Quasilinear systems of Jordan block type and the mKP hierarchy
aa r X i v : . [ n li n . S I] J a n Quasilinear systems of Jordan block type and the mKP hierarchy
Lingling Xue , E.V. Ferapontov , Department of MathematicsNingbo UniversityNingbo 315211, P.R. China Department of Mathematical SciencesLoughborough UniversityLoughborough, Leicestershire LE11 3TUUnited Kingdom Institute of Mathematics, Ufa Federal Research Centre,Russian Academy of Sciences, 112, Chernyshevsky Street,Ufa 450077, Russiae-mails: [email protected]@lboro.ac.uk
Abstract
We demonstrate that commuting quasilinear systems of Jordan block type are parametrisedby solutions of the modified KP hierarchy. Systems of this form naturally occur as hydrody-namic reductions of multi-dimensional linearly degenerate dispersionless integrable PDEs.MSC: 35Q51, 37K10.
Keywords: parabolic quasilinear systems, commuting flows, mKP hierarchy, hydrodynamicreductions. ontents Quasilinear systems of the form u t = v ( u ) u x (1)have been thoroughly investigated in the literature. Here u = ( u , . . . , u n ) T is a column vectorof the dependent variables and v is a n × n matrix. The main emphasis has always been on thestrictly hyperbolic case where the matrix v has real distinct eigenvalues. Under the additionalcondition that the Haantjes tensor of matrix v vanishes, any such system can be reduced to thediagonal form R it = λ i ( R ) R ix , (2) i = 1 , . . . , n , in specially adapted coordinates R , . . . , R n known as Riemann invariants. Systemsof type (2) govern a wide range of problems in pure and applied mathematics, see e.g. [15, 1, 13].It was shown by Tsarev [15] that under the so-called semi-Hamiltonian constraint, λ ij λ j − λ i ! k = (cid:18) λ ik λ k − λ i (cid:19) j , system (2) possesses infinitely many conservation laws and commuting flows, and can be solvedby the generalised hodograph method (here i = j = k and low indices indicate differentiationby the variables R j ).In this paper we study quasilinear systems (1) of Jordan block type. More precisely, weassume the existence of special coordinates (which we will also denote R , . . . , R n ) where theequations reduce to upper-triangular Toeplitz form R t = ( λ E + n − X i =1 λ i P i ) R x ; (3)here R = ( R , . . . , R n ) T , E is the n × n identity matrix, P is the n × n Jordan block with zeroeigenvalue (note that P n = 0), and λ , λ i are functions of R . Explicitly, a three-componentversion of system (3) is as follows: R R R t = λ λ λ λ λ λ R R R x . v is pointwise of Jordan block type;(b) the Haantjes tensor of matrix v vanishes.The vanishing of the Haantjes tensor makes systems (3) natural parabolic analogues ofhydrodynamic type systems (2) in Riemann invariants. Note that upper-triangular Toeplitzmatrices form a commutative family (cyclic Haantjes algebra in the terminology of [14]). Sys-tems of type (3) appear as degenerations of hydrodynamic type systems associated with multi-dimensional hypergeometric functions [5], in the context of parabolic regularisation of the Rie-mann equation [6], and as reductions of hydrodynamic chains and linearly degenerate disper-sionless PDEs in 3D [11]. The most well-studied case of system (3) corresponds to the choice λ = R , λ = 1 , λ i = 0 , i ≥ (cid:18) R R (cid:19) t = (cid:18) ψ ψ ψ (cid:19) (cid:18) R R (cid:19) x where ψ satisfies the Lax equation of the mKP hierarchy, ψ = ψ + ρψ ;here low indices indicate differentiation by R , R . Fixing the potential ρ and varying ψ we obtaincommuting flows of the corresponding hierarchy. Similarly, any three-component hierarchy oftype (3) can be parametrised in the form R R R t = ψ ψ ψ + w ψ ψ ψ ψ R R R x , where w solves the mKP equation4 w + 6 w w − w − w − w w = 0 , and ψ satisfies the corresponding Lax equations ψ = ψ + 2 w ψ , ψ = ψ + 3 w ψ + 32 ( w + w + w ) ψ . Fixing w and varying ψ we obtain commuting flows of the hierarchy. We show that the corre-sponding conserved densities are governed by the adjoint Lax equations.In section 3 we demonstrate that systems of Jordan block type naturally occur as hydrody-namic reductions of multi-dimensional linearly degenerate PDEs: the 3D Mikhalev system [10]is used as an illustrating example, see also [11]. The class of two-component Toeplitz systems (3), (cid:18) R R (cid:19) t = (cid:18) λ λ λ (cid:19) (cid:18) R R (cid:19) x ,
3s form-invariant under triangular changes of variables ( R , R ) ↔ ( r , r ) of the form R = F ( r , r ) , R = G ( r ) , (4)where F and G are arbitrary functions of the indicated arguments. Similarly, the class of three-component Toeplitz systems (3), R R R t = λ λ λ λ λ λ R R R x , is form-invariant under triangular changes of variables ( R , R , R ) ↔ ( r , r , r ) of the followingform: R = r ( ∂ r G ) ∂ r H + F ( r , r ) , R = G ( r , r ) , R = H ( r ) , (5)where F, G and H are arbitrary functions of the indicated arguments. This transformationfreedom will be utilised to simplify the classification results. Note that the group preservingthe class of diagonal systems (2) is far more narrow, generated by transformations of the form R i = F i ( r i ), functions of one variable only. Let us recall that commuting flows of the diagonal system (2) are governed by the equations w ij = a ij ( w j − w i ) (6)where a ij ( R ) = λ ij / ( λ j − λ i ) are fixed and solutions w i to linear system (6) vary [15]. Note thatthe consistency conditions of system (6), a ijk = a ij a jk + a ik a kj − a ik a ij , are equivalent to the integrable 2 + 1-dimensional n -wave system. Thus, commuting flows ofdiagonal form (2) are governed by the n -wave hierarchy.Below we demonstrate that commuting flows of Toeplitz type (3) are governed by the mKPhierarchy. Furthermore, the corresponding conserved densities of hydrodynamic type satisfy theadjoint Lax equations. Direct calculation shows that the compatibility (commutativity) of two-component systems ofthe form (cid:18) R R (cid:19) t = (cid:18) λ λ λ (cid:19) (cid:18) R R (cid:19) x , (cid:18) R R (cid:19) y = (cid:18) µ µ µ (cid:19) (cid:18) R R (cid:19) x is equivalent to the following two conditions: λ λ = µ µ ,λ − λ λ = µ − µ µ . λ λ = m, λ − λ λ = ρ ; note that the quantities m, ρ are shared by allcommuting flows. If m = 0, using symmetry (4) one can set m = 1. Denoting λ = ψ we obtain λ = ψ , ψ = ψ + ρψ . Thus, members of the commuting hierarchy can be parametrised inthe form (cid:18) R R (cid:19) t = (cid:18) ψ ψ ψ (cid:19) (cid:18) R R (cid:19) x , (7)where ψ solves the Lax equation of the mKP hierarchy, ψ = ψ + ρψ . Here the ‘potential’ ρ is fixed and the solution ψ of the Lax equation varies. The choice ρ =0 , ψ = R gives rise to the system (cid:18) R R (cid:19) t = (cid:18) R R (cid:19) (cid:18) R R (cid:19) x whose commuting flows are parametrised by solutions to the heat equation [5, 11]. Remark 1.
Conservation laws of system (7) are relations of the form φ t = g x which holdidentically modulo (7). This gives g = ψφ , g = ψφ + ψ φ , and the elimination of g results in the adjoint Lax equation for the conserved density φ : φ = − φ + ρφ . Direct calculation shows that the compatibility of three-component systems of the form R R R t = λ λ λ λ λ λ R R R x , R R R y = µ µ µ µ µ µ R R R x is equivalent to the following six conditions (we assume λ = 0 and µ = 0): m ≡ λ λ = µ µ , (8) q ≡ λ − mλ λ = µ − mµ µ , (9) p ≡ λ − mλ λ = µ − mµ µ , (10) s ≡ λ − λ − pλ λ = µ − µ − pµ µ , (11) r ≡ λ − λ + qλ λ = µ − µ + qµ µ , (12)5 ≡ λ − λ − ( s + r ) λ λ = µ − µ − ( s + r ) µ µ . (13)These conditions can be reduced to the mKP equation and its Lax pair as follows. Assuming m = 0 and denoting λ = ψ , from (8) and (9) one obtains λ = ψ m , λ = ψ m − (cid:16) qm + m m (cid:17) ψ . Inserting the above into (10) yields ψ = ψ m + (cid:18) p − qm − m m (cid:19) ψ , so that (11) gives m m ψ − (cid:20) sm − pq − p + m + m m q + (cid:16) m m (cid:17) (cid:21) ψ = 0 . This implies m = 0 , s = 1 m ( qp + p − m ) . As m = 0, modulo transformations (5) one can set m = 1 , s = qp + p , so that the formulae for λ , λ , λ simplify to λ = ψ, λ = ψ , λ = ψ − qψ where ψ = ψ + ( p − q ) ψ . (14)Then (12) yields ψ = ψ + ( p − q ) ψ + (cid:2) q + r + ( p − q ) (cid:3) ψ . (15)Finally, it follows from (13) that( p + q ) ψ − (cid:0) q p − h + rq + q + qq + 2 qp + r (cid:1) ψ = 0 , which allows us to set q = − p, h = ( p ) − rp − p − pp + r . The compatibility of (14) and (15), i.e., the condition ψ = ψ , leads to (cid:2) p − pp + 2 r − p (cid:3) ψ = (cid:2) p + 2( r − p ) p − pr + 2 pp − r + r + 2 p − p (cid:3) ψ , from which we obtain the following two equations: p − pp + 2 r − p = 0 , (16)2 p + 2( r − p ) p − pr + 2 pp − r + r + 2 p − p = 0 . (17)In order to solve (16) for r we introduce the potential variable w such that p = w . Thenintegrating (16) gives r = 12 w − w + 32 w . Finally from (17) we obtain4 w + 6 w w − w − w − w w = 0 , (18)6hich is the potential mKP equation. Note that the mKP hierarchy was introduced in [7], [4].The generalised Miura transformation connecting KP and mKP equations was constructed in[7], see also [8] for the first classification results of integrable equations in 2+1 dimensions. Exactsolutions of mKP equation were constructed in [9, 2, 3], see also references therein.To summarise, members of the three-component commuting hierarchy can be parametrisedin the form R R R t = ψ ψ ψ + w ψ ψ ψ ψ R R R x (19)where w satisfies the mKP equation (18) and ψ solves the corresponding Lax equations (14),(15): ψ = ψ + 2 w ψ , ψ = ψ + 3 w ψ + 32 ( w + w + w ) ψ . (20)Fixing w and varying ψ we obtain commuting flows of the hierarchy. Example.
Set w = 0, then equations for ψ and λ , λ , λ become ψ = ψ , ψ = ψ , λ = ψ, λ = ψ , λ = ψ . We can choose ψ = R , λ = R , λ = 1 , λ = 0 , or ψ = e kR + k R + k R , λ = ψ, λ = kψ, λ = k ψ, where k is an arbitrary constant (the former solution was considered in [11]). Remark 2.
Conservation laws of system (19) are relations of the form φ t = g x which holdidentically modulo (19). This gives g = ψφ , g = ψφ + ψ φ , g = ψφ + ψ φ + ( ψ + w ψ ) φ . The elimination of g results in the adjoint Lax equations for the conserved density φ : φ = − φ + 2 w φ , φ = φ − w φ + 32 ( w − w + w ) φ . Omitting details of calculations we present the final result: four-component commuting flows ofToeplitz type (3) can be parametrised in the form R R R R t = ψ ψ ψ + w ψ ψ + 2 w ψ + ( w + 3 w + w ) ψ ψ ψ ψ + w ψ ψ ψ ψ R R R R x (21)where w solves the first three equations of the mKP hierarchy,4 w + 6 w w − w − w − w w = 0 , w = 3 w − w + (3 w − w ) w + 6 w w w + ( w − w − w ) w , = 89 w − w + 29 w − w w w + 4 w w w + ( w − w ) w + 23 ( w + 2 w − w ) w + 43 w + (4 w w − w − w − w ) w , and ψ satisfies the corresponding Lax equations: ψ = ψ + 2 w ψ , ψ = ψ + 3 w ψ + ( w + w + w ) ψ ,ψ = ψ + 4 w ψ + (2 w + 4 w + 4 w ) ψ + ∆ ψ ;here ∆ = 43 w + 53 w + 23 w + 4 w w + w + 2 w w . Fixing w and varying ψ we obtain commuting flows of the hierarchy. Remark 3.
For the matrix elements of (21), λ = ψ, λ = ψ , λ = ψ + w ψ , λ = ψ + 2 w ψ + 12 ( w + w + 3 w ) ψ , we have simple recursive formulae: λ = ψ , λ + w λ = ψ , λ + w λ + w λ = ψ . This recurrence generalises to the general n -component case: λ k + w λ k − + w λ k − + · · · + w k − λ = ψ k , ≤ k ≤ n − . Solutions to system (3) can be obtained by the following recipe which is analogous to thegeneralised hodograph method of Tsarev [15]. Let R y = ( µ E + n − X i =1 µ i P i ) R x be a commuting flow of system (3). Then the matrix equation µ E + n − X i =1 µ i P i = Ex + ( λ E + n − X i =1 λ i P i ) t defines an implicit solution of (3). In components, this is equivalent to n implicit relations µ = x + λ t, µ = λ t, . . . , µ n = λ n t. It is remarkable that, although for ‘strongly nonlinear’ PDEs such as the dispersionless KP/Todaequations, the Jordan type reductions do not occur, they naturally arise in the context of multi-dimensional linearly degenerate
PDEs (such as linearly degenerate systems of hydrodynamictype, Monge-Amp´ere equations, etc). Below we illustrate this phenomenon for the 3D Mikhalevsystem [10]. 8 .1 3D Mikhalev system
Here we consider the system u t = v y + uv x − vu x , v x = u y . (22) Two-component hydrodynamic reductions (of Jordan block type) of system (22) are exactsolutions of the form u = u ( R , R ) , v = v ( R , R ) (23)where the variables R , R satisfy a pair of commuting 2 × (cid:18) R R (cid:19) t = (cid:18) ψ ψ ψ (cid:19) (cid:18) R R (cid:19) x , (cid:18) R R (cid:19) y = (cid:18) ϕ ϕ ϕ (cid:19) (cid:18) R R (cid:19) x , with ψ = ψ + ρψ , ϕ = ϕ + ρϕ . All such reductions can be described explicitly. Direct calculation shows that u and v must bepolynomial in R of degree 2 and 4, respectively: u = − a ′ ( R ) + bR + c, v = − u + au + d, where the coefficients a, b, c, d are functions of R satisfying a single relation d ′ + b + ca ′ = 0(prime denotes differentiation by R ). The functions ϕ, ψ and ρ are expressed in terms of u bythe formulae ϕ = − u + a, ψ = − v + aϕ, ρ = 2 u − a ′ u . A particular choice a = 0 , b = − , c = 0 , d = − R leads to u = − R , v = − R −
12 ( R ) , ϕ = R , ψ = R + 12 ( R ) , ρ = 0 , the case considered in [11]. Three component reductions of Jordan block type can be sought in the form R t = ψ ψ ψ + w ψ ψ ψ ψ R x , R y = ϕ ϕ ϕ + w ϕ ϕ ϕ ϕ R x , where w satisfies the mKP equation (18) and ψ , ϕ are two solutions of the corresponding Laxequations (20). In this case the formulae become more complicated. Direct calculation showsthat u and v must be polynomial in R , R : u = bR − a ′′ ( R ) + 23 γ ′ ( R ) + αR + β,v = − u + au − a ′ ( R ) − bγR + ζ, where b = γ − a ′ R and a , α , β , γ , ζ are functions of R with the condition ζ ′ + a ′ β + αγ = 0 . The functions ϕ, ψ and w are as follows: ϕ = − u + a, ψ = − v + aϕ, = − a ′ b ( R ) − b R (cid:0) a ′′ ( R ) − γ ′ R − α (cid:1) + f, where f is a function of R and R which satisfies the equation12 (3 γ − a ′ R ) f = x ( R ) + x ( R ) + x ( R ) + x ( R ) + 3 x . (24)Here the coefficients x i are functions of R defined as x ≡ a ′′′ a ′ − a ′′ , x ≡ a ′′ γ ′ − a ′′′ γ − γ ′′ a ′ ,x ≡ αa ′′ − α ′ a ′ + 2 γ ′′ γ − γ ′ , x ≡ α ′ γ − β ′ a ′ − αγ ′ + a ′ , x ≡ β ′ γ − a ′ γ − α . If a ′ = 0, then (24) gives f = 427 ( R ) (cid:18) γ ′ γ (cid:19) ′ + 13 ( R ) (cid:18) αγ (cid:19) ′ + R (cid:18) β ′ γ − α γ (cid:19) + η, otherwise, f = ( R ) a ′ x + ( R ) a ′ ( a ′ x + 3 γx ) + R a ′ (4 a ′ x + 12 a ′ γx + 27 γ x )+ 116 a ′ b (16 a ′ x + 8 a ′ γx + 12 a ′ γ x + 18 a ′ γ x + 27 γ x )+ ln b a ′ (4 a ′ x + 12 a ′ γx + 27 a ′ γ x + 54 γ x ) + η ;here η is an extra arbitrary function of R . Acknowledgements
We thank Maxim Pavlov and Vladimir Novikov for useful comments. EVF also thanks AlexeyBolsinov and David Calderbank for a discussion on quasilinear systems of Jordan block type andtheir role in the general method of hydrodynamic reductions. LX was supported by the NationalNatural Science Foundation of China (Grant No. 11501312) and the K.C. Wong Magna Fundin Ningbo University. LX also thanks Loughborough University for a kind hospitality.
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