Lagrangian Approach to Dispersionless KdV Hierarchy
We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct resu...
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2007
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irk-123456789-1472032019-02-14T01:23:33Z Lagrangian Approach to Dispersionless KdV Hierarchy Choudhuri, A. Talukdar, B. Das, U. We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation. 2007 Article Lagrangian Approach to Dispersionless KdV Hierarchy / A. Choudhuri, B. Talukdar, U. Das // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 17 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35A15; 37K05; 37K10 http://dspace.nbuv.gov.ua/handle/123456789/147203 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation. |
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Choudhuri, A. Talukdar, B. Das, U. |
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Choudhuri, A. Talukdar, B. Das, U. Lagrangian Approach to Dispersionless KdV Hierarchy Symmetry, Integrability and Geometry: Methods and Applications |
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Choudhuri, A. Talukdar, B. Das, U. |
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Choudhuri, A. |
| title |
Lagrangian Approach to Dispersionless KdV Hierarchy |
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Lagrangian Approach to Dispersionless KdV Hierarchy |
| title_full |
Lagrangian Approach to Dispersionless KdV Hierarchy |
| title_fullStr |
Lagrangian Approach to Dispersionless KdV Hierarchy |
| title_full_unstemmed |
Lagrangian Approach to Dispersionless KdV Hierarchy |
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lagrangian approach to dispersionless kdv hierarchy |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/147203 |
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Lagrangian Approach to Dispersionless KdV Hierarchy / A. Choudhuri, B. Talukdar, U. Das // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 17 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT choudhuria lagrangianapproachtodispersionlesskdvhierarchy AT talukdarb lagrangianapproachtodispersionlesskdvhierarchy AT dasu lagrangianapproachtodispersionlesskdvhierarchy |
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2025-07-11T01:36:41Z |
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2025-07-11T01:36:41Z |
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1837312573313122304 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 096, 11 pages
Lagrangian Approach to Dispersionless
KdV Hierarchy
Amitava CHOUDHURI †1, B. TALUKDAR †1 and U. DAS †2
†1 Department of Physics, Visva-Bharati University, Santiniketan 731235, India
E-mail: amitava ch26@yahoo.com, binoy123@bsnl.in
†2 Abhedananda Mahavidyalaya, Sainthia 731234, India
Received June 05, 2007, in final form September 16, 2007; Published online September 30, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/096/
Abstract. We derive a Lagrangian based approach to study the compatible Hamiltonian
structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that
our treatment of the problem serves as a very useful supplement of the so-called r-matrix
method. We suggest specific ways to construct results for conserved densities and Hamil-
tonian operators. The Lagrangian formulation, via Noether’s theorem, provides a method
to make the relation between symmetries and conserved quantities more precise. We have
exploited this fact to study the variational symmetries of the dispersionless KdV equation.
Key words: hierarchy of dispersionless KdV equations; Lagrangian approach; bi-Hamiltonian
structure; variational symmetry
2000 Mathematics Subject Classification: 35A15; 37K05; 37K10
1 Introduction
The equation of Korteweg and de Vries or the so-called KdV equation
ut = 1
4u3x + 3
2uux
in the dispersionless limit [1]
∂
∂t
→ ε
∂
∂t
and
∂
∂x
→ ε
∂
∂x
with ε→ 0
reduces to
ut = 3
2uux. (1.1)
Equation (1.1), often called the Riemann equation, serves as a prototypical nonlinear partial
differential equation for the realization of many phenomena exhibited by hyperbolic systems [2].
This might be one of the reasons why, during the last decade, a number of works [3] was envisaged
to study the properties of dispersionless KdV and other related equations with special emphasis
on their Lax representation and Hamiltonian structure.
The complete integrability of the KdV equation yields the existence of an infinite family of
conserved functions or Hamiltonian densities Hn’s that are in involution. All Hn’s that generate
flows which commute with the KdV flow give rise to the KdV hierarchy. The equations of the
hierarchy can be constructed using [4]
ut = Λnux(x, t), n = 0, 1, 2, . . . (1.2)
with the recursion operator
Λ = 1
4∂
2
x + u+ 1
2ux∂
−1
x .
mailto:amitava_ch26@yahoo.com
mailto:binoy123@bsnl.in
http://www.emis.de/journals/SIGMA/2007/096/
2 A. Choudhuri, B. Talukdar and U. Das
In the dispersionless limit the recursion operator becomes
Λ = u+ 1
2ux∂
−1
x . (1.3)
According to (1.2), the pseudo-differential operator Λ in (1.3) defines a dispersionless KdV
hierarchy. The first few members of the hierarchy are given by
n = 0 : ut = ux, (1.4a)
n = 1 : ut = 3
2uux, (1.4b)
n = 2 : ut = 15
8 u
2ux, (1.4c)
n = 3 : ut = 35
16u
3ux, (1.4d)
n = 4 : ut = 315
128u
4ux. (1.4e)
Thus the equations in the dispersionless hierarchy can be written in the general form
ut = Anu
nux, (1.5)
where the values of An should be computed using (1.3) in (1.2). We can also generate A1, A2,
A3 etc recursively using
An =
(
1 + 1
2n
)
An−1, n = 1, 2, 3, . . . and A0 = 1.
The Hamiltonian structure of the dispersionless KdV hierarchy is often studied by taking
recourse to the use of Lax operators expressed in the semi-classical limit [5]. In this work we
shall follow a different viewpoint to derive Hamiltonian structure of the equations in (1.5). We
shall construct an expression for the Lagrangian density and use the time-honoured method of
classical mechanics to rederive and reexamine the corresponding canonical formulation. A single
evolution equation is never the Euler–Lagrange equation of a variational problem. One common
trick to put a single evolution equation into a variational form is to replace u by a potential
function u = −wx. In terms of w, (1.5) will become an Euler–Lagrange equation. We can,
however, couple a nonlinear evolution equation with an associated one and derive the action
principle. This allows one to write the Lagrangian density in terms of the original field variables
rather than the w’s, often called the Casimir potential. In Section 2 we adapt both these
approaches to obtain the Lagrangian and Hamiltonian densities of the Riemann type equations.
In Section 3 we study the bi-Hamiltonian structure [6]. One of the added advantage of the
Lagrangian description is that it allows one to establish, via Noether’s theorem, the relationship
between variational symmetries and associated conservation laws. The concept of variational
symmetry results from the application of group methods in the calculus of variations. Here one
deals with the symmetry group of an action functional A[u] =
∫
Ω0
L
(
x, u(n)
)
dx with L, the
so-called Lagrangian density of the field u(x). The groups considered will be local groups of
transformations acting on an open subset M⊂ Ω0×U ⊂ X ×U . The symbols X and U denote
the space of independent and dependent variables respectively. We devote Section 4 to study
this classical problem. Finally, in Section 5 we make some concluding remarks.
2 Lagrangian and Hamiltonian densities
For u = −wx (1.5) becomes
wxt = An(−1)nwn
xw2x. (2.1)
Lagrangian Approach to Dispersionless KdV Hierarchy 3
The Fréchet derivative of the right side of (2.1) is self-adjoint. Thus we can use the homotopy
formula [7] to obtain the Lagrangian density in the form
Ln = 1
2wtwx +
An(−1)n+1
(n+ 1)(n+ 2)
wn+2
x . (2.2)
In writing (2.2) we have subtracted a gauge term which is harmless at the classical level. The
subscript n of L merely indicates that it is the Lagrangian density for the nth member of the
dispersionless KdV hierarchy. The corresponding canonical Hamiltonian densities obtained by
the use of Legendre map are given by
Hn =
An
(n+ 1)(n+ 2)
un+2. (2.3)
Equation (1.5) can be written in the form
ut +
∂ρ[u]
∂x
= 0 (2.4)
with
ρ[u] = − An
(n+ 1)
un+1. (2.5)
There exists a prolongation of (1.5) or (2.4) into another equation
vt +
δ(ρ[u]vx)
δu
= 0, v = v(x, t) (2.6)
with the variational derivative
δ
δu
=
m∑
k=0
(−1)k ∂
k
∂xk
∂
∂ukx
, ukx =
∂ku
∂xk
such that the coupled system of equations follows from the action principle [8]
δ
∫
Lc dxdt = 0.
The Lagrangian density for the coupled equations in (2.4) and (2.6) is given by
Lc = 1
2(vut − uvt)− ρ[u]vx.
For ρ[u] in (2.5), (2.6) reads
vt = Anu
nvx. (2.7)
For the system represented by (1.5) and (2.7) we have
Lc
n = 1
2(vut − uvt) +
An
(n+ 1)
un+1vx. (2.8)
The result in (2.7) could also be obtained using the method of Kaup and Malomed [9]. Referring
back to the supersymmetric KdV equation [10] we identify v as a fermionic variable associated
with the bosonic equation in (1.5). It is of interest to note that the supersymmetric system is
complete in the sense of variational principle while neither of the partners is. The Hamiltonian
density obtained from the Lagrangian in (2.8) is given by
Hc
n = − An
(n+ 1)
un+1vx. (2.9)
It remains an interesting curiosity to demonstrate that the results in (2.3) and (2.9) represent
the conserved densities of the dispersionless KdV and supersymmetric KdV flows. We demon-
strate this by examinning the appropriate bi-Hamiltonian structures of (1.5) and the pair (1.5)
and (2.7).
4 A. Choudhuri, B. Talukdar and U. Das
3 Bi-Hamiltonian structure
Zakharov and Faddeev [11] developed the Hamiltonian approach to integrability of nonlinear
evolution equations in one spatial and one temporal (1+1) dimensions and Gardner [12], in
particular, interpreted the KdV equation as a completely integrable Hamiltonian system with ∂x
as the relevant Hamiltonian operator. A significant development in the Hamiltonian theory is
due to Magri [6] who realized that integrable Hamiltonian systems have an additional structure.
They are bi-Hamiltonian, i.e., they are Hamiltonian with respect to two different compatible
Hamiltonian operators. A similar consideration will also hold good for the dispersionless KdV
equations and we have
ut = ∂x
(
δHn
δu
)
= 1
2 (u∂x + ∂xu)
(
δHn−1
δu
)
, n = 1, 2, 3 . . . . (3.1)
Here
H =
∫
Hdx. (3.2)
It is easy to verify that for n = 1, (2.3), (3.1) and (3.2) give (1.4b). The other equations of the
hierarchy can be obtained for n = 2, 3, 4, . . . . The operators D1 = ∂x and D2 = 1
2 (u∂x + ∂xu)
in (3.1) are skew-adjoint and satisfy the Jacobi identity. The dispersionless KdV equation, in
particular, can be written in the Hamiltonian form as
ut = {u(x),H1}1 and ut = {u(x),H0}2
endowed with the Poisson structures
{u(x), u(y)}1 = D1δ(x− y) and {u(x), u(y)}2 = D2δ(x− y).
Thus D1 and D2 constitute two compatible Hamiltonian operators such that the equations
obtained from (1.5) are integrable in Liouville’s sense [6]. Thus Hn’s in (2.3) via (3.2) give
the conserved densities of (1.5). In other words, Hn’s generate flows which commute with the
dispersionless KdV flow and give rise to an appropriate hierarchy. It will be quite interesting
to examine if a similar analysis could also be carried out for the supersymmetric dispersionless
KdV equations.
The pair of supersymmetric equations ut = unux and vt = unvx can be written as
ηt = J1
(
δHs
n
δη
)
= J2
(
δHs
n−1
δη
)
, (3.3)
where η =
(
u
v
)
, Hs
n = Hc
n
An
and Hc
n =
∫
Hc
ndx. In (3.3) J1 and J2 stand for the matrices
J1 =
(
0 1
−1 0
)
and J2 =
(
0 u
−u 0
)
. (3.4)
Since Hc
n for different values of n represent the conserved Hamiltonian densities obtained by the
use of action principle, the supersymmetric dispersionless KdV equations will be bi-Hamiltonian
provided J1 and J2 constitute a pair of compatible Hamiltonian operators. Clearly, J1 and J2
are skew-adjoint. Thus J1 and J2 will be Hamiltonian operators provided we can show that [5]
pr vJiθ(ΘJi) = 0, i = 1, 2. (3.5)
Lagrangian Approach to Dispersionless KdV Hierarchy 5
Here pr stands for the prolongation of the evolutionary vector field v of the characteristic Jiθ.
The quantity pr vJi
θ is calculated by using
pr vJiθ =
∑
µ,j
Dj
(∑
ν
(Ji)µνθ
ν
)
∂
∂ηµ
j
, Dj =
∂
∂xj
, µ, ν = 1, 2. (3.6)
In our case the column matrix θ =
(
φ
ψ
)
represents the basis univectors associated with the
variables η =
(
u
v
)
. Understandably, θν and ηµ denote the components of θ and η and (Ji)µν
carries a similar meaning. The functional bivectors corresponding to the operators Ji is given by
ΘJi = 1
2
∫
θT ∧ Jiθdx (3.7)
with θT , the transpose of θ. From (3.4), (3.6) and (3.7) we found that both J1 and J2 satisfy (3.5)
such that each of them constitutes a Hamiltonian operator. Further, one can check that J1 and J2
satisfy the compatibility condition
pr vJ1θ(ΘJ2) + pr vJ2θ(ΘJ1) = 0.
This shows that (3.3) gives the bi-Hamiltonian form of supersymmetric dispersionless KdV
equations. The recursion operator defined by
Λ = J2J−1
1 =
(
u 0
0 u
)
reproduces the hierarchy of supersymmetric dispersionless KdV equation according to
ηt = Λnηx.
for n = 0, 1, 2, . . . . This verifies that Hc
n
An
’s as conserved densities generate flows which commute
with the supersymmetric dispersionless KdV flow.
4 Variational symmetries
The Lagrangian and Hamiltonian formulations of dynamical systems give a way to make the re-
lation between symmetries and conserved quantities more precise and thereby provide a method
to derive expressions for the conserved quantities from the symmetry transformations. In its
general form this is referred to as Noether’s theorem. More precisely, this theorem asserts that
if a given system of differential equations follows from the variational principle, then a continu-
ous symmetry transformation (point, contact or higher order) that leaves the action functional
invariant to within a divergence yields a conservation law. The proof of this theorem requires
some knowledge of differential forms, Lie derivatives and pull-back [5]. We shall, however, carry
out the symmetry analysis for the dispersionless KdV equation using a relatively simpler mathe-
matical framework as compared to that of the algebro-geometric theories. In fact, we shall make
use of some point transformations that depend on time and spatial coordinates. The approach
to be followed by us has an old root in the classical-mechanics literature. For example, as early
as 1951, Hill [13] provided a simplified account of Noether’s theorem by considering infinitesimal
transformations of the dependent and independent variables characterizing the classical field.
We shall first present our general scheme for symmetry analysis and then study the variational
or Noether’s symmetries of the dispersionless KdV equation.
6 A. Choudhuri, B. Talukdar and U. Das
Consider the infinitesimal transformations
xi′ = xi + δxi, δxi = εξi(x, f) (4.1a)
and
f ′ = f + δf, δf = εη(x, f) (4.1b)
for a field variable f = f(x, t) with ε, an arbitrary small quantity. Here x = {x0, x1}, x0 = t
and x1 = x. Understandably, our treatment for the symmetry analysis will be applicable to
(1 + 1) dimensional cases. However, the result to be presented here can easily be generalized to
deal with (3 + 1) dimensional problems. For an arbitrary analytic function g = g(xi, f), it is
straightforward to show that
δg = εXg
with
X = ξi ∂
∂xi
+ η
∂
∂f
, (4.2)
the generator of the infinitesimal transformations in (4.1). A similar consideration when applied
to h = h(xi, f, fi) with fi = ∂f
∂xi gives
δh = εX ′h (4.3)
with
X ′ = X +
(
ηi − ξj
i fj
) ∂
∂fi
. (4.4)
Understandably, X ′ stands for the first prolongation of X. To arrive at the statement for the
Noether’s theorem we consider among the general set of transformations in (4.1) only those that
leave the field-theoretic action invariant. We thus write
L(xi, f, fi)d(x) = L′(xi′, f ′, fi
′)d(x′), (4.5)
where d(x) = dxdt. In order to satisfy the condition in (4.5) we allow the Lagrangian density
to change its functional form L to L′. If the equations of motion, expressed in terms of the
new variables, are to be of precisely the same functional form as in the old variables, the two
density functions must be related by a divergence transformation. We thus express the relation
between L′ and L by introducing a gauge function Bi(x, f) such that
L′(xi′, f ′, fi
′)d(x′) = L(xi′, f ′, fi
′)d(x′)− εdB
i
dxi′d(x
′) + o(ε2). (4.6)
The general form of (4.6) for the definition of symmetry transformations will allow the scale
and divergence transformations to be considered as symmetry transformations. Understandably,
the scale transformations give rise to Noether’s symmetries while the scale transformations in
conjunction with the divergence term lead to Noether’s divergence symmetries. Traditionally,
the concept of divergence symmetries and concommitant conservation laws are introduced by
replacing Noether’s infinitesimal criterion for invariance by a divergence condition [14]. However,
one can directly work with the conserved densities that follow from (4.6) because nature of the
vector fields will determine the contributions of the gauge term. For some of the vector fields
the contributions of Bi to conserved quantities will be equal to zero. These vector fields are
Lagrangian Approach to Dispersionless KdV Hierarchy 7
Noether’s symmetries else we have Noether’s divergence symmetries. In view of (4.5), (4.6) can
be written in the form
L(xi′, f ′, fi
′)d(x′) = L(xi, f, fi)d(x) + ε
dBi
dxi
d(x). (4.7)
Again using L for h in (4.3), we have
L(xi′, f ′, fi
′)d(x′) = L(xi, f, fi)
[
d(x) + εdξi(x, fi)
]
+ εX ′L(xi, f, fi)d(x). (4.8)
From (4.7) and (4.8), we write
dBi
dxi
=
dξi
dxi
L+X ′L. (4.9)
Using the value of X ′ from (4.4) in (4.9), dBi
dxi is obtained in the final form
dBi
dxi
=
dξi
dxi
L+ ξi ∂L
∂xi
+ η
∂L
∂f
+
(
ηi − ξj
i fj
) ∂L
∂fi
. (4.10)
Thus we find that the action is invariant under those transformations whose constituents ξ and η
satisfy (4.10). The terms in (4.10) can be rearranged to write
d
dxi
{
Bi − ξiL+
(
ξjfj − η
) ∂L
∂fi
}
+
(
ξjfj − η
) [∂L
∂f
− d
dxi
(
∂L
∂fi
)]
= 0. (4.11)
The expression inside the squared bracket stands for the Euler–Lagrange equation for the clas-
sical field under consideration. In view of this, (4.11) leads to the conservation law
dIi
dxi
= 0 (4.12)
with the conserved density given by
Ii = Bi − ξiL+
(
ξjfj − η
) ∂L
∂fi
. (4.13)
In the case of two independent variables (x0, x1) ≡ (t, x), (4.12) can be written in the explicit
form
dI0
dt
+
dI1
dx
= 0. (4.14)
From (2.2) the Lagrangian density for the dispersionless KdV equation is obtained as
L = 1
2wtwx + 1
4w
3
x. (4.15)
Identifying f with w we can combine (4.13), (4.14) and (4.15) to get
B0
t + wtB
0
w − 1
4ξ
0
tw
3
x − 1
4ξ
0
wwtw
3
x + 1
2ξ
1
tw
2
x + 1
2ξ
1
wwtw
2
x − 1
2ηtwx − ηwwtwx
+B1
x + wxB
1
w + 1
2ξ
1
xw
3
x + 1
2ξ
1
ww
4
x + 1
2ξ
0
xw
2
t + 1
2wxw
2
t ξ
0
w + 3
4ξ
0
xw
2
xwt − 3
4ηxw
2
x
− 1
2ηxwt − 3
4ηww
3
x + 3
4ξ
0
ww
3
xwt = 0. (4.16)
8 A. Choudhuri, B. Talukdar and U. Das
In writing (4.16) we have made use of (2.1) with n = 1. Equation (4.16) can be globally satisfied
iff the coefficients of the following terms vanish separately
w0
x or w0
t : B0
t +B1
x = 0, (4.17a)
wt : B0
w − 1
2ηx = 0, (4.17b)
w2
t : 1
2ξ
0
x = 0, (4.17c)
wx : B1
w − 1
2ηt = 0, (4.17d)
w2
x : 1
2ξ
1
t − 3
4ηx = 0, (4.17e)
w3
x : −1
4ξ
0
t − 3
4ηw + 1
2ξ
1
x = 0, (4.17f)
w4
x : 1
2ξ
1
w = 0, (4.17g)
wtwx : −ηw = 0, (4.17h)
wtw
2
x : 1
2ξ
1
w + 3
4ξ
0
x = 0, (4.17i)
wtw
3
x : 1
2ξ
0
w = 0, (4.17j)
w2
twx : 1
2ξ
0
w = 0. (4.17k)
Equations in (4.17) will lead to finite number of symmetries. This number appears to be
disappointingly small since we have a dispersionless KdV hierarchy given in (1.5). Further,
symmetry properties reflecting the existence of infinitely many conservation laws will require
an appropriate development for the theory of generalized symmetries. In this work, however,
we shall be concerned with variational symmetries only.
From (4.17c), (4.17j) and (4.17k) we see that ξ0 is only a function of t. We, therefore, write
ξ0(x, t, w) = β(t). (4.18)
Also from (4.17g), (4.17i) and (4.18) we see that ξ1 is not a function of w. In view of (4.17h)
and (4.18), (4.17f) gives
ξ1x − 1
2βt = 0
which can be solved to get
ξ1 = 1
2βtx+ α(t), (4.19)
where α(t) is a constant of integration. Using (4.19) in (4.17e) we have
ηx = 1
3βttx+ 2
3αt. (4.20)
The solution of (4.20) is given by
η = 1
6βttx
2 + 2
3αtx+ γ(t) (4.21)
with γ(t), a constant of integration. In view of (4.21), (4.17b) and (4.17d) yield
B0 = 1
6βttxw + 1
3αtw (4.22)
and
B1 = 1
12βtttx
2w + 1
3αttxw. (4.23)
Equations (4.22) and (4.23) can be combined with (4.17a) to get finally
βttt = 0 and αtt = 0. (4.24)
Lagrangian Approach to Dispersionless KdV Hierarchy 9
From (4.24) we write
β = 1
2a1t
2 + a2t+ a3 (4.25)
and
α = b1t+ b2, (4.26)
where a’s and b’s are arbitrary constants. Substituting the values of β and α in (4.18), (4.19),
(4.21) we obtain the infinitesimal transformation, ξ0, ξ1 and η, as
ξ0 = 1
2a1t
2 + a2t+ a3, (4.27a)
ξ1 = 1
2(a1t+ a2)x+ b1t+ b2, (4.27b)
η = 1
6a1x
2 + 2
3b1x+ b3. (4.27c)
In writing (4.27c) we have treated γ(t) as a constant and replaced it by b3. Implication of this
choice will be made clear while considering the symmetry algebra. In terms of (4.27), (4.2)
becomes
X = a1V1 + a2V2 + a3V3 + b1V4 + b2V5 + b3V6,
where
V1 = 1
2 t
2 ∂
∂t
+ 1
2xt
∂
∂x
+ 1
6x
2 ∂
∂w
, V2 = t
∂
∂t
+ 1
2x
∂
∂x
,
V3 =
∂
∂t
, V4 = t
∂
∂x
+ 2
3x
∂
∂w
, V5 =
∂
∂x
, V6 =
∂
∂w
. (4.28)
It is easy to check that the vector fields V1, . . . , V6 satisfy the closure property. The commutation
relations between these vector fields are given in Table 1.
Table 1. Commutation relations for the generators in (4.28). Each element Vij in the Table is represented
by Vij = [Vi, Vj ].
V1 V2 V3 V4 V5 V6
V1 0 −V1 −V2 0 −1
2V4 0
V2 V1 0 −V3
1
2V4 −1
2V5 0
V3 V2 V3 0 V5 0 0
V4 0 −1
2V4 −V5 0 −2
3V6 0
V5
1
2V4
1
2V5 0 2
3V6 0 0
V6 0 0 0 0 0 0
The symmetries in (4.28) are expressed in terms of the velocity field and depend explicitly
on x and t. Looking from this point of view the symmetry vectors obtained by us bear some
similarity with the so called ‘addition symmetries’ suggested independently by Chen, Lee and
Lin [15] and by Orlov and Shulman [16]. It is easy to see that V2 to V6 correspond to scaling,
time translation, Galilean boost, space translation and translation in velocity space respectively.
The vector field V1 does not admit such a simple physical realization. However, we can write V1
as V1 = 1
2 tV2 + 1
4xV4.
Making use of (4.15), (4.22), (4.23), (4.25) and (4.26) we can write the expressions for the
conserved quantities in (4.13) as
I0 = 1
6a1xw + 1
3b1w −
1
4ξ
0w3
x + 1
2ξ
1w2
x − 1
2ηwx, (4.29a)
I1 = 1
2ξ
0w2
t + 3
4ξ
0wtw
2
x + 1
2ξ
1w3
x − 1
2ηwt − 3
4ηw
2
x. (4.29b)
10 A. Choudhuri, B. Talukdar and U. Das
The expressions for I0 and I1 are characterized by ξi and η, the values of which change as we
go from one vector field to the other. The first two terms in I0 stand for the contribution of B0
and there is no contribution of the gauge term in I1 since from (4.23) and (4.24) B1 = 0. For
a particular vector field a1 and b1 may either be zero or non zero. One can verify that except for
vector fields V1 and V4, a1 = b1 = 0 such that V2, V3, V5 and V6 are simple Noether’s symmetries
while V1 and V4 are Noether’s divergence symmetries. Coming down to details we have found
the following conserved quantities from (4.29a) and (4.29b)
I0
V1
= 1
6xw −
1
8 t
2w3
x + 1
4xtw
2
x − 1
12x
2wx, (4.30a)
I1
V1
= 1
4xtw
3
x + 3
8 t
2wtw
2
x + 1
4 t
2w2
t − 1
12x
2wt − 1
8x
2w2
x, (4.30b)
I0
V2
= −1
4 tw
3
x + 1
4xw
2
x, (4.30c)
I1
V2
= 1
4xw
3
x + 3
4 twtw
2
x + 1
2 tw
2
t , (4.30d)
I0
V3
= −1
4w
3
x, (4.30e)
I1
V3
= 3
4wtw
2
x + 1
2w
2
t , (4.30f)
I0
V4
= 1
3w + 1
2 tw
2
x − 1
3xwx, (4.30g)
I1
V4
= 1
2 tw
3
x − 1
3xwt − 1
2xw
2
x, (4.30h)
I0
V5
= 1
2w
2
x, (4.30i)
I1
V5
= 1
2w
3
x, (4.30j)
I0
V6
= −1
2wx, (4.30k)
I1
V6
= −1
2wt − 3
4w
2
x. (4.30l)
It is easy to check that the results in (4.30) is consistent with (4.14). The pair of conserved
quantities corresponding to time translation, space translation and velocity space translation,
namely, {(4.30e),(4.30f)}, {(4.30i),(4.30j)} and {(4.30k),(4.30l)} do not involve x and t explicitly.
Each of the pair in conjunction with (4.14) give the dispersionless KdV equation in a rather
straightforward manner. As expected (4.30e) stands for the Hamiltonian density or energy
of (1.4b).
5 Conclusion
Compatible Hamiltonian structures of the dispersionless KdV hierarchy are traditionally ob-
tained with special attention to their Lax representation in the semiclassical limit. The deriva-
tion involves judicious use of the so-called r-matrix method [17]. We have shown that the
combined Lax representation–r-matrix method can be supplemented by a Lagrangian approach
to the problem. We found that the Hamiltonian densities corresponding to our Lagrangian rep-
resentations stand for the conserved densities for the dispersionless KdV flow. We could easily
construct the Hamiltonian operators from the recursion operator which generates the hierarchy.
We have derived the bi-Hamiltonian structures for both dispersionless KdV and supersymmetric
KdV hierarchies. As an added realism of the Lagrangian approach we studied the variational
symmetries of equation (1.4b). We believe that it will be quite interesting to carry out similar
analysis for the supersymmetric KdV pair in (1.4b) and for n = 1 limit of (2.7).
Acknowledgements
This work is supported by the University Grants Commission, Government of India, through
grant No. F.32-39/2006(SR).
Lagrangian Approach to Dispersionless KdV Hierarchy 11
References
[1] Zakharov V.E., Benney equations and quasiclassical approximation in the method of the inverse problem,
Funct. Anal. Appl. 14 (1980), 89–98.
[2] Olver P.J., Nutku Y., Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys.
29 (1988), 1610–1619.
Brunelli J.C., Dispersionless limit of integrable models, Braz. J. Phys. 30 (2000), 455–468, nlin.SI/0207042.
[3] Arik M., Neyzi F., Nutku Y., Olver P.J., Verosky J.M., Multi-Hamiltonian structure of the Born–Infeld
equation, J. Math. Phys. 30 (1989), 1338–1344.
Das A., Huang W.J., The Hamiltonian structures associated with a generalized Lax operator, J. Math. Phys.
33 (1992), 2487–2497.
Brunelli J.C., Das A., Properties of nonlocal charges in the supersymmetric two boson hierarchy, Phys.
Lett. B 354 (1995), 307–314, hep-th/9504030.
Brunelli J.C., Das A., Supersymmetric two-boson equation, its reductions and the nonstandard supersym-
metric KP hierarchy, Intern. J. Modern Phys. A 10 (1995), 4563–4599, hep-th/9505093.
Brunelli J.C., Hamiltonian structures for the generalized dispersionless KdV hierarchy, Rev. Math. Phys. 8
(1996), 1041–1054, solv-int/9601001.
Brunelli J.C., Das A., A Lax description for polytropic gas dynamics, Phys. Lett. A 235 (1997), 597–602,
solv-int/9706005.
Brunelli J.C., Das A., The sTB-B hierarchy, Phys. Lett. B 409 (1997), 229–238, hep-th/9704126.
Brunelli J.C., Das A., A Lax representation for Born–Infeld equation, Phys. Lett. B 426 (1998), 57–63,
hep-th/9712081.
[4] Calogero F., Degasperis A., Spectral transform and soliton, North-Holland Publising Company, New York,
1982.
[5] Olver P.J., Application of Lie groups to differential equation, Springer-Verlag, New York, 1993.
[6] Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162.
[7] Frankel T., The geometry of physics, Cambridge University Press, UK, 1997.
[8] Ali Sk.G., Talukdar B., Das U., Inverse problem of variational calculus for nonlinear evolution equations,
Acta Phys. Polon. B 38 (2007), 1993–2002, nlin.SI/0603037.
[9] Kaup D.J., Malomed B.A., The variational principle for nonlinear waves in dissipative systems, Phys. D 87
(1995), 155–159.
[10] Barcelos-Neto J., Constandache A., Das A., Dispersionless fermionic KdV, Phys. Lett. A 268 (2000), 342–
351, solv-int/9910001.
[11] Zakharov V.E., Faddeev L.D., Korteweg–de Vries equation: a completely integrable Hamiltonian systems,
Funct. Anal. Appl. 5 (1971), 18–27.
[12] Gardner C.S., Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as
a Hamiltonian system, J. Math. Phys. 12 (1971), 1548–1551.
[13] Hill E.L., Hamilton’s principle and the conservation theorems of mathematical physics, Rev. Modern Phys.
23 (1951), 253–260.
[14] Gelfand I.M., Fomin S.V., Calculus of variations, Dover Publ., 2000.
[15] Chen H.H., Lee Y.C., Lin J.E., On a new hierarchy of symmetries for the Kadomtsev–Petviashvilli equation,
Phys. D 9 (1983), 439–445.
[16] Orlov A.Yu., Shulman E.I., Additional symmetries for integral and conformal algebra representation, Lett.
Math. Phys. 12 (1986), 171–179.
[17] Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987.
http://arxiv.org/abs/nlin.SI/0207042
http://arxiv.org/abs/hep-th/9504030
http://arxiv.org/abs/hep-th/9505093
http://arxiv.org/abs/solv-int/9601001
http://arxiv.org/abs/solv-int/9706005
http://arxiv.org/abs/hep-th/9704126
http://arxiv.org/abs/hep-th/9712081
http://arxiv.org/abs/nlin.SI/0603037
http://arxiv.org/abs/solv-int/9910001
1 Introduction
2 Lagrangian and Hamiltonian densities
3 Bi-Hamiltonian structure
4 Variational symmetries
5 Conclusion
References
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