Nonlinear Analysis: Theory, Methods & Applications
Variational identities and applications to Hamiltonian structures of soliton equations
Introduction
Matrix spectral problems and zero curvature equations play an important role in exploring the mathematical properties of associated soliton equations [1], [2], [3]. If Lax pairs are taken from non-semisimple Lie algebras, soliton equations come in a triangular form [4], [5], due to the fact that general Lie algebras can be decomposed into semi-direct sums of semisimple Lie algebras and solvable Lie algebras [6]. Such soliton equations are called integrable couplings [7], [8]. There are plenty of examples of both continuous and discrete integrable couplings [4], [5], [7], [8], [9], [10], [11], [12], [13], and the existing results exhibit diverse mathematical structures that soliton equations possess. The semi-direct sum decomposition of Lie algebras provides a practical way to analyze soliton equations, particularly, multi-component soliton equations [11], [12], [14], allowing for more classifications of integrable equations supplementing existing theories [15], [16], for example, classifications within the areas of symmetry reductions [17], [18] and Lax pairs [19]. In this introductory report, we would like to discuss Hamiltonian structures of soliton equations associated with general Lie algebras.
Let be a matrix loop algebra. We assume that a pair of matrix continuous spectral problems where and denote the partial derivatives with respect to and , are called a Lax pair, is a spectral parameter and is a natural number indicating the differential order, determines (see, say, [20], [21]) a continuous soliton equation through their isospectral (i.e., ) compatibility condition (i.e., continuous zero curvature equation): This means that a triple () satisfies There exist rich algebraic structures for such triples in both the isospectral case [22], [23] and the non-isospectral case [24], [25], [26].
Similarly, it is assumed that a pair of matrix discrete spectral problems where is the potential, is the shift operator , , and are a Lax pair, determines (see, say, [27], [28]) a discrete soliton equation through their isospectral (i.e., ) compatibility condition (i.e., discrete zero curvature equation): This means that a triple () satisfies where denotes the Gateaux derivative as before. Algebraic structures for such triples were systematically discussed [28] and applied to the construction of isospectral flows [28] and non-isospectral flows [29].
A continuous (or discrete) Hamiltonian equation [30] is as follows: where is a Hamiltonian operator (see, say, [20], [28], [31] for details) and is a Hamiltonian functional . A Hamiltonian equation links its conserved functionals with its symmetries [32]: Moreover, there is a Lie homomorphism between the Lie algebra of functionals and the Lie algebra of vector fields [33], [34]:
If the Lie algebra is semisimple, then Hamiltonian structures of the soliton equations (1.2), (1.6) can be generated by the so-called trace identities [20], [27]. However, if we start from non-semisimple Lie algebras, the Killing forms involved in the trace identities are degenerate. Therefore, the trace identities, unfortunately, do not work all the time. To solve this problem, we get rid of some conditions required in the trace identities and present variational identities associated with general Lie algebras.
Let us now analyze the triangular form of soliton equations associated with general Lie algebras. An arbitrary Lie algebra takes a semi-direct sum of a semisimple Lie algebra and a solvable Lie algebra : and we begin with such a Lie algebra of square matrices. The notion of semi-direct sums means that and satisfy where . It is clear that is an ideal Lie sub-algebra of . The subscript indicates a contribution to the construction of integrable couplings. Then, choose a pair of enlarged continuous matrix spectral problems where the enlarged Lax pair is given as follows: Obviously, under the soliton equation (1.2), the corresponding enlarged continuous zero curvature equation equivalently gives rise to Similarly, we can have a pair of enlarged discrete matrix spectral problems where the enlarged Lax pair is given as in (1.13). We also require that the closure property between and under the matrix multiplication where , to guarantee that the discrete zero curvature equation over semi-direct sums of Lie algebras can generate coupling systems. Now, it is easy to see that under the soliton equation (1.6), the corresponding enlarged discrete zero curvature equation equivalently gives rise to
In the systems (1.14), (1.18), the first equations exactly present the soliton (1.2), (1.6), and thus, the systems (1.14), (1.18) provide the coupling systems for the (1.2), (1.6), respectively. The detailed algebraic structures on both continuous and discrete integrable couplings can be found in [35], [36]. If the solvable Lie algebra is zero, i.e., , then integrable couplings reduce to soliton equations associated with semisimple Lie algebras. More generally, semi-direct sums of block matrix Lie algebras yield soliton equations in block form. The analysis given here shows the basic idea of generating integrable couplings by using semi-direct sums of Lie algebras, proposed in [4], [5].
Now, the basic question for us is how to construct Hamiltonian structures for integrable couplings, namely soliton equations associated with semi-direct sums of Lie algebras. A bilinear form on a vector space is said to be non-degenerate when if for all vectors , then , and if for all vectors , then . The Killing form on a Lie algebra is non-degenerate iff is semisimple, and the Killing form satisfies for all iff is solvable, where denotes the Lie bracket of and . Semi-direct sums of Lie algebras with non-zero solvable Lie algebras are non-semisimple, and thus, the Killing forms are always degenerate on semi-direct sums of Lie algebras with . This is why the trace identities (see [37], [38], [27], [21]) can not be used to establish Hamiltonian structures for integrable couplings associated with semi-direct sums of Lie algebras with .
In this report, we would like to generalize the trace identities to semi-direct sums of Lie algebras to construct Hamiltonian structures of general soliton equations. The key point is that for a bilinear form on a given Lie algebra , we get rid of the invariance property under an isomorphism of the Lie algebra , but keep the symmetric property and the invariance property under the Lie bracket where is the Lie bracket of , or the invariance property under the multiplication where is assumed to be an algebra and and are two products in that algebra.
We can have plenty of non-degenerate bilinear forms satisfying the required properties on semi-direct sums of Lie algebras. In what follows, we would like to show that there exist variational identities under non-degenerate, symmetric and invariant bilinear forms, which allow us to generate Hamiltonian structures of soliton equations associated with semi-direct sums of Lie algebras. Applications to the AKNS case and the Volterra lattice case furnishes Hamiltonian structures of the AKNS hierarchy and the Volterra lattice hierarchy and Hamiltonian structures of two hierarchies of their integrable couplings associated with semi-direct sums of Lie algebras. The results also ensures that the algorithms [4], [5] to enlarge integrable equations using semi-direct sums of Lie algebras are efficient in presenting integrable couplings possessing Hamiltonian structures. A few of concluding remarks on coupling integrable couplings and super generalizations are given in the final section.
Section snippets
Variational identities on general Lie algebras
Variational identities:
Let be a loop algebra, either semisimple or non-semisimple, and and be taken from . Then the following continuous (or discrete) variational identity holds: where is a constant, is a non-degenerate, symmetric and invariant bilinear form on , and satisfy the stationary zero curvature equation The detailed proofs of the continuous and discrete variational
Continuous Hamiltonian structures
Let us focus on the case of the AKNS hierarchy. We will show how to use the continuous trace and variational identities to construct Hamiltonian structures of the AKNS hierarchy and a hierarchy of its integrable couplings.
Discrete Hamiltonian structures
Let us now consider the case of the Volterra lattice hierarchy. We will show how to use the discrete trace and variational identities to construct Hamiltonian structures of the Volterra lattice hierarchy and a hierarchy of its integrable couplings.
Concluding remarks
We have discussed variational identities associated with general matrix spectral problems and applied them to Hamiltonian structures of soliton equations associated with semisimple Lie algebras and integrable couplings associated with semi-direct sums of Lie algebras. The required conditions for the involved bilinear forms are the non-degenerate, symmetric and invariance properties under the Lie bracket or the multiplication. Illustrative examples includes the AKNS hierarchy and the Volterra
Acknowledgments
The work was supported in part by Zhejiang Normal University, the Established Researcher Grant of the University of South Florida, the CAS faculty development grant of the University of South Florida, Chunhui Plan of the Ministry of Education of China, and Wang Kuancheng foundation.
References (52)
- et al.
Semi-direct sums of Lie algebras and continuous integrable couplings
Phys. Lett. A
(2006) - et al.
Integrable theory of the perturbation equations
Chaos Solitons Fractals
(1996) Enlarging spectral problems to construct integrable couplings of soliton equations
Phys. Lett. A
(2003)- et al.
A unified expressing model of the AKNS hierarchy and the KN hierarchy as well as its integrable coupling system
Chaos Solitons Fractals
(2004) - et al.
The multi-component coupled burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions
Phys. A
(2004) - et al.
A hierarchy of nonlinear lattice soliton equations, its integrable coupling systems and infinitely many conservation laws
Chaos Solitons Fractals
(2006) - et al.
Classical integrable finite-dimensional systems related to lie algebra
Phys. Rep. C
(1981) New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry–Dym spectral problem
Phys. A
(1998)A simple scheme for generating nonisospectral flows from zero curvature representation
Phys. Lett. A
(1993)Application of hereditary symmetries to nonlinear evolution equations
Nonlinear Anal.
(1979)
Symplectic structures, their Bäcklund transformations and hereditary symmetries
Phys. D
Nonlinearization of spectral problems for the perturbation KdV systems
Phys. A
Solitons Nonlinear Evolution Equations and Inverse Scattering
Hamiltonian Methods in the Theory of Solitons
Semidirect sums of Lie algebras and discrete integrable couplings
J Math. Phys.
Lie Algebras
Integrable couplings of soliton equations by perturbations I — A general theory and application to the KdV hierarchy
Methods Appl. Anal.
Integrable couplings of vector AKNS soliton equations
J. Math. Phys.
Subalgebras of the Lie algebra and their applications
Internat. J. Modern Phys. B
The structure of Lie algebras and the classification problem for partial differential equations
Acta Appl. Math.
Calogero–Moser systems and Hitchin systems
Commun. Math. Phys.
On Liouville integrability of zero-curvature equations and the Yang hierarchy
J. Phys. A: Math. Gen.
A new family of Liouville integrable generalized Hamilton equations and its reduction
Chinese Ann. Math. Ser. A
Cited by (76)
Riemann–Hilbert problems and soliton solutions for a generalized coupled Sasa–Satsuma equation
2023, Communications in Nonlinear Science and Numerical SimulationMulti-component generalized Gerdjikov–Ivanov integrable hierarchy and its Riemann–Hilbert problem
2022, Nonlinear Analysis: Real World ApplicationsMulti-component Gerdjikov–Ivanov system and its Riemann–Hilbert problem under zero boundary conditions
2021, Nonlinear Analysis: Real World ApplicationsLong-time asymptotics of a three-component coupled nonlinear Schrödinger system
2020, Journal of Geometry and PhysicsRiemann–Hilbert problems and soliton solutions for a multi-component cubic–quintic nonlinear Schrödinger equation
2020, Journal of Geometry and PhysicsTwo integrable couplings of a generalized D-Kaup–Newell hierarchy and their Hamiltonian and bi-Hamiltonian structures
2020, Nonlinear Analysis, Theory, Methods and Applications
- 1
On sabbatical leave of absence from University of South Florida, Tampa, FL 33620, USA.