Bihamiltonian Systems of Hydrodynamic Type and Reciprocal Transformations

Abstract

We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian. Comment: 15 pages

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arXiv:math/0510250v1 [math.DG] 12 Oct 2005
Bihamiltonian Systems of Hydrodynamic Type and
Reciprocal Transformations
Ting Xue∗, Youjin Zhang†
Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
Abstract
We prove that under certain linear reciprocal transformation, an evo-
lutionary PDE of hydrodynamic type that admits a bihamiltonian struc-
ture is transformed to a system of the same type which is still bihamil-
tonian.
Mathematics Subject Classification (2000). 37K25; 37K10.
Key words. bihamiltonian system, reciprocal transformation
1Introduction
Systems of hydrodynamic type are a class of quasilinear evolutionary PDEs of
the form
ut= V ux,
u = (u1,...,un)T, V = (Vi
j(u))n×n.
(1.1)
Assume that the above system has two conservation laws
∂a(u)
∂t
=∂b(u)
∂x
,
∂p(u)
∂t
=∂q(u)
∂x
(1.2)
with a(u)q(u)−p(u)b(u) ?= 0, then we can perform a change of the independent
variables
(x,t) ?→ (y(x,t,u(x,t)),s(x,t,u(x,t))
by the following defining relations
(1.3)
dy = a(u)dx + b(u)dt,ds = p(u)dx + q(u)dt.
(1.4)
∗txue@math.mit.edu
†youjin@mail.tsinghua.edu.cn
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Such a change of the independent variables is called a reciprocal transformation
of the system (1.1). It originates from the study of gas dynamics, see [10] and
references therein. Under a reciprocal transformation the system (1.1) remains
to be a system of hydrodynamic type
us= (aV − bI)(q I − pV )−1uy.
(1.5)
Here I denotes the n × n unity matrix and the matrix qI − pV is assumed to
be nondegenerate.
The system (1.1) is called a Hamiltonian system of hydrodynamic type if
it has the representation
ut= J ∇h(u),
where J = (Jij) is a Hamiltonian operator of hydrodynamic type
(1.6)
Jij= ηij(u)d
dx− ηis(u)Γj
sk(u)uk
x,
1 ≤ i,j ≤ n,
(1.7)
the matrix η = (ηij) is nondegenerate and symmetric on certain open subset
U of Rn. Here and henceforth summations over repeated upper and lower
indices are assumed. It was proved by Dubrovin and Novikov [4, 5] that J is
a Hamiltonian operator if and only if the pseudo-Riemannian metric (ηij) =
η−1is flat, and Γj
skcoincide with the Christoffel symbols of the Levi-Civita
connection of (ηij). So we can assume that the dependent variables u1,...,un
of the system (1.6) are the flat coordinates of the metric (ηij), i.e., ηij(u) are
constants and Γj
sk= 0. In what follows we will also call a nondegenerate
symmetric bilinear form on T∗
metric.
A natural question is whether under a reciprocal transformation a system
of the form (1.6) remains to be a Hamiltonian system of hydrodynamic type.
In the special case when the reciprocal transformation is linear in x,t, i.e.
when a(u),b(u),p(u),q(u) are constants, Tsarev gave an affirmative answer to
the above question [11]. In fact, it was shown by Pavlov [9] that under the
linear reciprocal transformation
uU, such as the the one given by η = (ηij), a
y = ax + bt, s = px + qt,aq − bp ?= 0,
(1.8)
the Hamiltonian system (1.6) with Jij= ηij∂xis transformed to the following
Hamiltonian system of hydrodynamic type:
vs=¯J ∇¯h(v),
Here the new dependent variables viand the function¯h(v) are defined by
¯Jij= ηij∂y.
(1.9)
v = (v1,...,vn)T= η∇(qh0− ph),
?
h0=1
2ηijuiuj,
(1.10)
¯h(v) = aqh −1
2pηij∂h
∂ui
∂h
∂uj
?
− b
?
qh0− p(ui∂h
∂ui− h)
?
.
(1.11)
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In the general cases, the above transformation property of a Hamiltonian sys-
tem of hydrodynamic type no longer holds true as it was shown by Ferapontov
and Pavlov in [8]. Although under a reciprocal transformation (1.3), (1.4) the
transformed system of (1.6) can still be represented as a Hamiltonian system,
the transformed Hamiltonian operator becomes nonlocal, it contains terms
with integral operator ∂−1
y.
In this paper, we study the properties of a bihamiltonian system of hydro-
dynamic type under the action of a reciprocal transformation. We show that
under a linear reciprocal transformation of the form (1.8), a bihamiltonian sys-
tem (or a hierarchy of bihamiltonian systems) keeps to have a bihamiltonian
structure of hydrodynamic type.
The main motivation of this work comes from the program of classification
for certain class of bihamiltonian evolutionary PDEs that was initiated by
Boris Dubrovin and the second author in [6]. The dispersionless limits of
such evolutionary PDEs are bihamiltonian systems of hydrodynamic type.
Typical examples of this class of evolutionary PDEs include the KdV equation
and the interpolated Toda lattice equation, their dispersionless limits will be
considered in the examples of Sec.4. One of the important problems related
to this classification program is whether a bihamiltonian system of this class
remains to be bihamiltonian after a linear reciprocal transformation. The
present work is a first step toward answering this problem.
The plan of the paper is as follows: We first formulate the main results in
Sec.2, then give their proofs in Sec.3; in Sec.4 we present two examples to
illustrate the main results; the last section is a conclusion.
2The main results
A bihamiltonian system of hydrodynamic type is a system of the form (1.1)
that admits two compatible Hamiltonian structures of hydrodynamic type,
i.e., it has the following representation:
∂u
∂t= J1∇h(u),J1∇h(u) ≡ J2∇f(u) ≡ V (u)ux.
(2.1)
Here J1,J2are two Hamiltonian operators of hydrodynamic type
Jij
1= ηij∂x,Jij
2= gij(u)∂x− gik(u)Γj
kl(u)ul
x.
(2.2)
The symmetric matrices η = (ηij),g = (gij) have the properties that η is
constant and nondegenerate, g is nondegenerate on certain open subset U of
Rn, and det(g − λη) does not vanish identically for any constant parameter
λ. Compatibility of these two Hamiltonian operators means that any linear
combination J2− λJ1 also gives a Hamiltonian operator of the same type.
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Note that we have chosen the flat coordinates u1,...,unof the metric η as the
dependent variables of the above system.
Let us consider the effect of the linear reciprocal transformation (1.8) on
the bihamiltonian property of the system (2.1). To this end, we also take into
account the flow of translation along x. It is also a bihamiltonian system with
respect to the above bihamiltonian structure
∂u
∂t0
= J1∇h0(u),J1∇h0(u) ≡ J2∇f0(u) ≡∂u
∂x.
(2.3)
Here the functions h0,f0are defined by
h0(u) =1
2ηijuiuj,f0(u) =1
2ˆ gijˆ uiˆ uj,
(ηij) = (ηij)−1, (ˆ gij) = (ˆ gij)−1
with ˆ gijbeing the components of the metric g under its flat coordinates
ˆ u1,..., ˆ un.
We introduce the new dependent variables v = (v1,...,vn)Tby the rela-
tion (1.10), then the Jacobian is given by
Q :=
?∂vi
∂uj
?
= qI − pV.
(2.4)
Assume that Q is nondegenerate on U ⊂ Rn, denote
W := Q−1.
(2.5)
After the reciprocal transformation (1.8), the systems (2.1) and (2.3) are trans-
formed to
∂v
∂s= (aV − bI)Wvy,
∂v
∂t0
(2.6)
= (aq − bp)Wvy.
(2.7)
Note that when the transformation (1.8) satisfies the condition a = q = 0,b =
p = 1, the flows
∂sand
Define two metrics ¯ η = (¯ ηij(v)), ¯ g = (¯ gij(v)) whose components in the
local coordinates v1,...,vnare given by the following formulae
∂∂
∂t0coincide.
¯ ηij(v) = ηij,
¯ gij(v) = gij(u),i,j = 1,...,n,
(2.8)
where u and v are related by (1.10).
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Theorem 2.1. The metrics ¯ η, ¯ g are flat, the corresponding Hamiltonian op-
erators¯J1,¯J2with components
¯Jij
1= ¯ ηij∂y,
¯Jij
2= ¯ gij(v)∂y+¯Γij
k(v)vk
y
(2.9)
are compatible. Here¯Γij
symbols of the Levi-Civita connection of the metric ¯ g.
k(v) = −¯ gil¯Γj
lk(v) with¯Γj
lk(v) being the Christoffel
In other words, the metrics ¯ η, ¯ g form a flat pencil [2].
Theorem 2.2. The systems (2.6) is bihamiltonian with respect to the bihamil-
tonian structure¯J1,¯J2, i.e., it has the representation
∂v
∂s=¯J1∇¯h(v) ≡¯J2∇¯f(v),
(2.10)
where the functions¯h,¯f are defined by
∂¯h(v)
∂vi
=∂ (ah(u) − bh0(u))
∂ui
,
∂¯f(v)
∂vi
=∂ (af(u) − bf0(u))
∂ui
,
1 ≤ i ≤ n.
(2.11)
Now let us assume that there is given another bihamiltonian system
∂u
∂t1
= J1∇h1(u),J1∇h1(u) ≡ J2∇f1(u) ≡ A(u)ux,
(2.12)
we also assume that this flow commutes with the flow given by (2.1). By using
Corollary 4.2 of [3], we know that the commutativity of these two flows holds
true automatically when the bihamiltonian structure J1,J2is semisimple, i.e.
when the characteristic polynomial det(gij−ληij) has pairwise distinct roots.
Theorem 2.3. Under the reciprocal transformation (1.8), the system (2.12)
is transformed to the form
∂v
∂t1
= (aq − bp)AWvy,
(2.13)
it is bihamiltonian with respect to the bihamiltonian structure¯J1,¯J2and has
the representation
∂v
∂t1
Here the functions¯h1(v),¯f1(v) are defined by
=¯J1∇¯h1(v) ≡¯J2∇¯f1(v).
(2.14)
∂¯h1(v)
∂vi
= (aq−bp)∂h1(u)
∂ui
,
∂¯f1(v)
∂vi
= (aq−bp)∂f1(u)
∂ui
,
1 ≤ i ≤ n. (2.15)
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