Hamiltonian structure of reductions of the Benney system

Abstract

We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dKP) starting from the family of conformal maps associated to it. Comment: 35 pages, 2 figures

Full text

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arXiv:0802.1984v3 [nlin.SI] 28 Feb 2008
Hamiltonian structure of reductions of the Benney
system
John Gibbons∗, Paolo Lorenzoni∗∗, Andrea Raimondo∗
* Department of Mathematics, Imperial College
180 Queen’s Gate, London SW7 2AZ, UK
j.gibbons@imperial.ac.uk, a.raimondo@imperial.ac.uk
** Dipartimento di Matematica e Applicazioni
Universit` a di Milano-Bicocca
Via Roberto Cozzi 53, I-20125 Milano, Italy
paolo.lorenzoni@unimib.it
Abstract
Weshow how to construct the Hamiltonian structures of any reduction of the Benney
chain (dKP) starting from the family of conformal maps associated to it.
Introduction
The Benney moment chain [4], given by the equations
Ak
t= Ak+1
x
+ kAk−1A0
x,k = 0,1,...,
with Ak=Ak(x,t), is themost famous exampleofa chain ofhydrodynamictype, which gen-
eralizes the classical systems of hydrodynamictype in the case when the dependent variables
(and the equations they have to satisfy) are infinitely many.
A n−component reduction of the Benney chain is a restriction of the infinitedimensional
system to a suitable n−dimensional submanifold, that is
Ak= Ak(u1,...,un),k = 0,1,...
The reduced systems are systems of hydrodynamic type in the variables (u1,...,un) that
parametrize the submanifold:
ui
t= vi
j(u)uj
x,i = 1,...,n.
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Benney reductions were introduced in [11], and there it was proved that such systems are
integrable via the generalized hodograph transformation [20]. In particular, this method
requires the system to be diagonalizable, that is, there exists a set of coordinates λ1,...,λn,
called Riemann invariants, such that the reduction takes diagonal form:
λi
t= vi(λ)λi
x.
The functions viare called characteristic velocities.
A more compact description of the Benney chain can be given by introducing the formal
series
+∞
?
In this picture, as follows from [16, 17], the Benney chain can be written as the single equa-
tion
λt= pλx− A0
which is the equation of the second flow of the dispersionless KP hierarchy. This equation
related with the Benney chain also appears in [18].
Clearly, in the case ofa reduction, thecoefficients ofthis series depend on afinite number
of variables (u1,...,un). In this case, the series can be thought as the asymptotic expansion
for p ?→ ∞ of a suitable function λ(p,u1,...,un) depending piecewise analytically on the
parameter p. It turns out [11, 12] that such a function satisfies a system of chordal Loewner
equations, describing families of conformal maps (with respect to p) in the complex upper
half plane. The analytic properties of λ characterize the reduction. More precisely, in the
case of an n−reduction the associated function λ possess n distinct critical points on the real
axis, these are the characteristic velocities viof the reduced system, and the corresponding
critical values can be chosen as Riemann invariants.
Someexamplesofsuch reductions, discussedbelow,haveknownHamiltonianstructures,
but the most general result is far weaker, all such reductions are semi-Hamiltonian [20, 11].
The aim of this paper is to investigate the relations between the analytic properties of the
function λ(p,u1,...,un) and the Hamiltonian structures of the associated reduction. As is
well known, such structures are associated to pseudo-riemannian metrics, and in particular,
local Hamiltonian structures are associated to flat metrics.
Our approach is general, in the sense that it applies to all Benney reductions. Conse-
quently, it reveals a unified structure for the Hamiltonian structure of such reduced systems.
The main result of the paper provides the Hamiltonian structures of a Benney reduction di-
rectly in terms of the function λ(p,u1,...,un) and its inverse with respect to p, denoted by
p(λ,u1,...,un). The Hamiltonian operator then takes the form
λ = p +
k=0
Ak
pk+1.
xλp,
Πij= ϕiλ
′′(vi)δijd
dx+ Γij
kλk
x+
1
2πi
n
?
k=1
?
Ck
∂p
∂λλi
x
(p(λ) − vi)2
?d
dx
?−1
∂p
∂λλj
x
(p(λ) − vj)2ϕk(λ)dλ,
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where
Γij
kλk
x=ϕjλi
x− ϕiλj
(vi− vj)2
x
i ?= j,
Γii
kλk
x= ϕi
?1
6
λ
λ
′′′′(vi)
′′(vi)−1
4
λ
λ
′′′(vi)2
′′(vi)2
?
λi
x+1
2ϕ
′
iλi
x−
?
k?=i
λ
λ
′′(vi)
′′(vk)
ϕiλk
(vi− vk)2.
x
Here ϕ1,...,ϕnare arbitrary functions of a single variable, Ckare suitable closed contours
on a complex domain, and
λ
′′(p) =∂2λ
∂p2(p),λ
′′′(p) =∂3λ
∂p3(p), ...
The paper is organized as follows. In Section 1 we review the concepts of integrability
for diagonalizable systems of hydrodynamic type and the Hamiltonian formalism for these
systems, both in the local and nonlocal case. In Section 2 we introduce the Benney chain,
its reductions, and we discuss the properties of these systems. Section 3 is dedicated to the
representation of Benney reductions in the λ picture and to the relations with the Loewner
evolution. The study of the Hamiltonian properties of reductions of Benney is addressed in
Sections 4 and 5: in the former we use a direct approach, starting from the reduction itself,
in the latter we describe these results from the point of view of the function λ associated
with the reduction. In the last secion we discuss two examples where calculations can be
expressed in details.
1 Systems of hydrodynamic type
1.1Semi-Hamiltonian systems
In (1+1) dimensions, systems of hydrodynamic type are quasilinear first order PDE of the
form
ui
x,
Here and below sums over repeated indices are assumed if not otherwise stated. We say
that the system (1.1) is diagonalizable if there exist a set of coordinates λ1,...,λn, called
Riemann invariants, such that the matrix vi
t= vi
j(u)uj
i = 1,...,n.
(1.1)
j(λ) takes diagonal form:
λi
t= vi(λ)λi
x.
(1.2)
The functions viare called characteristic velocities. We recall that the Riemann invariants
λiare not defined uniquely, but up to a change of coordinates
˜λi=˜λi(λi).
(1.3)
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A diagonal system of PDEs of hydrodynamic type (1.2) is called semi-Hamiltonian [20] if
the coefficients vi(u) satisfy the system of equations
∂j
?
∂kvi
vi− vk
?
= ∂k
?
∂jvi
vi− vj
?
∀i ?= j ?= k ?= i,
(1.4)
where ∂i=
∂
∂λi. The equations (1.4) are the integrability conditions both for the system
∂jwi
wi− wj=
∂jvi
vi− vj,
(1.5)
which provides the characteristic velocities of the symmetries
ui
τ= wi(u)ui
x
i = 1,...,n
of (1.1), and for the system
(vi− vj)∂i∂jH = ∂ivj∂jH − ∂jvi∂iH,
which provides the densities H of conservation laws of (1.1). The properties of being diag-
onalizable and semi-Hamiltonian imply the integrability of the system:
Theorem 1 [20](Generalized hodograph transformation)
Let
λi
t= viλi
x
(1.6)
be a diagonal semi-Hamiltonian system of hydrodynamic type, and let (w1,...,wN) be the
characteristicvelocities of one of its symmetries. Then, the functions (λ1(x,t),...,λN(x,t))
determined by the system of equations
wi= vix + t,i = 1,...,N,
(1.7)
satisfy(1.6). Moreover, every smooth solution of this system is locally obtainablein this way.
1.2 Hamiltonian formalism
A class of Hamiltonian formalisms for systems of hydrodynamic type (1.1) was introduced
by Dubrovinand Novikovin [6, 7]. They considered local Hamiltonianoperators of the form
Pij= gij(u)d
dx− gisΓj
sk(u)uk
x
(1.8)
and the associated Poisson brackets
{F,G} :=
?
δF
δuiPijδG
δujdx
(1.9)
where F =
ux, uxx,...
?g(u)dx and G =
?g(u)dx are functionals not depending on the derivatives
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Theorem 2 [6] If detgij?= 0, then the formula (1.9) with (1.8) defines a Poisson bracket if
and only if the tensor gijdefines a flat pseudo-riemannian metric and the coefficients Γj
the Christoffel symbols of the associated Levi-Civita connection.
skare
Non-local extensions of the bracket (1.9), related to metrics of constant curvature, were
considered by Ferapontov and Mokhov in [19]. Further generalizations were considered by
Ferapontov in [9], where he introduced the nonlocal differential operator
Pij= gijd
dx− gisΓj
skuk
x+
?
α
ǫα(wα)i
kuk
x
?d
dx
?−1
(wα)j
huh
x,ǫα= ±1.
(1.10)
The index α can take values on a finite or infinite – even continuous – set.
Theorem 3 If detgij?= 0, then the formula (1.9) with (1.8) defines a Poisson bracket if
and only if the tensor gijdefines a pseudo-riemannian metric, the coefficients Γj
Christoffel symbols of the associated Levi-Civita connection ∇, and the affinors wαsatisfy
the conditions
skare the
?wα,wβ?= 0,
gik(wα)k
j= gjk(wα)k
i,
∇k(wα)i
j= ∇j(wα)i
k,
Rij
kh=
?
α
?
(wα)i
k(wα)j
h− (wα)j
k(wα)i
h
?
,
where Rij
kh= gisRj
skhare the components of the Riemann curvature tensor of the metric g.
In the case of zero curvature, operator (1.10) reduces to (1.9). Let us focus our atten-
tion on semi-Hamiltonian systems. In [9] Ferapontov conjectured that any diagonalizable
semi-Hamiltonian system is always Hamiltonian with respect to suitable, possibly non lo-
cal, Hamiltonian operators. Moreover he proposed the following construction to define such
Hamiltonian operators:
1. Consider a diagonal system (1.2). Find the general solution of the system
∂jln√gii=
∂jvi
vj− vi,
(1.11)
which is compatible for a semi-Hamiltonian system, and compute the curvature tensor
of the metric g.
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