Classification of Three-Dimensional Integrable Scalar Discrete Equations

Abstract

We classify all integrable 3-dimensional scalar discrete quasilinear equations Q 3 = 0 on an elementary cubic cell of the lattice Z 3 . An equation Q 3 = 0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the lattice Z 4 . Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. MSC: 37K10, 52C99

On the Classification of Discrete Hirota-Type Equations in 3D

“…It was shown in [37] that the Schwarzian BKP equation is the only non-linearizable affine linear discrete equation consistent around a 4D cube. Taking log of both sides we get △ 3 ln sin(ǫ△ 12 u + △ 1 u + △ 2 u) sin(△ 1 u − △ 2 u) = △ 1 ln sin(ǫ△ 23 u + △ 2 u + △ 3 u) sin(△ 3 u − △ 2 u) .…”

“…BKP equation in Hirota form[29]:αT 1 τ T1τ + βT 2 τ T2τ + γT 3 τ T3τ + δT 123 τ T123τ = 0.Dividing by τ 2 and setting τ = e u/ǫ 2 we getαe △ 11 u + βe △ 22 u + γe △ 33 u + δe ǫ(△ 123 u−△123u)+S = 0, where S = (△ 11 u + △ 22 u + △ 33 u) + (△ 12 u + △12u) + (△ 13 u + △13u) + (△ 23 u + △23u).Its dispersionless limit is αe u 11 + βe u 22 + γe u 33 + δe u 11 +u 22 +u 33 +2u 12 +2u 13 +2u 23 = 0.Schwarzian BKP equation[34,24,37]:(T 1 u − T 2 u)(T 123 u − T 3 u) (T 2 u − T 3 u)(T 123 u − T 1 u) = (T 13 u − T 23 u)(T 12 u − u) (T 12 u − T 13 u)(T 23 u − u) .…”

Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

“…In equivalent form, the last example is known as the 2 + 1 dimensional analogue of the modified Volterra lattice [30]. Although fully discrete 3D equations are outside the scope of this paper, they can be dealt with in a similar way, suggesting an alternative to the standard approach based on the multidimensional consistency [2,3,23,29]. The only constraint required by our method is that of the non-degeneracy of the dispersionless limit (see Sect.…”

О Совместности Детерминанта На Кубических Решетках

“…[1]- [5]). Классификации дис-кретных уравнений вида (1), совместных на кубической решетке, изучались в [4] и [8] при некоторых дополнительных условиях (см.…”