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Integrability of Exceptional Hydrodynamic Type Systems

In this paper we consider non-diagonalisable hydrodynamic type systems integrable by the Extended Hodograph Method. We restrict our consideration to non-diagonalisable hydrodynamic reductions of the Mikhalev equation. We show that families of these hydrodynamic type systems are reducible to the Heat hierarchy. Then we construct new particular explicit solutions for the Mikhalev equation.


Introduction
The theory of two-dimensional integrable quasilinear systems of first order by the Tsarev Generalised Hodograph Method was established in [12]. Such systems we call semi-Hamiltonian hydrodynamic type systems. They have all pairwise distinct roots of characteristic polynomials of velocity matrices 1 and they are diagonalisable.
In this paper we deal with integrable hydrodynamic type systems, which have just one root. These systems are non-diagonalisable. However, they have infinitely many hydrodynamic conservation laws, commuting flows and particular solutions. Instead of Generalised Hodograph Method we apply the Extended Hodograph Method presented in [11].
In a general case even diagonalisable hydrodynamic type systems with pairwise distinct roots are not semi-Hamiltonian, i.e. they are non-integrable. Classification and integration of hydrodynamic type systems with double roots, triple roots, etc. become much more complicated from combinatorial point of view. For instance, in the two-component case we have two different sub-cases: two distinct roots and a one common root; in the three-component case we have three different sub-cases: three distinct roots, a one double root and a one triple root; in the four-component case we have five different sub-cases: four distinct roots, a one double root, two double roots, a one triple root and a one quadruple root, etc. This is one of many reasons: why integrable hydrodynamic type systems with multiple roots were not investigated earlier.
Another important reason was absence of any interesting examples known in the theory of integrable systems. Just very recently (see [5]), such a family of hydrodynamic type systems was extracted from integrable hydrodynamic chain (see [8]) It was shown in [8] that this chain possesses infinitely many integrable diagonalisable hydrodynamic reductions with pairwise distinct roots. The authors of [5] proved that this chain also possesses infinitely many integrable hydrodynamic reductions with multiple roots. Then the case with a single root was deeply investigated in [6], where the authors constructed a link to the Heat hierarchy. In this paper we continue to consider hydrodynamic reductions with a single root. We apply the Extended Hodograph Method for linearisation of these hydrodynamic type systems to the Heat hierarchy, i.e. u t = u xx , u y = u xxx , u z = u xxxx , ..., and we derive corresponding solutions for the remarkable Mikhalëv equation (see [7]). The paper is organised as follows: in Section 2 we discuss integrable hydrodynamic chains (2), which possess infinitely many non-diagonalisable hydrodynamic reductions with multiple roots. In Section 3 we apply the Extended Hodograph Method for linearisation of multi-component hydrodynamic reductions with a single multiple root to the remarkable Heat hierarchy. In Subsection 3.1 we consider the simplest two-component case integrable by the classical hodograph transformation. In Subsection 3.2 we reduce two commuting three-component hydrodynamic type systems with a single triple root to the Heat equation together with its first higher commuting flow. In Subsection 3.3 we reduce an arbitrary number of commuting multi-component hydrodynamic type systems with a single quadruple root to the Heat hierarchy. In Section 4 we show that the integrable three-dimensional linearly degenerate Mikhalëv system simultaneously possesses multi-component non-diagonalisable hydrodynamic reductions with a single multiple root. In the three-component case we construct corresponding particular solutions presented in explicit forms. Finally in Conclusion 5 we discuss existence of such non-diagonalisable hydrodynamic reductions for other integrable hydrodynamic chains.
Thus, (2) can be written in the form where Φ is a potential function for conservation laws (7), i.e. a k = Φ k . Here Φ m ≡ ∂Φ/∂t m . Remark: Infinitely many conservation law densities σ k (c) can be found from the expansion (λ → 0) This means, that (cf. (3)) Taking into account (4), one can obtain again Bell polynomials (with another choice of multipliers) Corresponding conservation laws (see [8]) are The crucial observation made in [5] is that the reduction c N +1 = 0 reduces hydrodynamic chain (1) to N component non-diagonalisable hydrodynamic type systems, which have infinitely many conservation laws, commuting flows and infinitely many particular solutions.
In these cases, (3) and (9) reduce to the form, respectively The potential function Φ (N ) can be found in quadratures

The Extended Hodograph Method
Integration of N component hydrodynamic type systems by the Extended Hodograph Method is based on the preliminary computation of auxiliary N − 2 commuting flows. In a general case, this is very complicated task even for semi-Hamiltonian hydrodynamic type systems, because usually commuting hydrodynamic flows can be found solving some linear system of equations with variable coefficients in partial derivatives (for further details, see [12]). However, in our case they already are extracted from higher commuting flows (2) by the same reduction c N +1 = 0. Examples: I. If N = 2, then c 3 = 0. Then we have just the two-component hydrodynamic type system (see (5)) II. If N = 3, then c 4 = 0. Then we have two three-component commuting hydrodynamic type systems (see (5) III. If N = 4, then c 5 = 0. Then we have three four-component commuting hydrodynamic type systems (see (5), (6)) where (we remind, see (4)) a 1 = c 1 , a 2 = c 2 + 1 2 c 2 1 and a 3 = c 3 + c 1 c 2 + 1 6 c 3 1 . So, now the construction of first N −1 commuting hydrodynamic type systems becomes obvious. Due to a triangular form of all above square matrices, each of them possesses just a single root (i.e. the Nth matrix has a single root a N ).

The Hodograph Method
Here we apply a classical hodograph method for integration of two-component hydrodynamic type system (12). As usual such a system can be written in the potential form Then one can introduce the auxiliary potentials Now field variables c 1 and c 2 became new independent variables, while independent variables x and t became new field variables. Existence of the potential functionsξ 1 andξ 2 means that their second order derivatives must coincide, i.e.
Thus, here we show that two-component hydrodynamic type system (12) is connected with the Heat equation (15) by the hodograph transformation.

The Three Component Case
Here we apply the Extended Hodograph Method for integration of two commuting threecomponent hydrodynamic type systems (13). These systems also can be written in the potential form Then one can introduce the auxiliary potentials dξ 1 = x + tc 1 + yc 2 + 1 2 yc 2 1 dc 1 + (t + yc 1 )dc 2 + ydc 3 , Now field variables c 1 , c 2 and c 3 became new independent variables, while independent variables x, t and y became new field variables. Existence of the potential functionsξ 1 ,ξ 2 andξ 3 means that their second order derivatives must coincide, i.e.

The Multi Component Case
Once some particular common solution of these equations is found, all other time variables can be found in quadratures, i.e.
Moreover, the dependencies t k (c) can be expressed explicitly. Indeed, where the function U N satisfies the reduced Heat hierarchy (17) Thus, the potential function Φ (N ) (see (11)) also can be found explicitly, i.e.
Mikhalëv system (20) also can be obtained from (5). Indeed, one can take first two equations from the first hydrodynamic chain and just one equation from the second hydrodynamic chain Eliminating the field variable c 3 , one can obtain the two-component three-dimensional system which is equivalent to Mikhalëv system (20), where (see (4)) a 1 = c 1 , a 2 = c 2 + c 2 1 /2. Taking into account (see (8) In a general case, computation of particular solutions based on the method of twodimensional diagonalisable reductions is a very complicated task, because some intermediate calculations contain integrations of linear systems with variable coefficients in partial derivatives (for further details, see [2]). However, in the case of non-diagonalisable hydrodynamic reductions considered in the previous Section, corresponding solutions of Mikhalëv equation (22) can be written in implicit form.
Without loss of generality and for simplicity here we consider the three-component case (13) only. For better presentation below we denote c 1 = X, c 2 = T and c 3 = Y . Now we utilise formulas (18) and (19).
Lemma: The particular solution of three-dimensional Mikhalëv equation (22) selected by non-diagonalisable hydrodynamic reduction (13) can be presented in the implicit form where the function U(X, T, Y ) is a solution of the pair of commuting Heat equations and the functions X(x, t, y), T (x, t, y), Y (x, t, y) can be found from Example I: If (κ is an arbitrary constant)

then Mikhalëv equation (22) has a particular solution
Φ(x, t, y) = y κ 3 ln Remark: While the Extended Hodograph Method admits linearisation of sets of N − 1 commuting hydrodynamic flows, the Tsarev Generalised Hodograph Method allows to extract infinitely many particular solutions from higher commuting dispersive chains (8). For instance, non-diagonalisable commuting hydrodynamic type systems (13) have infinitely many hydrodynamic conservation laws and infinitely many higher commuting flows where the function U(X, T, Y ) satisfies (23). Statement: The Heat hierarchy has infinitely many polynomial solutions U k (c) which are Bell polynomials, determined by (see (3) and (9)) Example II: A general solution of Mikhalëv equation (22) is determined by (see (11), where (we remind that the function U(X, T, Y ) satisfies (23)) So, infinitely many polynomial solutionsŨ k (c) can be extracted from the expansion (cf. For instance (here we again use the notation c 1 = X, c 2 = T, c 3 = Y ), Corresponding solutions of Mikhalëv equation (22) become (k = 4, 5, ...) In the simplest nontrivial caseŨ 4 , we have three commuting hydrodynamic type system (cf.
According to [4], [12] one can obtain the system 2 Then (see (11) So, the existence of non-diagonalisable hydrodynamic reductions for integrable threedimensional quasilinear systems of first order is also a new interesting phenomenon. Thus, examination of multi-dimensional quasilinear systems of first order should be continued by this reason in further publications.

Conclusion
Strictly hyperbolic hydrodynamic type systems have pairwise distinct roots. In this paper we considered the opposite situation: hydrodynamic type systems with multiple roots. The most degenerate case, just a one multiple root. This means that such hydrodynamic type systems no longer are hyperbolic. They are parabolic. The transformation to the Heat equation hierarchy is a natural illustration of this parabolic phenomenon.
An effective mechanism of integration of hydrodynamic type systems with pairwise distinct roots based on existence of special coordinate system, i.e. the Riemann invariants. In the case considered in this paper the coordinate system c k is a most convenient from computational point of view. These coordinates have a natural interpretation, in these coordinates the Heat hierarchy is written as an infinite set of commuting linear equations in partial derivatives with constant coefficients.
A straightforward computation shows that, for instance, the Benney hydrodynamic chain does not possess such non-diagonalisable hydrodynamic reductions (hydrodynamic type systems with multiple roots). So, we come to the observation: just linearly degenerate hydrodynamic chains (they have at least one N component linearly degenerate hydrodynamic reduction for each N, for further details see [1], [3], [9]) possess hydrodynamic reductions with multiple roots. Inspection of other linearly degenerate hydrodynamic chains should be made in a separate paper. was partially supported by the RFBR grant No. 16-51-55012. The author also appreciates kind hospitality at the Tsinghua University (Beijing, China), where the main part of this research was completed.