On some integrable lattice related by the Miura-type transformation to the Itoh–Narita–Bogoyavlenskii lattice

Abstract

We show that by Miura-type transformation the Itoh-Narita-Bogoyavlenskii lattice, for any n ≥ 1, is related to some differential-difference (modified) equation. We present corresponding integrable hierarchies in its explicit form. We study the elementary Darboux transformation for modified equations.

On some classes of discrete polynomials and ordinary difference equations

“…One can also consider lreduction of polynomials (18) and (19) defined by the condition x r j ≡ 0, for r ≥ l + 1. One sees that in the degenerate case n = 0 and h = 1 (28) and (29) are nothing but elementary and complete symmetric polynomials, respectively [20]. We could expect that polynomials given by (7) and (14) keep some nice properties of symmetric ones.…”

“…It is worth mentioning that 1-reduced discrete polynomials T k s given by (28) in the particular case h = 1 and n = 1 first appeared in [29], where they were used for description of the integrals of the mKdV and sine-Gordon discrete equations. In this connection see also [12], [29] and [30].…”

Integrable Möbius-invariant evolutionary lattices of second order

“…Theorem 3. Any integrable lattice (6) belongs to one of the following types (types I, II correspond to the case l = 0 and the rest ones to the case l = 1):…”

“…. , u m ), u j = u(t, n + j) (1) include, first of all, the Bogoyavlensky lattices [1,2,3,4] and their modifications related by Miura type substitutions [5,6,7]. More general families of the lattices were considered in [8,9,10,11], relations with other discrete models were studied in [12,13,5,14,15].…”

Generalized Volterra lattices: Binary Darboux transformations and self-consistent sources

“…Here k > 0 and l are integers and we recover (1.1) for k = 1 and l = 2. The family of equations (1.2) has been explored, mostly only in the scalar case, in [7][8][9][10][11][12][13][14][15][16][17][18][19][20], for example. For k = 1 and l = −1, (1.2) yields the modified Volterra lattice equation…”