New hierarchies of integrable positive and negative lattice models and Darboux transformation

A deformed reduced semi-discrete Kaup–Newell equation, the related integrable family and Darboux transformation

“…Thus, we conclude that the transformation (15) and (22) can change the Lax pair (4) and (11) into another Lax pair with same form, and then we obtain that two Lax pairs lead to the same Eq. (3).…”

“…Many integrable differential-difference equations have been presented [2][3][4][5][6]. Much investigation on the integrable differential-difference equations has been obtained , such as the inverse scattering transformation [2], the symmetries and master symmetries [7][8][9], Hamiltonian structures [10][11][12][13][14][15][16], integrable coupling systems [13][14][15][16], nonlinearization of the Lax pairs [17,18], constructing complexiton solutions by the Casorati determinant [19], the Darboux transformations [20][21][22], and so on. For a function f n ¼ f ðn; tÞ,the shift operator E, the inverse of E and the difference operator D are defined by…”

Two 3×3 discrete matrix spectral problems and associated Liouville integrable lattice soliton equations

“…In a lot of literature, the U n (u n ,k) in Eq. (2) is usually a 2 Â 2 matrix [9][10][11][12]. In fact, the well-known method can also be used for the discrete matrix spectral problems with 3 Â 3 matrices.…”

Constructing integrable coupling systems of Hamiltonian lattice equations using semi-direct sums of Lie algebras

“…It is well known that such systems arise and play an important role in a very large number of contexts and have an extensive range of applications: mathematical physics, chaos, fractals, disordered systems, biology, optics, economics, statistical physics, numerical analysis, discrete geometry, cellular automata, quantum field theory, and so on. Much research has been systematically carried out on lattice systems [ [1][2][3][4][5][6][7][8][9][10], and the references therein].…”