The lattice Schwarzian KdV equation and its symmetries

Abstract

Abstract. In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the lSKdV equation to construct non-autonomous non-integrable generalized symmetries.

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

“…This can be done effectively only in the case of point symmetries as in the generalized case we have a nonlinear differential-difference equation for which we cannot find the general solution , but, at most, we can construct particular solutions. Equation (16) is equivalent to the request that the ε-derivative of the equation E = 0, written for u 0,0 (ε), is identically satisfied on its solutions when the ε-evolution of u 0,0 (ε) is given by equation (18). This is also equivalent to say that the flows (in the group parameter space) given by equation (18) are compatible or commute with E = 0.…”

“…Equation (16) is equivalent to the request that the ε-derivative of the equation E = 0, written for u 0,0 (ε), is identically satisfied on its solutions when the ε-evolution of u 0,0 (ε) is given by equation (18). This is also equivalent to say that the flows (in the group parameter space) given by equation (18) are compatible or commute with E = 0.…”

“…We refer to the papers [3,24,28,17,18,27] for some recents results about these equations. Our main purpose is the analysis of their transformation properties.…”

Integrable Möbius-invariant evolutionary lattices of second order

“…, 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]. The classification of integrable equations (1) at m = 1 was obtained by Yamilov [16,17,18].…”

Analytic investigation of the (2+1)-dimensional Schwarzian Korteweg–de Vries equation for traveling wave solutions

“…As a result, being of great interest in both mathematics and physics, a great deal of research work has been invested for SKdV type equations [6][7][8][9][10][11][12][13]. Our objective in this study, by using two of the most recent expansion methods, is to perform an analytic study on the system (3) in order to derive as widest families of solutions as possible.…”