Abstract
Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.
Highlights
Consider an evolution equation ut = K (u) = K (u, ux, . . .), (1)where the field function u(x, t) is in a linear space S and K(u) = K(u, ux, . . .) is a suitable C∞ vector field
It is interesting to search for new soliton hierarchies associated with a particular Lie algebra
To obtain a soliton hierarchy, we introduce a new spectral problem p φ1 φx = U (u, λ) φ, u = ( ), q φ = (φ2), (10)
Summary
(8) = {∑Miλn−i | Mi ∈ so (3, R) , i ≥ 0, n ∈ Z} , i≥0 that is, the space of all Laurent series in λ with a finite number of nonzero terms of positive powers of λ and coefficient matrices in so(3, R) Particular examples of this matrix loop algebra so(3, R) contain the following linear combinations: λme1 + λne2 + λle. Soliton hierarchies possess many nice properties, for instance, Lax representations or zero curvature representations, Hamiltonian structures, infinitely many conservation laws, and infinitely many symmetries. It is interesting to search for new soliton hierarchies associated with a particular Lie algebra. A hierarchy of commuting bi-Hamiltonian soliton equations will be generated from associated zero curvature equations.
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