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

Abstract

Abstract. In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples.

Linear degree growth in lattice equations

“…In fact it can be seen as the (1, −1) reduction u 1 = u 2 of the associated lattice equation derived from the linear equation v+av 1 +bv 2 +cv 12 = pπ, where p ∈ Z and a = −1, b = −2, c = 1 or a = −2, b = −1, c = 1 via the transformation u = tan(v). We also note that the other two QRT type equations in [7] derived from the cases where a = −3, b = 3, c = 1 and a = −2, b = 1, c = 1 by the same transformation given in Remark 4 have exponential growth. This shows that algebraic entropy is not preserved under non-rational transformation.…”

New Integrable and Linearizable Nonlinear Difference Equations

“…It is appropriate to mention that Levi and Scimiterna [9] have derived a set of necessary conditions for a nonlinear PΔΔE (2.4) to be linearizable [9] and we checked that equation (2.18) satisfies those conditions [9].…”

Darboux integrable discrete equations possessing an autonomous first-order integral

“…These conditions allow us to express any argument of the function F in terms of the others for rewriting (1), after appropriate shifts in i and j, in any of the following forms (3) u −1,−1 = F (u, u −1,0 , u 0,−1 ), (4) u 1,−1 =F (u, u 1,0 , u 0,−1 ), (5) u −1,1 =F (u, u −1,0 , u 0,1 ).…”

“…Using (1)- (5) and their consequences derived by shifts in i and j, we can express any 'mixed shift' u m,n (for both positive and negative non-zero n and m) in terms of the functions u k,0 , u 0,l . Thus, we can (and will) formulate reasonings of this article only in terms of an arbitrary solution u to (1) and its 'canonical shifts' u k,0 , u 0,l , , k, l ∈ Z, which are called dynamical variables.…”