 0 then we get respectively. In fact, the latter two systems are seen to be the same by swapping u ↔ v and identifying parameters suitably. Taking a 1 = 1, a 8 = 0 and c 6 = −1 in (50), we get the two-component CH system (26) in [25].
For case (v), we let m = (1 − D 2 x )u. Then we get the integrable system For case (vi), we let n = (1 − D 2 x )v. Then we get the integrable system Similarly to before, equations (51) and (52) are seen to be the same by swapping the dependent variables. After taking c 1 = 0, c 3 = 0 and rescaling suitably, the system (51) becomes the known two-component CH equation (5) from [3].
With a change of notation, the transformation (7) presented in the introduction is which implies that Thus equation (51) when c 5 = −1, c 3 = 1 and c 1 = 2 becomes which is system (28) when a 1 = 2. Thus we obtain the bi-Hamiltonian structure of system (28) by using the result for equation (51), as follows: Corollary 2. Define m and n as in (43). System (28) is a bi-Hamiltonian system, given by where the compatible Hamiltonian operators H 1 and H 2 are given by x n x and the Hamiltonian functions are given by the densities ρ 1 = 2u x n and Proof: To clarify the notation, we put hats on all variables in equation (51), that is, we writem,n etc. Thus the transformation (54) becomes m = 1 With a 1 = 2, c 5 = −1, c 3 = 1 and c 1 = 2 in equation (51), the compatible Hamiltonian operators are Under the transformation (56), the Hamiltonian operator J is sent to and H is transformed to H 2 in the statement.
Remark 3. The inverse operator of Hamiltonian operator 2H 2 in Corollary 2 is of the form Thus the local symmetries for system (28) can be generated by the recursion operator H 1 H −1 2 .

Compatible quadratic Hamiltonian operators
In this section, we consider antisymmetric differential operators that are quadratic in the dependent variables m and n, instead of linear as in the previous subsection. We assume that they are of the form where, as before, † denotes the adjoint operator, with and b i , i = 1, . . . , 14 being constants.
Theorem 6. Let the operators H and J be given by (57) and (34), respectively.
There are no non-trivial CH Hamiltonian pairs.
Proof: We prove this statement in the same way as we did for Theorem 5. Due to the large degree of similarity, we avoid tedious repetition and only write down the necessary steps and results. The operator H is compatible with the Hamiltonian operator J if and only if the constants in H and J satisfy an overdetermined algebraic system of the same type as (41). When c 4 = 1, we solve it and obtain only one solution after applying our elimination requirements: the nonzero constants in (57) should satisfy −b 14 = b 9 = b 5 = b 13 = b 1 , and c 2 = c 6 = 0. We denote the operators H and J under the above constraints by H (c) and J (c) . By direct computation, we are able to show the operator H (c) is Hamiltonian, and thus we obtain the Hamiltonian pair in the statement. For other cases, there are no solutions for the above system after applying our elimination requirements.
For the Hamiltonian pair given by (58) and (59) we can immediately write down the integrable two-component equation We introduce the same notation for u and v as in (43). It follows that For equation (60), if we take c 1 = c 5 = 0 and c 3 = 1 and rescale u and v, then we get the system (31) in Theorem 3. Thus we have the following result.
Corollary 3. Define m and n as in (43). System (31) is a bi-Hamiltonian system, that is, it takes the form where the compatible Hamiltonian operators H 1 and H 2 are given by and the corresponding Hamiltonian functions are specified by the densities ρ 1 = un and ρ 2 = u 2 vn + uv.
Notice that we did not get system (30) in Theorem 3. This is due to the assumptions we made on the Hamiltonian operators. Indeed, it is also bi-Hamiltonian, but does not have a Hamiltonian operator of the form (57); this shows that the classification using Hamiltonian pairs is not equivalent to the symmetry approach. Here we just state the relevant result without proof, since the proof uses the same method as for Theorem 5.
Proposition 2. Define m and n as in (43). System (30) is a bi-Hamiltonian system, that is, where the compatible Hamiltonian operators H 1 and H 2 are given by and the corresponding Hamiltonian densities are ρ 1 = 2un x and ρ 2 = u 2 vn x + u 2 v x n + uv x .
Remark 4. The inverse of the Hamiltonian operator 2H 2 in Proposition 2 takes the form Thus the local symmetries for system (30) can be generated using the recursion operator H 1 H −1 2 .
It follows from Corollary 1 and Corollary 3 that both systems possess the same Hamiltonian operator H 1 (in fact, J (3) = J (c) ). So both Hamiltonian operators H (3) and H (c) form Hamiltonian pairs with the same operator. We are able to directly verify that any linear combination of H (3) and H (c) is also Hamiltonian, and forms a Hamiltonian pair with J (3) . Thus we can construct the integrable system which contains both equations (46) and (60). If we take c 1 = c 5 = 0, c 3 = 1, a 8 = 2α, a 1 = 2β, and b 1 = 2γ, then we get the system (32) in Theorem 4. Thus we have the following result. where the compatible Hamiltonian operators H 2 and H 1 are given by and the corresponding Hamiltonian densities are ρ 1 = un and ρ 2 = uv(2αm + 2βn + γun + 1).
The same situation arises for systems (28) and (30), upon comparing Corollary 2 to Proposition 2. We present the result immediately, as follows.
Corollary 5. Define m and n as in (43). System (33) takes the bi-Hamiltonian form where the compatible Hamiltonian operators H 1 and H 2 are given by x n x ) and the corresponding Hamiltonian densities are ρ 1 = 2un x and

Reciprocal links, Lax pairs and exact solutions
In this section we describe reciprocal transformations relating the coupled Camassa-Holm type systems to negative flows in other known integrable hierarchies. We also present Lax pairs, and provide some exact solutions in certain cases.
Before we proceed, it is worth commenting on the linear terms appearing in the systems under consideration. It is necessary to include linear dispersion terms in order to be able to apply the perturbative symmetry approach. However, given a system in the form (27), we can rescale the dependent variables and time so that where d is the common degree of f and g in u, v and their derivatives, and then take the limit → 0, to obtain the system in the form m t =f , n t =ĝ, where m = u − u x , n = v + v x andf ,ĝ are homogeneous of degree d. In general, the latter system is not isomorphic to the original system (27), although this is the case for the first quadratic system (28). Indeed, if we perform a combination of shifting the dependent variables with a Galilean transformation, that is then with u 0 = v 0 and a suitable choice of c it is possible to remove the u x and v x terms on the right-hand side of (28). However, for the second quadratic system (29) this is not the case, because applying (62) creates a mixture of u x and v x terms on the right-hand side of the system; and for the systems with cubic nonlinearity, applying (62) produces additional quadratic and linear terms.

First quadratic system
We have already seen that the first quadratic system is related by a Miura map to the system (5) of Chen-Liu-Zhang. This means that we can immediately obtain a Lax pair for (28), by using the results in [3].
Proposition 3. The first quadratic system (28) has the Lax representation In fact, as already mentioned, the linear dispersion terms can be removed from this particular system without taking any scaling limit, by using (62), and after rescaling time the system becomes (4), which can be written in the form In order to obtain solutions of the system, it is helpful to make use of the second equation of the system (5), which is in conservation form, and leads to the introduction of new independent variables X, T via the reciprocal transformation dX = q dx + pq dt, dT = dt, with p = u + v, q = m + n.
As explained in [3], this change of independent variables transforms (5) to the first negative flow of the AKNS hierarchy (which, at the level of the Lax pair, is equivalent to the classical Boussinesq hierarchy, up to a gauge transformation). Under the reciprocal transformation, we have the following system of four equations relating u, v, m, n: Solutions of this system, as functions of X, T , lead to parametric solutions of the original system (4). However, it turns out that it is more convenient to first obtain solutions of the second quadratic system, as described in the next subsection, and then exploit a Miura map between the two systems, rather than attempting to solve (64) directly.

Second quadratic system
In this subsection we consider the second quadratic system (29) without linear dispersion terms, which (after rescaling t) can be written as where all the dependent variables are given upper case letters to distinguish them from the variables in the first quadratic system. The need to make this distinction here is due to the following result.

Proposition 4.
A solution of the first quadratic system (4) gives rise to a solution of the second quadratic system (65) via the Miura map Proof: From (66) it follows that U = u + n and U + V = u + v, so that Upon taking the time derivative of the latter two equations and using (4), we see that the difference and sum of M and N evolve according to which is equivalent to (65).
From the above, we see that the system (65) is intermediate between (4) and (5), and we can write the Miura map from (65) to (5) directly as By taking the Lax pair in [3], or by shifting/scaling the coefficients of the Lax pair in Proposition 3 and using (66), we immediately have the following.
Proposition 5. The second quadratic system (65) has the Lax representation Next, observe that the first equation in (67) is just the conservation law q t = (pq) x . This means that the same reciprocal transformation (63) can be used to link (65) to the first negative AKNS flow. The equations (67) and the relations Upon adding and subtracting the equations that involve only X derivatives, and using (68), we see that the relations hold. With the introduction of a potential f (X, T ) into the conservation law for q −1 , such that q −1 = f X , p = −f T , it is possible to use (68) and (70) to express U, V purely in terms of derivatives of f . Moreover, all of the terms in the conservation law for (M + N )q −1 in (69) can also be rewritten in terms of f , to yield a single equation for this potential, namely The latter equation is equivalent to equation (2.16) in [3]; below we rewrite it in a form which makes it more easily identifiable as such.  Theorem 7. Let f (X, T ) be a solution of the equation and gives a solution (U (x, t), V (x, t)) of the system (65) in parametric form.

Corollary 6.
A solution (u(x, t), v(x, t)) of the system (4) is given in parametric form by taking Proof of Corollary: Applying the reciprocal transformation (63) to the second equation in The expression for v then follows by using the formula for V in (72) and integrating with respect to X (which leaves a function of time unspecified); u is then found by noting that u Example: travelling waves. Travelling waves of the system (65) depend on x, t via the combination z = x − ct, where c is the wave velocity. They are obtained in parametric form by taking z =f (Z), f (X, T ) =f (Z) + cT, Z = X − CT, which gives solutions of (71) corresponding to travelling waves with velocity C in the reciprocally transformed system (69). If we set ρ =f , then (71) becomes an ordinary differential equation of third order for ρ, and after integrating twice this yields where K 1 , K 2 are arbitrary constants. The general solution of the latter equation is an elliptic function ρ(Z). In general, from (72), U and V are then given in parametric form in terms of ρ(Z) and ρ (Z) according to Here we consider single soliton solutions, which are obtained by choosing the quartic in (73) to have a double root. In that case, the solutions take the form where the values of C and c are fixed by the choice of parameters k and r 0 , δ > 0. To be more precise, substituting the solution (75) into (73) determines C, c, K 1 , K 2 , and r 0 must be chosen to ensure that dz/dZ = ρ(Z) > 0 everywhere, in order for the parametric solution for U, V to be single-valued. Upon integrating (75), the similarity variable z = x − ct is obtained as z = r 0 Z ± 2k log 1 + tanh(δZ/2) 1 + tanh(δZ/2) + log 1 ∓ 2ktanh(δZ/2) + tanh 2 (δZ/2) 1 ± 2ktanh(δZ/2) + tanh 2 (δZ/2) , up to shifting by an arbitrary constant. The field ρ has the shape of a dark soliton (a wave of depression) when the plus sign is chosen in (75), while with a minus sign it is a bright soliton; in Figure 1 the corresponding fields U, V given by (74) are plotted in these two different cases.

First cubic system
For simplicity, we consider the system (30) in the absence of linear terms on the right-hand sides, in which case (with suitable scaling) it can be written as In that case, it is useful to consider the first non-trivial symmetry of the system, which (up to rescaling) takes the form The quantity F = mn is a conserved density for both (76) and the latter symmetry, which satisfies In order to find the Lax pair for the cubic system, it is helpful to consider a simultaneous reciprocal transformation in the independent variables x, t, τ , by setting dX = F dx + uvF dt − Gdτ, dT = dt, ds = dτ.
(Of course, this could be extended to include the whole hierarchy of symmetries of (76), but the symmetry ∂τ is sufficient for our purposes.) The partial derivatives transform as ∂ x = F ∂ X , ∂ t = ∂ T + uvF ∂ X and ∂ τ = ∂ s − G∂ X . To begin with, we identify the symmetry (77) by introducing new dependent variables and find that under the reciprocal transformation (79) it yields a system of derivative nonlinear Schrödinger type, namely which is the Chen-Lee-Liu system [2]. For the latter system, we take the Lax pair in the form If the same reciprocal transformation (79) is applied to (76), then in terms of the variables p, q, u, v we find a system given by two pairs of equations, that is which is symmetrical under the involution The latter system corresponds to a negative flow in the hierarchy of symmetries of the Chen-Lee-Liu system [2], and its Lax pair is found by taking the same X part as in (81) and a T part which is linear in the inverse of the spectral parameter λ.
Proposition 6. The system (82) has the Lax pair where F is as in (81), and .
Remark 5. Upon taking the first component of the vector Ψ to be ψ 1 = √ qφ, the X part of the Lax pair implies that the function φ is a solution of the energy-dependent Schrödinger equation where U, V are certain functions of p, q and their derivatives. This shows that the system (82) is related by a Miura transformation to the first negative flow in the classical Boussinesq hierarchy.
Corollary 7. The system (76) has the Lax pair Proof of Corollary: This follows immediately by applying the inverse of the reciprocal transformation (79) to the vector wave function Ψ = (ψ 1 , ψ 2 ) T in (84).
the inverse of the reciprocal transformation (79), as required. The same parametric solution (u(x, t), v(x, t)), with x = f (X, t), can be obtained in a different way by exploiting the symmetry (83). Indeed, from (82), or by applying the involution to (86), the dependent variables p, q can be rewritten in terms of u, v and their derivatives as with Π = uv and r * = v u . The involution (83) swaps w ↔ Π and r ↔ r * , and this leads to the alternative system (89), from which u, v are recovered directly.
Furthermore, for the independent variables we have where the prime denotes d/dZ. To describe these travelling waves, it is most convenient to obtain a single equation for π(Z), which is achieved by first using the definition of B * to write then putting this and (91) into (89), to obtain a pair of quadratic equations in B * with coefficients depending only on π and its derivatives. After eliminating B * to find (a) The quartic curve (96) in the (π, π ) plane.
(b) Parametric plot of (z(Z), π(Z)). then removing a prefactor, a single equation of second order and second degree for π results: The latter equation has a first integral: if π satisfies the first order equation for any constant value K, then it satisfies (95). The generic solution of (96) is an elliptic function of Z, but to have bounded periodic solutions for real c, C, K requires that the curve (π ) 2 = Q(π) in the real (π, π ) phase plane should have a compact component (see Figure 3a), otherwise solutions are generically unbounded with simple poles on the real Z axis.
(b) Parametric plot of (z(Z), v(Z)). up to shifting z by an arbitrary constant. It is necessary to impose the conditions in order to have a real single-valued solution in z, otherwise dz dZ = w(Z) will vanish for some Z. So in this solution, corresponding to the plus sign in (97), u is constant and v is a kink-shaped travelling wave -see Figure 2; with the opposite choice of sign, the roles of u and v are reversed.

Second cubic system
After removing the linear dispersion terms and rescaling for the sake of simplicity, the system (31) becomes Both equations in this system are in conservation form, but in order to apply a reciprocal transformation we pick the conservation law For what follows, we also note the equation Now from (102) we can define new independent variables according to so that derivatives transform according to ∂ x = q ∂ X , ∂ t = ∂ T + pq ∂ X . Since this is a reciprocal transformation, the equation (102) becomes a conservation law in the new variables, that is while the evolution of κ in (103) becomes This means we can write the quantities m and n in terms of q as where the prefactor κ ∓1 depends only on the new independent variable X. The question is now how to find an equation for q = q(X, T ) and thence obtain the fields u and v in terms of functions of X and T , and thence obtain solutions u(x, t), v(x, t) in parametric form.
To begin with note that, in view of (104) and (106), we can use u x = u − m, v x = n − v and transform the derivatives to find This means that from (105) we obtain ∂ T (q −1 ) = −(u X v + uv X ) = −κ u + κ −1 v, and hence The above expression for v can be substituted back into (105) to yield In order to get a single equation involving only κ and q, it is necessary to write u in terms of κ,q and their derivatives, and this is achieved by substituting (108) into the second equation in (107), so that the latter becomes a linear system for u and u X , which is readily solved. However, it turns out that it is most convenient to introduce a new function ϑ(X, T ), which is defined by In terms of ϑ and κ, u and v are then given by so that the product p = uv is independent of κ, and so (105), or equivalently (109), becomes an autonomous partial differential equation for ϑ alone, namely Upon introducing a potential f (X, T ) such that ϑ = f X , this equation can be integrated with respect to X, and an arbitrary function of T that appears can be absorbed into f without loss of generality, so that an equation of third order for f results, that is Theorem 9. Let f = f (X, T ) be a solution of (113), let κ = κ(X) be an arbitrary function, and let ϑ(X, T ) = f X (X, T ). Then setting together with (111) gives a solution (u(x, t), v(x, t)) of the system (101) in parametric form.
Proof: Comparison of (113) with (111) shows that p = uv = −2f T . Then taking the differential of x above gives dx = (2f X (X, t) − ∂ X log κ) dX + 2f T (X, t) dt = q −1 dX − p dT , in accordance with the inverse of the reciprocal transformation (104). By reversing the reciprocal transformation, the equations (102) and (103) result, and together these imply the system (101) for u and v.
In order to find solutions of the equation (113), it is instructive to consider the behaviour near singularities. The equation has two types of expansions near a movable singularity manifold ϕ(X, T ) = 0, with leading order behaviour f ∼ ± log ϕ, corresponding to simple poles in the solution of (112). This suggests that one can apply the two-singular-manifold method introduced in [4], leading to the following result.
where Y is a solution of the Riccati system with an arbitrary parameter λ. The Riccati system is linearized via the transformation to yield a scalar Lax pair for (112), given by Corollary 8. The system (101) has the scalar Lax pair where with p = uv, q = √ mn, κ = n/m as above.
Proof of Corollary: The Lax pair follows from (115) by setting ψ = √ q φ and applying the inverse of the reciprocal transformation (104). The compatibility conditions for this linear system consist of (102) together with r t = 1 2 p xxx + 2p x r + pr x + 2q 2 w x + 2qq x w and w xxx + 4rw x + 2r x w = 0, where the last one is a consequence of the definition of r in (117). These conditions are best checked with computer algebra.
The form of the Lax pair (115) reveals that ϑ corresponds to the dependent variable for the modified KdV equation, and the standard Miura map V = ϑ X − ϑ 2 relates (112) to the first negative flow of the KdV hierarchy, as considered in [10] (see also [12]), which takes the form in terms of the variables U, V. If r and w were constants, then (116) would reduce to the Lax pair for the Camassa-Holm equation, as presented in [1].  we find that W (Z) =f (Z) satisfies the following ordinary differential equation of second order and second degree:

Example
The latter equation is solved in elliptic functions: for any value of the constant c 2 , W is a solution of (118) whenever it satisfies For such a solution, Theorem 9 gives x = log ρ(X − µt) 2 κ(X) − νt, with ρ(Z) = exp W (Z) dZ, while (111) becomes so in order to avoid singularities in u and v, we require that W should be a bounded, positive periodic function of Z; this is achieved by choosing the quartic on the right-hand side of (119) to have three positive real roots, 0 < w 1 < w 2 < w 3 , whence the fourth root is w 0 = −(w 1 + w 2 + w 3 ) < 0. Using a Möbius transformation W = α(℘ − β) −1 + w 1 to send the first positive root to infinity leads to the solution in terms of Weierstrass functions, similarly to the previous example for the system (76).
The behaviour of the solutions u(x, t), v(x, t) obtained in this way depends crucially on the choice of function κ(X). In order to have singled-valued soutions it is necessary that the derivative ∂x/∂X should never vanish, which requires that the logarithmic derivative κ /κ should be suitably bounded. In particular, if κ = constant then this is so, and in that case travelling wave solutions of (101) result, and both u and v are periodic functions. More generally, taking κ = exp k(X) in (121), where both the function k and its first derivative are bounded, gives bounded deformations of these periodic solutions -see Figure 5 for the comparison between the cases κ = 1 and κ = exp sin X. However, if κ = exp k(X) with k(X) being a linear function of X, then unbounded solutions result, exhibiting similar profiles to the solutions of (76) with exponential growth/decay on a periodic background, as illustrated in Figure 4.

Conclusions
The perturbative symmetry approach has yielded a classification of integrable two-component systems of the form (3), producing two systems with quadratic nonlinearities (Theorem 2), two systems with cubic nonlinearities (Theorem 3), and two mixed quadratic/cubic systems (Theorem 4); the systems with mixed nonlinear terms include the others as limiting cases, by sending suitable parameters to zero. At the same time, an alternative approach via compatible Hamiltonian operators has provided a different set of two-component systems, and has allowed us to find bi-Hamiltonian structures for all of the systems obtained from the symmetry approach. We have also found Lax pairs for all of the systems in Theorems 2 and 3, at least in the absence of linear dispersion terms, as well as reciprocal transformations linking them to known integrable hierarchies, and this has allowed us to construct some simple solutions explicitly.
As far as we know, integrable systems of the form (3) have not been considered in detail before, apart from Falqui's system (6). However, while we were completing this work we learned of a three-component system in which two of the equations involve nonlocal terms of this type; the system was constructed as a dispersive version of the WDVV associativity equations [21]. There are several issues still to be resolved regarding the systems introduced here. In particular, for the systems (29), (30) and (31), as well as the systems in Theorem 4, we have not presented Lax pairs that include the linear dispersion terms. Also, the system (44), or equivalently (45), is worthy of further analysis, since it is outside the class (3).
In the near future, we further intend to classify two-component systems with the nonlocal terms (1 − D 2 x )u t , (1 − D 2 x )v t on the left-hand side, which include (48). Recently, various different systems of this kind have been proposed [22,27], which deserve to be studied further.