Simple Lax Description of the ILW Hierarchy
In this note, we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structu...
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| description | In this note, we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structure in terms of a single Lax difference-differential operator.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 120, 7 pages
Simple Lax Description of the ILW Hierarchy
Alexandr BURYAK †‡ and Paolo ROSSI §
† School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
E-mail: a.buryak@leeds.ac.uk
URL: http://sites.google.com/site/alexandrburyakhomepage/home/
‡ Faculty of Mechanics and Mathematics, Lomonosov Moscow State University,
Moscow, GSP-1, 119991, Russia
§ Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova,
Via Trieste 63, 35121 Padova, Italy
E-mail: paolo.rossi@math.unipd.it
URL: http://www.math.unipd.it/~rossip/
Received September 07, 2018, in final form November 06, 2018; Published online November 10, 2018
https://doi.org/10.3842/SIGMA.2018.120
Abstract. In this note we present a simple Lax description of the hierarchy of the interme-
diate long wave equation (ILW hierarchy). Although the linear inverse scattering problem
for the ILW equation itself was well known, here we give an explicit expression for all higher
flows and their Hamiltonian structure in terms of a single Lax difference-differential operator.
Key words: intermediate long wave hierarchy; ILW; Lax representation; integrable systems;
Hamiltonian
2010 Mathematics Subject Classification: 37K10
1 Introduction
The intermediate long wave (ILW) equation was introduced in [7] to describe the propagation
of waves in a two-layer fluid of finite depth. The model represents the natural interpolation
between the Benjamin–Ono deep water and Kortweg–de Vries shallow water theories. The ILW
equation for the time s evolution of a function w = w(x) defined on the real line has the following
form:
ws + 2wwx + T (wxx) = 0, T (f) := p.v.
∫ +∞
−∞
1
2δ
(
sgn(x− ξ)− coth
π(x− ξ)
2δ
)
f(ξ)dξ,
where δ is a parameter and p.v.
∫
·dξ represents the principal value integral. This equation can
be rewritten in the formal loop space formalism (see for instance [3], from which we borrow
notations for the rest of the paper) as
ut = uux +
∑
g≥1
µg−1ε2g
|B2g|
(2g)!
u2g+1, (1.1)
where B2g are the Bernoulli numbers. Indeed the action of the operator T on a function f can
be written in terms of the derivatives of f as T (f) =
∑
n≥1
δ2n−122n |B2n|
(2n)! ∂
2n−1
x f , and, by setting
w =
√
µ
ε u, s = − ε
2
√
µ t and δ =
ε
√
µ
2 , we see how the above equations are equivalent.
In [10] the integrability of the ILW equation was established, finding an infinite number of
conserved densities and giving a corresponding inverse scattering problem. From this inverse
scattering problem a Lax representation can of course be deduced. Also, through a simple
mailto:a.buryak@leeds.ac.uk
http://sites.google.com/site/alexandrburyakhomepage/home/
mailto:paolo.rossi@math.unipd.it
http://www.math.unipd.it/~rossip/
https://doi.org/10.3842/SIGMA.2018.120
2 A. Buryak and P. Rossi
Hamiltonian structure, the conserved densities generate an infinite number of commuting flows.
However the explicit relation between the Lax representation of the ILW equation, its higher
flows and the Hamiltonian structure was never, to the best of our knowledge, clarified in the
literature.
In this paper we give an explicit and remarkably simple Lax description of the full hierarchy
of commuting flows of the ILW equation (the ILW hierarchy) and their Hamiltonian structure
in terms of a single Lax mixed difference-differential operator.
As it turns out, this Lax description corresponds to the equivariant bi-graded Toda hierarchy
of [9, Section B.2] in the somewhat degenerate case of bi-degree (1, 0), hence a reduction of the
2D-Toda hierarchy. In fact, in [9], because of the geometric origin of their problem, the authors
always work with strictly positive bi-degree, but this is an unnecessary restriction. What we do
here is prove that the bi-degree (1, 0) case corresponds to the ILW hierarchy in its Hamiltonian
formulation. A similar question is raised in [5, Remark 31], where the author notices that the
definition of all but the extended flows of the extended bi-graded Toda hierarchy survives when
one of the bi-degrees vanishes. Since in the equivariant case there is no need of extended flows,
this problem does not arise in the Lax representation of the equivariant bi-graded Toda hierarchy.
Of course this means that equivariant bi-graded Toda hierarchies of any bidegree (N, 0), with
N ≥ 1, are well defined (see also [1] for their relation with the rational reductions of the 2D-
Toda hierarchy). We will study the relation of such hierarchies with the geometry of equivariant
orbifold Gromov–Witten theory in an upcoming publication.
2 ILW hierarchy
Here and in what follows we will use the formal loop space formalism in the notations of [3].
The Hamiltonian structure of the ILW equation (1.1) is given by the Hamiltonian
h
ILW
1 =
∫ u3
6
+
∑
g≥1
µg−1ε2g
|B2g|
2(2g)!
uu2g
dx
and the Poisson bracket {·, ·}∂x , associated to the operator ∂x. The Hamiltonians h
ILW
d ∈ Λ̂
[0]
u ,
d ≥ 2, of the higher flows of the ILW hierarchy are uniquely determined by the properties
h
ILW
d
∣∣∣
ε=0
=
∫
ud+2
(d+ 2)!
dx,
{
h
ILW
d , h
ILW
1
}
∂x
= 0.
For example,
h
ILW
2 =
∫ u4
4!
+
ε2
48
u2uxx +
∑
g≥2
|B2g|
(2g)!
ε2g
(
µg−2
g + 1
2
uu2g + µg−1
1
4
u2u2g
) dx.
We refer the reader to the paper [2, Section 8], which explains a relation between the Hamilto-
nians h
ILW
d and the conserved quantities of the ILW equation, constructed in [10]. It is convenient
to introduce an additional Hamiltonian h
ILW
0 :=
∫
u2
2 dx, which generates spatial translations.
So the flows of the ILW hierarchy are given by
∂u
∂td
= ∂x
δh
ILW
d
δu
, d ≥ 0,
where we identify the times t1 and t.
Simple Lax Description of the ILW Hierarchy 3
3 Lax description of the ILW hierarchy
Our Lax description of the ILW hierarchy is presented in Section 3.2, see Theorem 1. Before
that, in Section 3.1, we recall necessary definitions from the theory of shift operators.
3.1 Shift operators
Let Λ := eiε∂x . We will consider formal series of the form
A =
∑
n≤m
anΛn, an ∈ Âu, m ∈ Z.
Via the operation of composition ◦, the vector space of such formal operators is endowed with
the structure of a non-commutative associative algebra. The positive part A+, the negative
part A− and the residue resA of the operator A are defined by
A+ :=
m∑
n=0
anΛn, A− := A−A+, resA := a0.
Let z be a formal variable. The symbol  of the operator A is defined by
 :=
∑
n≤m
anenz.
For an operator L of the form
L = Λ +
∑
n≥0
anΛ−n, an ∈ Âu,
one can define the dressing operator P ,
P = 1 +
∑
n≥1
pnΛ−n,
by the identity
L = P ◦ Λ ◦ P−1.
Note that the coefficients pn of the dressing operator do not belong to the ring Âu, but to
a certain extension of it (see, e.g., [6, Section 2]). The dressing operator P is defined up to
the multiplication from the right by an operator of the form 1 +
∑
n≥1
p̂nΛ−n, where p̂n are some
constants.
The logarithm logL is defined by
logL := P ◦ iε∂x ◦ P−1 = iε∂x − iεPx ◦ P−1,
where Px =
∑
n≥1
(pn)xΛ−n. The ambiguity in the choice of dressing operator is cancelled in the
definition of logL and, moreover, the coefficients of logL do belong to Âu (see the proof of
Theorem 2.1 in [6]). To be more precise, one has the commutation relations[
logL,Lm
]
= 0, m ≥ 1, (3.1)
which imply that
res
[
iεPx ◦ P−1, Lm
]
= iε∂x resLm, m ≥ 1.
4 A. Buryak and P. Rossi
These relations allow to compute recursively all the coefficients of the operator iεPx ◦ P−1. As
a result, if we write
logL = iε∂x +
∑
n≥1
fnΛ−n,
then the coefficient fn can be expressed as a differential polynomial in the coefficients a0, a1, . . .,
an−1 of the operator L, fn = fn(a0, . . . , an−1) ∈ Â[0]
a0,...,an−1 . For example,
f1 =
iε∂x
Λ− 1
a0 =
∑
n≥0
Bn
n!
(iε∂x)na0. (3.2)
3.2 Lax description
Let τ be a formal variable and
L := Λ + u− τ iε∂x.
From the discussion of the construction of the logarithm logL in the previous section it is easy
to see that there exists a unique operator L of the form
L = Λ +
∑
n≥0
anΛ−n, an ∈ Â[0]
u [τ ],
satisfying
L− τ logL = L. (3.3)
Since
[
Ld+1,L
]
= 0, d ≥ 0, the commutator
[(
Ld+1
)
+
,L
]
doesn’t contain terms with non-zero
powers of Λ. Consider the following system of PDEs:
∂u
∂Td
=
∂L
∂Td
=
1
τ iε(d+ 1)!
[(
Ld+1
)
+
,L
]
, d ≥ 0. (3.4)
The following theorem is the main result of our paper.
Theorem 1.
1. The flows ∂
∂Td
, given by (3.4), pairwise commute.
2. The system of Lax equations (3.4) possesses a Hamiltonian structure given by the Hamil-
tonians
h
Lax
d =
∫ (
resLd+2
(d+ 2)!
− τ
d+ 1
resLd+1
(d+ 1)!
)
dx, d ≥ 0,
and the Poisson bracket associated to the operator ∂x.
3. Let y be a formal variable and define polynomials Pd(y) ∈ Q[y, τ ], d ≥ 1, by
Pd(y) := y
d−1∏
i=1
(
y +
τ
i
)
=
d∑
j=1
Pd,jy
jτd−j , Pd,j ∈ Q.
The ILW hierarchy is related to the hierarchy (3.4) by the following triangular transfor-
mation:
h
Lax
d =
d∑
j=0
Pd+1,j+1τ
d−j h
ILW
j
∣∣∣µ=−τ−1
ε7→ε
√
−τ
, d ≥ 0. (3.5)
Simple Lax Description of the ILW Hierarchy 5
Proof. 1. Hereafter, for simplicity, we will use Lm+ to denote (Lm)+. Let
Hd :=
1
τ iε(d+ 1)!
Ld+1
+ , d ≥ 0.
Let us first check that
∂L
∂Td
= [Hd, L],
∂ logL
∂Td
= [Hd, logL]. (3.6)
Equations (3.1) and (3.3) imply that
∂L
∂Td
− τ ∂ logL
∂Td
= [Hd, L]− τ [Hd, logL], (3.7)
res
[
∂ logL
∂Td
, Lm
]
+
∑
a+b=m−1
res
[
logL,La ◦ ∂L
∂Td
◦ Lb
]
= 0, m ≥ 1. (3.8)
We consider these equations as a system of equations for the pair of operators ∂L
∂Td
, ∂ logL∂Td
.
Similarly to the discussion of the computation of the logarithm logL in the previous section,
equations (3.7) and (3.8) allow to compute recursively all the coefficients of the operators ∂L
∂Td
and ∂ logL
∂Td
. Then it remains to note that the operators ∂L
∂Td
= [Hd, L] and ∂ logL
∂Td
= [Hd, logL]
satisfy system (3.7)–(3.8). This completes the proof of equations (3.6).
When we know formulas (3.6), the commutativity of the flows ∂
∂Td
is proved by a standard
computation:
τ iε(d1 + 1)!τ iε(d2 + 1)!
(
∂
∂T1
∂u
∂T2
− ∂
∂T2
∂u
∂T1
)
=
[[
Ld1+1
+ , Ld2+1
]
+
,L
]
+
[
Ld2+1
+ ,
[
Ld1+1
+ ,L
]]
−
[[
Ld2+1
+ , Ld1+1
]
+
,L
]
−
[
Ld1+1
+ ,
[
Ld2+1
+ ,L
]]
Jacobi
identity
=
[[
Ld1+1
+ , Ld2+1
]
+
,L
]
−
[[
Ld2+1
+ , Ld1+1
]
+
,L
]
+
[[
Ld2+1
+ , Ld1+1
+
]
,L
]
= −
[[
Ld1+1
− , Ld2+1
+
]
+
,L
]
−
[[
Ld2+1
+ , Ld1+1
]
+
,L
]︸ ︷︷ ︸
=−
[[
L
d2+1
+ ,L
d1+1
+
]
+
,L
] +
[[
Ld2+1
+ , Ld1+1
+
]
,L
]
= 0.
2. Note that the flows ∂
∂Td
can be written as
∂u
∂Td
=
1
(d+ 1)!
∂x resLd+1.
Let us compute the flow ∂
∂T1
. For the coefficients of the operator L, one can immediately see
that a0 = u and then, using formula (3.2), we get
a1 = τ
iε∂x
Λ− 1
u.
This allows to compute
∂u
∂T1
=
1
2
∂x resL2 = ∂x
(
u2
2
+
τ iε∂x
2
Λ + 1
Λ− 1
u
)
= uux + τux − τ
∑
g≥1
|B2g|
(2g)!
ε2gu2g+1
= ∂x
δ
δu
u3
6
+ τ
u2
2
− τ
∑
g≥1
|B2g|
2(2g)!
ε2guu2g
.
6 A. Buryak and P. Rossi
The local functionals h
Lax
d are conserved quantities for the flow ∂
∂T1
. Indeed,
∂
∂T1
∫
resLddx =
1
2τ iε
∫
res
[
L2
+, L
d
]
dx,
which is zero because∫
res
[
fΛm, gΛn
]
dx = δm+n,0
∫ (
f · Λmg − g · Λnf
)
dx = 0, f, g ∈ Âu, m, n ∈ Z.
Therefore, the local functionals h
Lax
d together with the Poisson bracket {·, ·}∂x generate the
flows which commute with the flow ∂
∂T1
. Then these flows are uniquely determined by their
dispersionless parts (see [8, Lemma 3.3] or [4, Lemma 4.14]). Hence, it is sufficient to check the
equation
∂x
δh
Lax
d
δu
= ∂x
resLd+1
(d+ 1)!
at the dispersionless level.
Denote L̂0 := L̂
∣∣
ε=0
. We see that it is sufficient to check that
∂x
∂
∂u
(
res L̂d+2
0
(d+ 2)!
− τ
d+ 1
res L̂d+1
0
(d+ 1)!
)
= ∂x res
(
L̂d+1
0
(d+ 1)!
)
, d ≥ 0. (3.9)
For this we compute
L̂0 − τ log L̂0 = ez + u− τz ⇒ ∂L̂0
∂u
− τ
∂L̂0
∂u
L̂0
= 1 ⇒ ∂L̂0
∂u
=
1
1− τL̂−10
.
Therefore,
∂
∂u
(
res L̂d+1
0
(d+ 1)!
)
=
1
d!
res
(
L̂d0
∂L̂0
∂u
)
=
1
d!
d∑
j=0
τ j res
(
L̂d−j0
)
, d ≥ 0,
which gives
∂
∂u
(
res L̂d+1
0
(d+ 1)!
)
=
res L̂d0
d!
+
τ
d
∂
∂u
(
res L̂d0
d!
)
, d ≥ 1. (3.10)
This implies equation (3.9).
3. We see that
∂u
∂T1
=
∂u
∂t1
∣∣∣∣µ=−τ−1
ε7→ε
√
−τ
+ τux.
Using again the result of [8, Lemma 3.3] (see also [4, Lemma 4.14]), we conclude that it is
sufficient to prove equation (3.5) at the dispersionless level, namely,
res
(
L̂d+2
0
(d+ 2)!
− τ
d+ 1
L̂d+1
0
(d+ 1)!
)
=
d∑
j=0
Pd+1,j+1τ
d−j uj+2
(j + 2)!
, d ≥ 0.
Simple Lax Description of the ILW Hierarchy 7
Using formula (3.10) and the property L̂0
∣∣
u=0
= ez, the last equation can be equivalently written
as
res L̂d+1
0
(d+ 1)!
=
d∑
j=0
Pd+1,j+1τ
d−j uj+1
(j + 1)!
. (3.11)
Recursion (3.10) implies that
res L̂d+1
0
(d+ 1)!
=
d∏
j=1
(
∂−1u +
τ
j
)u,
where we define the action of the operator ∂−1u in the polynomial ring Q[u, τ ] by ∂−1u uj := uj+1
j+1 ,
j ≥ 0. Since we obviously have
d∑
j=0
Pd+1,j+1τ
d−j uj+1
(j + 1)!
=
d∏
j=1
(
∂−1u +
τ
j
)u,
identity (3.11) becomes clear. This completes the proof of the theorem. �
Acknowledgements
We would like to thank Andrea Brini, Guido Carlet, Oleg Chalykh, Allan Fordy, Alexander
Mikhailov and Vladimir Novikov for useful discussions. The work of the first author (Theorem 1,
parts 1 and 3) was supported by the grant no. 16-11-10260 of the Russian Science Foundation.
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1 Introduction
2 ILW hierarchy
3 Lax description of the ILW hierarchy
3.1 Shift operators
3.2 Lax description
References
|
| id | nasplib_isofts_kiev_ua-123456789-209837 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:53:20Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Buryak, A. Rossi, P. 2025-11-27T14:46:18Z 2018 Simple Lax Description of the ILW Hierarchy / A. Buryak, P. Rossi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10 arXiv: 1809.00271 https://nasplib.isofts.kiev.ua/handle/123456789/209837 https://doi.org/10.3842/SIGMA.2018.120 In this note, we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structure in terms of a single Lax difference-differential operator. We would like to thank Andrea Brini, Guido Carlet, Oleg Chalykh, Allan Fordy, Alexander Mikhailov, and Vladimir Novikov for useful discussions. The work of the first author (Theorem 1, parts 1 and 3) was supported by the grant no. 16-11-10260 of the Russian Science Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Simple Lax Description of the ILW Hierarchy Article published earlier |
| spellingShingle | Simple Lax Description of the ILW Hierarchy Buryak, A. Rossi, P. |
| title | Simple Lax Description of the ILW Hierarchy |
| title_full | Simple Lax Description of the ILW Hierarchy |
| title_fullStr | Simple Lax Description of the ILW Hierarchy |
| title_full_unstemmed | Simple Lax Description of the ILW Hierarchy |
| title_short | Simple Lax Description of the ILW Hierarchy |
| title_sort | simple lax description of the ilw hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209837 |
| work_keys_str_mv | AT buryaka simplelaxdescriptionoftheilwhierarchy AT rossip simplelaxdescriptionoftheilwhierarchy |