An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.
We introduce the multiparameter universal characters of B-type (B-type universal character), which contain special factorial Schur Q-functions, classical Schur Q-functions, and classical B-type universal characters. Then, we prove that multiparameter B-type universal characters are solutions of the universal character hierarchy of B-type.
A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.
Введены ряды двухкомпонентных симметрических функций, которые применяются к исследованию иерархий Кадомцева-Петвиашвили и Кадомцева-Петвиашвили типа B. С использованием некоторых результатов классической теории симметрических функций показано, что уравнения Плюккера, получающиеся из билинейных тождеств для этих двух иерархий, выражаются через составные функции Шура. Дано комбинаторное доказательство того факта, что двухкомпонентные полиномы Шура и двухкомпонентные Q-полиномы Шура являются решениями связанных иерархий Кадомцева-Петвиашвили и Кадомцева-Петвиашвили типа B соответственно.
We define a Frobenius-coupled integrable lattice hierarchy including its Lax pair.Then the bi-Hamiltonian structures of this hierarchy are constructed, which show that this hierarchy possesses Liouville integrability. We also obtained N-fold Darboux transformation of the Frobenius-coupled integrable lattice hierarchy. Finally, the explicit solutions after onefold Darboux transformation of this hierarchy are investigated when we choose appropriate seed solutions.
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