Lattice and q-difference Darboux-Zakharov-Manakov systems via $\bar{\partial}$-dressing method

A general scheme is proposed for introduction of lattice and q-difference variables to integrable hierarchies in frame of $\bar{\partial}$-dressing method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov systems of equations are derived. Darboux, Bäcklund and Combescure transformations and exact solutions for these systems are studied.

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