On the conservation laws for nonlinear partial difference equations

Abstract

In this paper, a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations (PΔEs). The identified PΔEs can be classified into QRT type, KdV type, mKdV type and sG type. The integrability of the identified partial difference equations is also discussed.

Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

“…In the verification of the examples presented in [13], we discovered an accidental misprint in (5.27), as in the original article the signs of v 1,0 and v 0,1 are inverted.…”

“…As a final example we consider, as suggested by one of the referees, the four QRT -type linearizable nonlinear partial difference equations recently presented in [13] u 1,1 = u 1,0 + u 0,1 − (1 − u 1,0 u 0,1 )u 0,0 1 − u 1,0 u 0,1 + (u 1,0 + u 0,1 )u 0,0 , u 1,1 = u 1,0 − u 0,1 + (1 + u 1,0 u 0,1 )u 0,0 1 + u 1,0 u 0,1 − (u 1,0 − u 0,1 )u 0,0 , u 1,1 = f 2 (n, m) + f 1 (n, m)u 0,0 f 1 (n, m) − f 2 (n, m)u 0,0 , f 1 (n, m) . = (1 + u 1,0 u 0,1 ) u 2 1,0 u 2 0,1 − 3u 2 1,0 − 3u 2 0,1 + 8u 1,0 u 0,1 + 1 , f 2 (n, m) .…”

Linear degree growth in lattice equations

“…A symmetry method given in [8] was used to linearize the RJGT equation. Moreover, we have also noted that there are linearizable equations with exponential growth [3,11]. This is because the transformations to bring these equations to linear equations are not rational.…”

“…Suppose that initial values for equation ( 9) are di 0 , di 0 +1 , di 0 +2 and di 0 +4 . Solving the system of linear equations for c i , we get c 4 = 0 if and only if di 0 + di 0 +3 − di 0 +1 − di 0 +2 = 0, which is equation (11) when k = 0.…”

“…equations can be brought to linear equations or systems after some transformations cf. [1,7,8,11,12]). One quick test for linearization is using the algebraic entropy test.…”