Abstract
The paper aims to construct nonlocal PT-symmetric integrable equations of fourth-order, from nonlocal integrable reductions of a fourth-order integrable system associated with the Lie algebra so(3,R). The nonlocalities involved are reverse-space, reverse-time, and reverse-spacetime. All of the resulting nonlocal integrable equations possess infinitely many symmetries and conservation laws.
Highlights
Matrix spectral probems associated with matrix Lie algebras are used to construct and classify integrable equations [1,2,3], which possess infinitely many symmetries and conservation laws
Hamiltonian structures that guarantee the Liouville integrability can be established through the trace identity [4,5]
The bi-Hamiltonian structure (21) leads to infinitely many symmetries and conservation laws for the integrable system (18), which can often be generated through symbolic computation by computer algebra systems
Summary
Matrix spectral probems associated with matrix Lie algebras are used to construct and classify integrable equations [1,2,3], which possess infinitely many symmetries and conservation laws. We will use the special orthogonal Lie algebra g = so(3, R) over the field of real numbers This Lie algebra can be presented by all 3 × 3 trace-free, skew-symmetric real matrices, and a basis can be chosen as follows:. Thereafter, the subscripts denote the partial derivatives with respect to the independent variables This zero curvature equation is the compatibility condition of the following two matrix spectral problems:. If we start from non-semisimple Lie algebras, matrix spectral problems can yield so-called integrable couplings [17], and the variational identity [18] is a powerful tool for furnishing their Hamiltonian structures and hereditary recursion operators in block matrix form [19]. Mathematics 2021, 9, 2130 non-Hermitian physics has been the subject of intense study and broad interest over the past decades in both classical optics and quantum mechanics (see, e.g., [21])
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