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Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with \(4\times4\) Lax pairs. (English) Zbl 1384.37093

Summary: The Darboux transformation method with \(4\times4\) spectral problem has more complexity than \(2\times2\) and \(3\times3\) spectral problems. In this paper, we start from a new discrete spectral problem with a \(4\times4\) Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53 KdV equations (Korteweg-de Vries equations)
39A14 Partial difference equations
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