Abstract
We obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are the ε-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limit ε → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ε → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ε → 0 and ε → ∞.
1 Introduction
Coupled Korteweg-de Vries (KdV) systems describe several physical interactions of interest. Hirota and Satsuma [1] proposed a model that describes interactions of two long waves with different dispersion relations. Gear and Grimshaw [2, 3] considered a coupled KdV system to describe linearly-stable internal waves in a stratified fluid.
More recently Lou, Tong, Hu and Tang [4] proposed models which may be used in the description of atmospheric and oceanic phenomena. Coupled KdV systems were also analyzed in [5-7]. An important area of interest of high energy physics related to coupled systems is provided by the supersymmetric extensions of KdV equations [8-14] and more generally by operator and Clifford valued extensions of KdV equation [15, 16].
In this work we consider a parametric coupled KdV system. For some values of the parameter, λ < 0, the system corresponds to the complexification of KdV equation. For λ = 0 the system corresponds to one of the HirotaSatsuma coupled KdV systems, while for λ > 0 the system is equivalent to two decoupled KdV equations. We analyze the Hamiltonian formulation and the associated Poisson bracket structure of the system. Although some properties of the complexification of KdV arise directly from the analogous ones on the solutions of the KdV equation there are new properties, in particular, the full Hamiltonian structure, which does not have an analogous on the original real equation. In fact, the complexification approach gives rise only to holomorphic observables on phase space. The full Hamiltonian structure of the complex system give rise to self-adjoint Hamiltonian functionals, whose Hamiltonian flow are the complex KdV equations, and it provides the full structure of observables on phase space, not only the holomorphic ones.
The goal of this paper is to analyze the different Poisson structures associated with the coupled KdV system in order to determine which of them are compatible, in the sense that a linear combination of them is also a Poisson structure. This goal is important from the point of view of the integrability of the system. Moreover, it may also be an important first step in order to understand the quantization of the system in the sense of quantization deformation of Poisson structures [17].
The approach we will follow in our analysis is to construct a family of lagrangians from which the coupled KdV system is obtained by taking independent variations of the associated functional action with respect to the fields defining the lagrangian function. It turns out that these lagrangians are singular ones. This implies that the Hamiltonian construction, via a Legendre transformation is formulated on a constrained phase space. In all the cases we will consider the constraints turn out to be primary constraints and of the second class. The unconstrained phase space is equipped with a Poisson bracket structure, however since there are second class constraints we must obtain the Poisson bracket structure on the constrained submanifold of the phase space. This Poisson bracket is provided by the Dirac brackets [18]. It satisfies all the properties of a Poisson bracket, in particular the Jacobi identity. In this way starting from a lagrangian for the system we can construct a Poisson bracket structure, together with a Hamiltonian functional. This approach was followed for the KdV equation in [19, 20]. It provides a geometrical picture on phase space of the Hamiltonian structure of the integrable system. The other way to proceed is to find a Hamiltonian operator together with a Hamiltonian functional. Afterwards we may construct a Poisson bracket structure provided the Hamiltonian operator satisfies a differential restriction [21] ensuring that the Jacobi identity is satisfied. In this approach the set of allowed observables is only a subset of the space of observables of the more general formulation in terms of the constrained phase space approach.
In Section 2 we introduce the parametric coupled KdV system we wish to analyze. In Section 3 we obtain two lagrangian densities. The functional Gateaux derivatives of the corresponding actions give rise to the parametrically coupled KdV system. By a Legendre transformation we obtain the corresponding hamiltonians and associated Poisson structures. In Section 4 we introduce a Miura transformation which allows the construction of two additional lagrangian densities with the corresponding Hamiltonian and Poisson structures. In Section 5 we study the compatibility of the Poisson structures obtained in Sections 3 and 4. We obtain a pencil of Poisson bracket structures each of them associated to a Hamiltonian functional. In particular this implies compatibility between some of the Poisson structures. In Section 6 we construct a duality relation among the Hamiltonian structures. We construct two master lagrangians for the parametric coupled system whose field equations are the Gardner equations obtained from the coupled KdV system though a ε-parametric Gardner transformation. In the weak limit ε → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ε → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ε → 0 and ε → ∞.
2 The parametric coupled KdV system
We consider a coupled Korteweg-de Vries (KdV) system, formulated in terms of two real differentiable functions u(x,t) and v(x,t), given by the following partial differential equations:
where λ is a real parameter.
Here and in the sequel u and v belong to the real Schwartz space defined by
By a redefinition of v given by
The case λ = +1 corresponds to two decoupled KdV equations.
The system (1),(2) for λ = −1 describes a two-layer liquid model studied in references [2–4, 22]. It is a very interesting evolution system. It is known to have solutions developing singularities on a finite time [23]. Also, a class of solitonic solutions was reported in [24] via the Hirota approach [25].
The system (1),(2) (introduced in [26]) for λ = 0 corresponds to the ninth Hirota-Satsuma [1] coupled KdV system given in [6] (for the particular value of k = 0) (see also [5]) and is also included in the interesting study which relates integrable hierarchies with polynomial Lie algebras [7].
A Bäcklund transformation, the permutability theorem, the Gardner transformation as well as the Gardner equations for the coupled KdV system (1), (2), were obtained in [27]. Also a class of multisolitonic solutions, a class of periodic solutions and a regular static solution, which can be interpreted as a non-trivial background of the theory, were found in [27].
3 Poisson structures
In this and in the following section we will show that there exists four basic Hamiltonians and four associated basic Poisson structures for the coupled KdV system we are considering. We will use the method of Dirac for constrained systems to deduce them. The Hamiltonian as defined in quantum physics must be a selfadjoint operator conjugate to the time, hence our four Hamiltonians will be four real functionals in terms of the real fields which define them.
We rewrite (1) and (2) in terms of the Casimir potentials w and y given by
and we obtain
We use the Helmholtz procedure to obtain a lagrangian density from the field equations. We notice that the matrix constructed from the Gateaux derivatives of P and λQ is self-adjoint. We then get
which explicitly gives
for λ ≠ 0.
We define the functional action
By taking independent variations of L2 with respect to w and y we obtain the same field equations.
which are the same as equations (1), (2). Here
We now introduce a second functional action
which explicitly gives
for any λ
By taking independent variations of L2 with respect to w and to y we obtain the field equations.
We will now construct the Hamiltonian structure associated to each of the lagrangians 𝓛i; i = 1, 2. We start by considering the lagrangian 𝓛1. We introduce the conjugate momenta associated to w and y, we denote them p and q respectively, we have
We define
Hence ϕ1 = ϕ2 = 0. ϕ1 and ϕ2 do not have any wt nor any yt dependence, they are constraints on the phase space. It turns out that these are the only constraints on the phase space.
The Hamiltonian may be obtain directly from 𝓛1 by performing a Legendre transformation,
We obtain
and the corresponding Hamiltonian is
We introduce a Poisson structure on the phase space defined by
with all other brackets between these variables being zero.
We then have that ϕ1, ϕ2 are second class constraints. In fact,
The lagrangian system is degenerate (singular), such system can be hamiltonized only by the use of Dirac’s theory of constraints [28, 29].
Since we have a constrained phase space we must introduce the Dirac brackets corresponding to a Lie bracket structure on the constrained submanifold of phase space. For the second and third order degenerate lagrangians considered in this paper, one first needs to look for some kind of order reduction and then use the Dirac’s theory of constraints to hamiltonize the system [19, 30, 31]. Equivalently, one can formulated the Dirac theory of constraints for higher order lagrangians using the above Poisson bracket structure and the relations
The Dirac brackets between two functionals F and G on phase space is defined as
where <>x′ denotes integration on x′ from −∞ to +∞. The indices i, j = 1, 2 and the 𝕔ij(x′, x″) are the components of the inverse of the matrix whose components are {ϕi(x′), ϕi(x″)}PB.
This matrix becomes
and its inverse, satisfying
is given by
It turns out, after some calculations, that
We notice that this Poisson bracket is not well defined for λ = 0. We have already assumed λ ≠ 0.
From them we obtain the Hamilton equations, which are of course the same as (1), (2):
We notice that adding any function of the constraints to H1 does not change the result, since the Dirac bracket of the constraints with any other local function of the phase space variables is zero.
Moreover, we may obtain directly the Dirac bracket of any two functionals F (u, v) and G (u, v) from (3) using the above bracket relations for u and v . We notice that the observables F and G in (3) may be functionals of w, y, p and q, not only of u and v . In this sense the phase space approach for singular lagrangians provides the most general space of observables. The same comment will be valid for the phase space construction using the action L2 and
The lagrangians 𝓛1 and 𝓛2 do not differ by a gauge term, despite that both of these lead to same equations of motion. Thus 𝓛2 represents an alternative lagrangian representation of the system. First, there can arise ambiguities in the association of symmetries with constants of the motion. Secondly, the same classical system, via alternative lagrangian description, can give rise to entirely different quantum mechanical systems. These points have been developed in the context of classical mechanics [32, 33]. The Noether theorem assures that associated to a differentiable symmetry of the lagrangian one has a conserved quantity. A symmetry of the action is also a symmetry of the field equations. However there may be symmetries of the field equations which are not symmetries of the action. In [27] we obtained an infinite sequence of polynomial local conserved quantities for equations (1) and (2). Most of them are not symmetries of the action L1 nor of L2.
We now consider the action L2 and its associated Hamiltonian structure. In this case we denote the conjugate momenta to w and y by p̂ and q̂ respectively. We have
The constraints become in this case
The corresponding Poisson brackets between ϕi and ϕj, i, j = 1, 2, are given by
The corresponding construction of the Dirac brackets yields
The Hamiltonian
The Hamilton equations
now using the corresponding Dirac brackets yields the same fields equations (1),(2) for any . We have thus constructed two Hamiltonian functionals and associated Poisson bracket structures. These two Hamiltonian structures arise directly from the basic actions L1 and L2. We will now construct two additional Hamiltonian structures by considering the Miura transformation.
The Hamiltonians H1 and H2,
4 The Miura transformation
We consider the Miura transformation
The corresponding modified KdV system (MKdVS) is given by
These equations may be obtained from two actions, which we will denote
The lagrangian densities
and
respectively. We will now construct the Hamiltonian structure associated to
We denote by α and β the conjugate momenta associated to σ and ρ respectively. We have
These are constraints on the phase space.
The Hamiltonian
where u and v are given in terms of μ and by the Miura transformation.
The construction of the Dirac brackets follows in the usualway.We end up with the following Poisson structure on the constrained submanifold,
From these Poisson bracket structure we obtain for the original u and v fields
which defines the Poisson structure on the original fields inherited from the Poisson structure on the constrained submanifold on the phase space associated to the modified KdV system. This Poisson bracket is not well defined for λ ≠ 0. We have already assumed λ ≠ 0.
From the Dirac brackets of u and v we may obtain directly the Hamiltonian field equations
which, as it should be, coincide with system (1), (2).
We have then obtained the Poisson structure associated to the Hamiltonian
We now proceed to obtain a second Poisson structure starting from the Lagrangian
The Hamiltonian obtained via a Legendre transformation is given by
We denote by α̂ and β̂ the conjugate momenta associated to σ and ρ respectively.
The constraints on phase space become now
The Dirac brackets are
We then obtain, for any λ,
This is the Poisson bracket structure inherited from the second Poisson structure on the modified phase space. One may directly verify that the corresponding Hamilton equations exactly coincide with equations (1), (2). We have then constructed four basic lagrangians and associated Hamiltonian functionals together with four basic Poisson structures.
5 Two pencils of Poisson structures for the coupled system
In this section we show the existence of two pencils of Poisson structures for the system (1) and (2). The strategy will be to introduce two parametric lagrangian densities 𝓛k and
We now introduce the parametric lagrangian density 𝓛k, where k is a real parameter, associated to the two basic actions L1 and L2.
We define the lagrangian density
The field equations obtained from this lagrangian density are equivalent to (1) and (2) in the following cases: If λ < 0 for any k. If λ < = 0, for k ≠ 1. If λ > 0 for
and the primary constraints by
These are the only constraints on phase space.
The Poisson brackets on the unconstrained phase space are
Hence they are second class constraints.
We will denote by
The Dirac brackets are then given by
where the denominator is different from zero for the values of k we are considering. They define the Poisson structure for the Hamiltonian
The associated Hamilton equations coincide with the coupled equations (1), (2). It is interesting to notice that the above Poisson structure is a linear combination of the Dirac brackets associated to Hamiltonians H1 and H2. In the present notation H2 corresponds to k = 0.
We then have
where F ,G are any functionals of u and v and
Consequently, since
We also notice that for any k and λ = −1, using the above Poisson bracket structure, one gets
We emphasize that only (13) arises from the complexification of the corresponding Poisson structure for real KdV. The relation (12) follows in our approach from first principles. It is not imposed by hand. The existence of a local real Hamiltonian Hk for each k is a non-trivial feature of the system (1), (2) and is not an algebraic consequence of the complexification of the real KdV equation.
We may now consider the case λ = 0. The Poisson bracket for any k ≠ 1 becomes
in particular for
We have thus constructed a pencil of Poisson structures with an associated local real Hamiltonian
We now introduce, as we have already done with 𝓛1and 𝓛2, a parametric lagrangian density
these constraints are the only ones on the phase space. The Poisson brackets on the unconstrained phase space are
Hence they are second class constraints.
The Dirac brackets are then given by
It follows from the construction that the Hamilton equations in terms of the corresponding Poisson structure,
are equivalent to the coupled KdV system (1),(2).
As in the previous case the pencil of Poisson structures can be rewritten in terms of the basic Poisson structures which corresponds to k = 1 and k = 0 in (14):
where
We notice that this decomposition is the same as in previous case, however the basic Poisson structure are different.
In particular for
We notice that by construction ϕ1 and ϕ2 as well as any functional of them, in all the cases we have considered, are Casimirs of the Poisson structure defined in terms of the Dirac brackets. In fact, for any functional F on phase space. This is a general property of the Dirac bracket.
It is a non-trivial feature that for each real k, the parameter of the pencil of Poisson structures, there are Hamiltonians Hk and
6 Duality among the Hamiltonian structures
In this section we show the existence of two ϵ-deformed Hamiltonians and their corresponding ϵ-deformed Poisson structures. In the limit ϵ → 0 they reduced to the Hamiltonians and Poisson structures obtained in Section 3. In the limit ϵ → ∞ they reduce to the Hamiltonians and Poisson structures obtained in Section 4. The ε-deformation for the coupled KdV system is analogous to the duality transformation in quantum field theory. One formulation can be analyzed in perturbation theory for small values of the coupling constant ϵ , the weak coupling limit, while the dual formulation can be analyzed perturbatively as an expansion in
The Hamiltonian, constraints and Poisson structures we will present in this section are obtained following the same approach we have developed in previous sections. We will only provide the results.
The associated Gardner transformation and Gardner equations for the system (1), (2) are given by [27]
and
Any solution of the Gardner equations define through Gardner transformation a solution of the system (1), (2). It follows as in the case of KdV equation and the corresponding Gardner transformation [36] that if we take the limit ϵ → 0 we obtain from (16), (17)
and from (18), (19) after replacing (20), (21) we get the original KdV coupled system (1), (2).
On the other hand, if we redefine
Taking the limit ϵ → 1 we have
which are exactly equations given by (5).
If we proceed in the same way with (18), (19) we obtain the modified KdV system equations (6). That means that if u, v satisfies the MKdV then û, v̂ satisfies equations (1), (2).
This result is analogous to the one known for the KdV equation and its Miura transformation.
We go now one step forward, beyond this relation between field equations, and prove that there exists a master lagrangian form which by taking variations with respect to r and s we obtain the ϵ -parametric Gardner equations (18), (19). Moreover, we will construct two different lagrangians with such property and the associated Hamiltonian structures. The corresponding ϵ -parametric Poisson structures are compatible. In the limit ϵ → 0 the ε-parametric Hamiltonian structures reduce to the compatible Hamiltonian structures obtained in Section 3 and with the ϵ → ∞ limit, after suitable redefinition of fields, we obtain the two compatible Hamiltonian structures obtained in Section 4. There is a duality relation between the Hamiltonian structure of the parametric coupled KdV system (1), (2) and the modified parametric system (MKdV) (6). In the weak limit ϵ → 0 we get one Hamiltonian structure and in the strong limit ϵ → ∞ we get the modified Hamiltonian structure.
By freezing the fieldv to zero we obtain from (1), (2) the KdV equation and the same occurs with the ϵ-parameter Hamiltonian structure of system (1), (2). It reduces to aϵ- parametric Hamiltonian structure of KdV equation (which to our knowledge had not been constructed before).
In particular, the Poisson structure of the KdV system (1), (2) is a ϵ -deformation of the Poisson structure of MKdV system which is equivalent to the Virasoro algebra.
We now proceed to the construction of the master Gardner lagrangians.
We introduce the fields w(x, t), y(x, t) through
The first master Gardner lagrangian density, which works for λ ≠ 0, is given by
and the second master Gardner lagrangian density is given, for any λ, by
If we take the limit ϵ → 0 of the above expressions, we obtain lagrangians densities 𝓛1 and 𝓛2 respectively given in Section 3.
If we redefine
and take the limit ϵ → ∞ we get
The Hamiltonian structure associated to
We introduce the conjugate momenta associated to w andy. We denote them p and q. We have
and define
ϕ1 = 0 and ϕ2 = 0 are constraints on the phase space. They are second class constraints. Notice that this is the same structure of phase space as in Section 3 because the ϵ-parameter does not appear in the definition of the conjugate momenta. The Hamiltonian density may be obtained by performing a Legendre transformation
We get
The Dirac brackets between the canonical variables w, p, y, q are the same as was obtained in section 3 since the ϵ-parameter does not appear in the constraints. However the corresponding brackets among u, v are different because the Gardner transformation involves the parameter ϵ . In the limit ϵ → 0 we obtain the same Dirac brackets for u and v as in Section 3. The same analysis can be performed for LG2. The conjugate momenta and contraints as well as the Dirac brackets among the canonical variables w, p, y, q are the same as in second part of Section 3. The Hamiltonian is given by
In the limit ϵ → 0 it reduces to the Hamiltonian 𝓗2 in Section 3. In the same way by redefining ϵ2𝓗G1, ϵ2𝓗G2 and taking ϵ → ∞ we get the Hamiltonians
The Poisson structures in both limits are the ones explicitly given in in Sections 3 and 4.
The construction of the ϵ -parametric Gardner Hamiltonian and its Poisson structures unifies all the Hamiltonian structures of the coupled KdV system and make manifest the duality relation among the KdV and the modified KdV systems.
7 Conclusions
We obtained the full Hamiltonian structure for a coupled parametric KdV system. We started from four basic singular lagrangians. The associated Hamiltonian formulation on phase space is restricted by second class constraints. The Poisson structure on the constrained variety of phase space was obtained using the Dirac approach. The Dirac brackets on the constrained phase space yields the most general structure of observables. A subset of them are functionals of the original fieldsu(x,t ), v(x,t ) of the coupled KdV system. We then constructed two pencils of Poisson brackets each of them with an associated parametric Hamiltonian in terms of the same parameter of each pencil.
Each pencil of Poisson brackets is obtained from two compatible Poisson brackets of the same dimension. Consequently it is not possible to construct a hierarchy of higher dimensional Hamiltonians from them. However the two pencils of Poisson brackets are of different dimensions, hence one may construct a hierarchy of higher order Hamiltonians as in the KdV case.
Finally we constructed two master lagrangians for the parametric coupled KdV system whose field equations are the Gardner equations obtained from the coupled KdV through a ϵ -parametric Gardner transformation. The Gardner transformation is a non-linear transformation of the fields for each value of the parameter ϵ. It defines a duality transformation between the theories obtained in the two limits ϵ → 0 and ϵ → ∞.
In the weak limit ϵ → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ϵ → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ϵ → 0 and ϵ → ∞.
We have thus disentangled all the Hamiltonian structure (a very rich one) associated with the parametric coupled KdV system. This goal which we have fulfilled is the first step towards the quantization of the coupled KdV system in the sense of quantization deformation of the Poisson structures. The duality relation may be very important to relate the quantization of all the Poisson structures.
Acknowledgement
The authors sincerely thank the reviewer for their valuable suggestions and useful comments.
A. R. and A. S. are partially supported by Projects Fondecyt 1121103 and Mecesup ANT398 (Chile).
References
1 R. Hirota, J. Satsuma, Phys. Lett. 85A, 8, 9 407 (1981)10.1016/0375-9601(81)90423-0Search in Google Scholar
2 J.A. Gear, R. Grimshaw, Stud. Appl. Math. 70, 235 (1984)10.1002/sapm1984703235Search in Google Scholar
3 J.A. Gear, Stud. Appl. Math. 72, 95 (1985)10.1002/sapm198572295Search in Google Scholar
4 S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, J. Phys. A: Math. Gen. 39, 513 (2006)10.1088/0305-4470/39/3/005Search in Google Scholar
5 A.K. Karasu, J. Math. Phys. 38, 3616 (1997)10.1063/1.532056Search in Google Scholar
6 S. Yu. Sakovich, J. Nonlin. Math. Phys. 6, 255 (1999)10.2991/jnmp.1999.6.3.2Search in Google Scholar
7 P. Casati, G. Ortenzi, J. Geom. Phys. 56, 418 (2006)10.1016/j.geomphys.2005.02.010Search in Google Scholar
8 B.A. Kupershmidt, Phys. Lett. A 102, 213 (1984)10.1016/0375-9601(84)90693-5Search in Google Scholar
9 P. Mathieu, J. Math. Phys. 29, 2499 (1988)10.1063/1.528090Search in Google Scholar
10 P. Labelle, P. Mathieu, J. Math. Phys. 32, 923 (1991)10.1063/1.529351Search in Google Scholar
11 S. Belucci, E. Ivanov, S. Krivonos, J. Math. Phys. 34, 3087 (1993)10.1063/1.530064Search in Google Scholar
12 F. Delduc, E. Ivanov, Phys. Lett. B 309, 312 (1993)10.1016/0370-2693(93)90939-FSearch in Google Scholar
13 F. Delduc, E. Ivanov, S. Krivonos, J. Math. Phys. 37, 1356 (1996)10.1063/1.531796Search in Google Scholar
14 Z. Popowicz, Phys. Lett. B 459, 150 (1999)10.1016/S0370-2693(99)00633-4Search in Google Scholar
15 P.J. Olver, V.V. Sokolov, Commun. Math. Phys. 193, 245 (1998)10.1007/s002200050328Search in Google Scholar
16 A. Restuccia, A. Sotomayor, J. Math. Phys. 54, 113510 (2013)10.1063/1.4834056Search in Google Scholar
17 B. Fedosov, Deformation quantization and index theory, Mathematical topics: 9 (Akademic-Verl. Berlin, Germany, 1996) 325p.Search in Google Scholar
18 P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School Monograph Series No.2 (Yeshiva University, New York, 1964)Search in Google Scholar
19 Y. Nutku, J. Math. Phys. 25, 2007 (1984)10.1063/1.526395Search in Google Scholar
20 G.W. Kentwell, J. Math. Phys. 29, 46 (1988)10.1063/1.528133Search in Google Scholar
21 P.J. Olver, Y. Nutku, J. Math. Phys. 29, 1610 (1988)10.1063/1.527909Search in Google Scholar
22 V.A. Brazhnyi, V.V. Konotop, Phys. Rev. E 72, 026616 (2005)10.1103/PhysRevA.72.033615Search in Google Scholar
23 J.L. Bona, S. Vento, F.B. Weissler, Discret. Contin. Dyn. S. 33, 4811 (2013)10.3934/dcds.2013.33.4811Search in Google Scholar
24 J.R. Yang, J.J. Mao, Commun. Theor. Phys. 49, 22 (2008)10.1088/0253-6102/49/1/04Search in Google Scholar
25 R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)10.1103/PhysRevLett.27.1192Search in Google Scholar
26 L.C. Vega, A. Restuccia, A. Sotomayor, J. Phys. Conf. Series 490, 012024 (2014)10.1088/1742-6596/490/1/012024Search in Google Scholar
27 L.C. Vega, A. Restuccia, A. Sotomayor, arXiv:1407.7743 v3 [mathph]Search in Google Scholar
28 E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley and Sons, New York, 1974)Search in Google Scholar
29 J. Shamanna, B. Talukdar, U. Das, Phys. Lett. A 305, 93 (2002)10.1016/S0375-9601(02)01355-5Search in Google Scholar
30 P.J. Olver, J. Math. Phys. 27, 2495 (1986)10.1063/1.527315Search in Google Scholar
31 B. Talukdar, U. Das, Higher-order Systems in Classical Mechanics (Norasa Publishing House, New Delhi, 2008)Search in Google Scholar
32 G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo, C. Rubano, Phys. Rep. 188, 149 (1990)10.1016/0370-1573(90)90137-QSearch in Google Scholar
33 A. Choudhury, S. Ghosh, B. Talukdar, Pramana, J. Phys. 70, 657 (2008)10.1007/s12043-008-0027-ySearch in Google Scholar
34 A. Restuccia, A. Sotomayor, J. Phys. Conf. Series 563, 012028 (2014)10.1088/1742-6596/563/1/012028Search in Google Scholar
35 Sk. Golam Ali, B. Talukdar, Acta Phys. Pol. B, 38, 6 (2007)Search in Google Scholar
36 R.M. Miura, J. Math. Phys. 9, 1202 (1968)10.1063/1.1664700Search in Google Scholar
© 2016 A. Restuccia and A. Sotomayor, published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.