Hereditary Symmetry Research Articles

Symmetries and their lie algebra of a variable coefficient Korteweg-de Vries hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painleve integrable Korteweg-de Vries (KdV for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries (vcKdV for short) hierarchy.

Lie algebras and Hamiltonian structures of multi-component Ablowitz–Kaup–Newell–Segur hierarchy

Isospectral and non-isospectral hierarchies of multi-component Ablowitz–Kaup–Newell–Segur (AKNS) are obtained from a matrix spectral problem, then by means of the zero curvature representations of the isospectral flows {Km} and non-isospectral flows {σn}, we construct the symmetries and their algebraic structures for isospectral multi-component AKNS hierarchies, demonstrate the recursive operator L is a strong and hereditary symmetry for the isospectral hierarchy. We also derive that there are implectic operator θ and symplectic operator J such that L = θJ, and discuss the multi-Hamiltonian structures and the Liouville integrability of the isospectral hierarchies.

The Second mKdV Equation and Its Hereditary Symmetryand Hamiltonian Structure

The isospectral problem of the second mKdV equation is found outfirstly. It follows that the strong hereditary symmetry and theHamiltonian structure of the second mKdV equation are presented.

A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

A subalgebra of loop algebra A 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given.

On the Structure of Solutions to the Periodic Hunter--Saxton Equation

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

A unified expressing model of the AKNS hierarchy and the KN hierarchy, as well as its integrable coupling system

A new subalgebra of loop algebra A ̃ 1 is first constructed. Then a new Lax pair is presented, whose compatibility gives rise to a new Liouville integrable system(called a major result), possessing bi-Hamiltonian structures. It is remarkable that two symplectic operators obtained in this paper are directly constructed in terms of the recurrence relations. As reduction cases of the new integrable system obtained, the famous AKNS hierarchy and the KN hierarchy are obtained, respectively. Second, we prove a conjugate operator of a recurrence operator is a hereditary symmetry. Finally, we construct a high dimension loop algebra G to obtain an integrable coupling system of the major result by making use of Tu scheme. In addition, we find the major result obtained is a unified expressing integrable model of both the AKNS and KN hierarchies, of course, we may also regard the major result as an expanding integrable model of the AKNS and KN hierarchies. Thus, we succeed to find an example of expanding integrable models being Liouville integrable.

Hamiltonian structure of discrete soliton systems

We describe an approach for investigating the Hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem. By using the so-called implicit representations of the isospectral flows, we demonstrate the existence of the recursion operator L, which is a strong and hereditary symmetry of the flows. It is then proven that every equation in the isospectral hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation (ut = K(0)) in the hierarchy satisfies some simple condition. We obtain related properties, such as the implectic-symplectic factorization of L, the Liouville complete integrability and the multi-Hamiltonian structures of the isospectral hierarchy. Four examples are given.

Two Coupled Integrable Hierarchies Possessing Hereditary Symmetry

Two coupled integrable hierarchies associated with a proper isospectral problem are obtained. These hierarchies possess hereditary symmetry structure but each hierarchy is not Hamiltonian.

A coupled AKNS–Kaup–Newell soliton hierarchy

A coupled AKNS–Kaup–Newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from Hamiltonian pairs. Zero curvature representations and tri-Hamiltonian structures are established for all coupled AKNS–Kaup–Newell systems in the hierarchy. Therefore all systems have infinitely many commuting symmetries and conservation laws. Two reductions of the systems lead to the AKNS hierarchy and the Kaup–Newell hierarchy, and thus those two soliton hierarchies also possess tri-Hamiltonian structures.

Extension of hereditary symmetry operators

Two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commutative symmetries may be generated. Some concrete structures of hereditary symmetry operators are carefully analyzed on the base of the resulting general conditions and several corresponding nonlinear systems are explicitly given out as illustrative examples.

Higher-Dimensional Integrable Models with a Common Recursion Operator**The project supported by National Natural Science Foundation of China and Natural Science Foundation of Zhejiang Province of China

Starting from any one of hereditary symmetries, we can construct a type of integrable models with arbitrary dimensions. The models with different dimensions obtained from a same hereditary symmetry possess a common recursion operator. The symmetry structures of the models are studied in their potential forms. Using the formal series symmetry approach, we can get various sets of formal series symmetries with some arbitrary functions. Generally, these sets of series symmetries are not truncated for arbitrary functions. The series symmetries will all be truncated if the arbitrary functions are fixed as polynomials. Some sets of nontruncated symmetries constitute generalized Virasoro algebras. The more details about the symmetries and algebras are discussed for a concrete -dimensional KdV equation.

A formula for obtaining new hereditary symmetries and new integrable equations

In this paper we give a simple formula to obtain hereditary symmetries related to the isospectral eigenvalue problem of integrable equations. Using this formula, (i) we prove that eigenfunctions of hereditary strong symmetry are symmetries for the whole hierarchy, which improves the result of Fokas and Anderson [J. Math. Phys. 23, 1066 (1982)]; (ii) we find new integrable equations; and (iii) we give strong symmetries of these new integrable equations and various equations for eigenfunctions of the isospectral eigenvalue problems.

On the multi-component extensions of the Burgers equation

We construct a multi-component Burgers equation by means of a hereditary symmetry. This n-component system has n parameters.

The Lie algebraic structure of symmetries generated by hereditary symmetries

Based on the works of Li, Zhu and co-workers (1985), a general Lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries.

On the symmetries of a multicomponent water wave equation

A strong and hereditary symmetry operator for a multicomponent water wave equation is found which yields a hierarchy of classical symmetries. Furthermore it is shown that Eq. (3.1) possesses new symmetries which depend explicitly on the time-variablet and all of the symmetries for Eq. (3.1) form an infinitely dimensional Lie algebra.

Infinitely many commuting symmetries and constants of motion in involution for explicitly time-dependent evolution equations

The existing symmetry approach to exactly solvable time-evolution equations in one spatial dimension is extended to the case that the equation depends explicitly on the time and space coordinates. In this context (commuting) symmetries, constants of motion (in involution), strong and hereditary symmetries (squared-eigenfunction operators), and Hamiltonian structures are discussed. The cylindrical Korteweg–de Vries equation is used as an illustrative example.

(M2, T5) On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

(M2, T5) On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

The existence of infinitely many Lie-Bäcklund symmetries for a new derivative nonlinear schrödinger equation

The new derivative nonlinear Schrödinger equation considered by Chen et al., is shown to possess strong and hereditary symmetries, and hence infinitely many commuting Lie-Bäcklund (L-B) symmetries. Further, we derive the corresponding constants of motion, which are in involution.

On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared-eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.

Comments on the symmetry structure of bi-Hamiltonian systems

The algebraic structure of exactly solvable equations is reviewed and results are reported which 1) establish that isospectral eigenvalue problems yield hereditary symmetries for bi-Hamiltonian equations and 2) show that if both an equation and its “modified” equation have known Hamiltonian formulations then their hereditary symmetries and bi-Hamiltonian formulations are readily obtained via their Miura transformation.