Narrow Results

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

A new higher-dimensional loop algebra is given for which a Lax isospectral problem is set up whose compatibility condition gives rise to a Liouville integrable soliton hierarchy along with eight-component potential functions. Specially, the hierarchy of evolution equations has a tri-Hamiltonian structure obtained by the trace identity.

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

Based on the Tu scheme [G.-Z. Tu, J. Math. Phys.30 (1989) 330], we construct a counterpart of the Boiti–Pempinelli–Tu soliton hierarchy from a matrix spectral problem associated with the Lie algebra $\mathrm{\text{so}}(3,ℝ)$, and formulate Hamiltonian structures for the resulting soliton equations by means of the trace identity. We then show that the newly presented equations possess infinitely many commuting symmetries and conservation laws. Finally, we derive the well-known combined KdV-mKdV equation from the new hierarchy.

In this paper, we discuss some problems in PI-theory. Some developments of this theory are presented.