## 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
Full Text: