On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems

The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A dispersionless Lax formulation is found.


Introduction
Quasilinear (2+1)-dimensional systems of the first order, play an important role in the description of variety of physical phenomena. The method of the hydrodynamical reductions (see e.g. [1]) enables us to pick from this class the integrable systems which possess sufficiently many hydrodynamic reductions, and thus infinitely many particular solutions. Recently, a system in involution describing the integrable (2+1)-dimensional hydrodynamical type systems where A ik and B ik are functions of v and w, was derived in [2] using the method of hydrodynamic reductions. In a particular case where α and β are constants, the corresponding system in involution for two functions r(v, w) and q(v, w) simplifies to the form q vv = (qr) w , q vw = q v q w q + q 2 r, q ww = q 2 w q + 2qq v − q w r w r , r ww = (qr) v , r vw = r v r w r + r 2 q, and a general solution therefore depends on 6 arbitrary constants. Two other functions p(v, w) and s(v, w) can be found in quadratures dp = rqdv + q v dw, ds = r w dv + rqdw.
A general solution of system (3) was presented in [2] in a parametric form. Under a simple linear transformation of independent variables (y, t) such that Our first result is that the equations (5) reduces to the most compact form Our second result is that we present a general solution of system in involution (7) in a new parametric form (33), which is more convenient for the investigation in the special cases. This approach is universal, i.e. any other (2+1)-dimensional quasilinear equations can be investigated effectively in the same fashion (see [6,5]). In the general case (1), a dispersionless Lax representation has the form (see [2]) where a(µ, v, w) and c(µ, v, w) are some functions. The function ψ is called a pseudopotential.
Equations (3) and (4) can be derived directly from the compatibility condition (ψ t ) y = (ψ y ) t . Our third result is that the dispersionless Lax representation (8) for (2+1)-dimensional hydrodynamical type system (5) reduces to the most compact form i.e., each of the functions a and c depends on two arguments only. Moreover, we were able to integrate the corresponding nonlinear system for the functions a(µ, w) and c(µ, v), i.e. we found the dispersionless Lax pair (9) for the (2+1)-dimensional quasilinear equations (6). In general, the compatibility condition (ψ t ) y = (ψ y ) t for the system where a = a(µ, u 1 , u 2 , . . . , u N ) and c = c(µ, u 1 , u 2 , . . . , u N ) are some functions (cf. (9)), yields (2+1)-dimensional quasilinear systems of the nonlinear equations of the first order. Recently, a complete classification of pseudopotentials that satisfy the single equation ψ t = a(ψ x , w) was given in [6]. Our fourth result is that we were able to extract the functions a(µ, w) and c(µ, v) such that (9) yields a system of the form (6).
Let us mention that the particular case, when was considered in [1] and a complete classification of all admissible functions h(v, w) was presented in [4]. The paper is organized as follows. In Section 2, we prove that quasilinear system (6) admits the dispersionless Lax representation (9), which is a special case of (8), and conversely, a dispersionless Lax representation (9) yields an integrable (2+1)-dimensional quasilinear system (6). We also derive the associated system in involution (3) which in the case under study takes the form (7). The problem of computation of a single function H(v, w) that solves (7) is reduced to quadratures. In Section 3, we present an effective method for direct integration of the system (3) and the corresponding equations are integrated in the parametric form. The last section contains the conclusion.

System in involution
A classification of integrable (2+1)-dimensional two-component Hamiltonian hydrodynamictype systems (10) was obtained in [1] using the method of hydrodynamic reductions. In this paper, we use dispersionless Lax representations following the original papers [2] and [7] (see [5] for further details). It was proved in [2] that the pseudopotentials ψ(x, y, t) for (2+1)dimensional hydrodynamic-type systems (1) must satisfy the dispersionless Lax representation of the form (8). Also, it was proved in [4] that there exist the dispersionless Lax representations for (10) of the form (9). Moreover, in our case the dispersionless Lax representation (9) remains valid for more general case (5). Indeed, the following assertion holds. Lemma 1. The hydrodynamic-type system (5) can be obtained from the compatibility condition (ψ t ) y = (ψ y ) t , where the pseudopotential ψ satisfies dispersionless Lax representation (9).
Proof . The compatibility condition for (8) implies

Substituting (5) into this equation yields
where µ = ψ x . Since we are interested in a general solution, the coefficients at the t-and x-derivatives v and w on the left-and right-hand sides of the above equations must be identical, that is, and a v = 0, c w = 0. Thus, a is independent of v, and c is independent of w. This means that (8) reduces to (9). The lemma is proved.
Proof . The compatibility condition (ψ t ) y = (ψ y ) t implies where the dispersionless Lax representation (9) is re-written in the form Since a w = 0, differentiating the equation (11) cannot depend on µ. This means that (see (5)) we have for some functions p(v, w) and q(v, w) that do not depend on µ. Then the above equation (13) reduces to the form (cf. (5)) Since c v = 0, two other relations of the above type, where r(v, w) and s(v, w) are some other functions independent of µ, can be derived in analogy with the above, along with the second equation of the hydrodynamic-type system (5). Thus, the hydrodynamic-type system (5) is indeed uniquely determined by the compatibility condition (ψ t ) y = (ψ y ) t . The theorem is proved.
Substituting (5) into (11) implies the relations among the first derivatives of the functions a(µ, w) and c(µ, v).
The compatibility conditions (a w ) µ = (a µ ) w , (a w ) v = 0 and (a µ ) v = 0 yield Similar formulas can be obtained from the other compatibility conditions, namely, ( The last equation can be replaced by the pair of equations which can be obtained from the compatibility conditions (p vw ) v = (p vv ) w and (s vw ) w = (s ww ) v . The second and fifth equations of (16) can be integrated once to yield where ϕ 1 (v) and ϕ 2 (w) are arbitrary functions. However, without loss of generality these functions can be set equal to 1, because these functions can be eliminated from all of the above equations upon using the scaling Thus, substituting p v = rq, s w = rq into (16), we finally obtain the system in involution (3) together with (4) (cf. [2]). Thus, integrable (2+1)-dimensional system (5) can be written in the most compact form (6), and the reduction to the Hamiltonian case (10) is given by the symmetric constraint r = q. The function H can be reconstructed via the complete differentials, dH vv = rdv + sdw, dH vw = sdv + pdw, dH ww = pdv + qdw, Remark 1. The above choice (17) uniquely fixes system (5) in conservative form (6). This means that the integrable (2+1)-dimensional hydrodynamic-type system (5) is reducible to (6) under appropriate choice of functions ϕ 1 (v) and ϕ 2 (w) in (17).

Dispersionless Lax representation
In this section, we obtain a general solution of the system in involution (7) and simultaneously reconstruct the functions a(µ, w) and c(µ, v) (see (9)). Rewrite system (15) for the function c(µ, v) in the form , where all the coefficients A k , B n , D m depend on v alone and are to be determined. The compatibility conditions (c vv ) µ = (c µv ) v , (c µµ ) v = (c µv ) µ give rise to the following system for the coefficients A k , B n , D m : where the prime denotes the derivative with respect to v.
The derivative c v can be expressed from the last equation of (18). Plugging this expression into the l.h.s. of the second equation of (18) yields which can be expanded into simple fractions (see [6] for the general case) where k i (v) are some functions such that k 1 + k 2 + k 3 = 2, and the functions b k (v) are roots of the cubic polynomial in the last equation of (18): Conversely, upon comparing (20) with (21) the above coefficients A k , B n , D m can be expressed via the functions b k (v) in the symmetric form Lemma 2. k i are constants.
Proof . Integrating (21) yields the following equation: where b(v) is a function of v alone. Then (see the last equation in (18)) we have The compatibility condition (c v ) µµ = (c µµ ) v is satisfied only if k i are constants. The lemma is proved.
Theorem 2. Under the above substitutions, the system (19) reduces to the form In this case we have Proof . The coefficient A 2 in the form (24) can be expressed from the second equation of (19). The last three equations of (19) are linear with respect to the first derivatives b ′ k . Solving this linear system with respect to b k immediately yields (23). Moreover, the third and fourth equations of (19) are then automatically satisfied. The theorem is proved.
where ξ is an arbitrary constant we obtain the following theorem: Theorem 3. General solution of system (25) can be written in the form Proof . Introduce the auxiliary functions b 12 = b 2 − b 1 and b 13 = b 3 − b 1 . Then the system (25) reduces to the form The ratio of the last two equations is nothing but a first-order ODE. Substituting the intermediate function V = 1 − b 12 /b 13 into this ODE reduces to the following quadrature: Taking into account that b 12 = (1 − V )b 13 , one can obtain the equality b 12 = ξV where the above quadrature is integrated in the parametric form b 13 = ξV k 1 −1 (1 − V ) k 3 −1 . Substituting these expressions for b 12 and b 13 into the first equation of (27), b ′ 1 = (1 − k 1 )b 12 b 13 , yields the quadrature The remaining equations in (27) yield (26). The theorem is proved. coincides with (31) up to the replacement c µ ↔ s. Since D 1 = 1, the compatibility condition (c v ) µµ = (c µµ ) v implies b(v) = 1 in (22). Thus, the derivative c v can also be found. Finally, substituting (33) and just obtained expressions for c µ , c v into (34) yields the corresponding dispersionless Lax representation (see (12), (14), (18)) which is now determined by means of the formulas where the auxiliary variable ǫ(µ) is determined by the formula (cf. (32))

Conclusion
As it was mentioned in the Introduction, the integrable (2+1)-dimensional quasilinear systems of the nonlinear first-order equations (see [5] for details) are determined by the compatibility condition (ψ t ) y = (ψ y ) t , where in general ψ t = a(µ, u 1 , u 2 , . . . , u N ) and ψ y = c(µ, u 1 , u 2 , . . . , u N ), cf. (8) and (9). An open problem is whether it is possible to construct hydrodynamic chains (see, for instance, [3]) associated with such (2+1)-dimensional quasilinear systems. The theory of integrable hydrodynamic chains is much simpler than the theory of integrable (2+1)-dimensional quasilinear equations, because the former still is a theory of integrable (1+1)-dimensional hydrodynamic-type systems with just one nontrivial extension -allowing for infinitely many components. Thus, an integrable hydrodynamic chain possesses the properties that are well-known in the theory of finite-component systems (dispersive or dispersionless), such as infinite series of conservation laws, infinite series of commuting flows, and infinite series of Hamiltonian structures. At least, we can answer the above question regarding the construction of the associated hydrodynamic chain for the case of (9), but this will be the subject of a separate paper. Moreover, if we fix the first equation in (9), then an associated integrable hierarchy can be found from the dispersionless Lax representations (cf. (8) and (9)) ψ t = a(ψ x , w), where the first member of this hierarchy is given by (6) and uniquely determined by the dispersionless Lax representation (9). This means that infinitely many commuting flows (numbered by the "times" y k ) will be determined. It would be interesting to find the associated hydrodynamic chains for the case when, instead of the above dispersionless Lax representation, one has a more general ansatz ψ t = a ψ x , w 1 , w 2 , . . . , w M , ψ y N = c ψ x , v 1 , v 2 , . . . , v N , where M and N are arbitrary positive integers.