An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.

https://shell.cas.usf.edu/~wma3/MaF-CSF1996.pdf

Integrable theory of the perturbation equations

The trace variational identity is generalized to zero curvature equations associated with non-semi-simple Lie algebras or, equivalently, Lie algebras possessing degenerate Killing forms. An application of the resulting generalized variational identity to a class of semi-direct sums of Lie algebras in the AKNS case furnishes Hamiltonian and quasi-Hamiltonian structures of the associated integrable couplings. Three examples of integrable couplings for the AKNS hierarchy are presented: one Hamiltonian and two quasi-Hamiltonian.

https://shell.cas.usf.edu/~wma3/MaC-JPA2006.pdf

Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras

An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows (λt=λl, l⩾0) from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.

https://shell.cas.usf.edu/~wma3/MaF-JMP1999.pdf

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

https://shell.cas.usf.edu/~wma3/MaC-AMC2009.pdf

Direct search for exact solutions to the nonlinear Schrödinger equation

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.

http://iopscience.iop.org/article/10.1088/0305-4470/35/33/316/pdf

Hamiltonian structure of discrete soliton systems

On the basis of Tu's work (1951), the author establishes a more general skeleton on K symmetries and tau symmetries of evolution equations and their Lie algebraic structures and discuss carefully the corresponding symmetries and Lie algebras for KdV and Jaulent-Miodek hierarchies.

http://iopscience.iop.org/article/10.1088/0305-4470/23/13/011/pdf

Wen-Xiu Ma

Papers

K symmetries and tau symmetries of evolution equations and their Lie algebras