On the Integrability of Lattice Equations with Two Continuum Limits
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ON THE INTEGRABILITY OF LATTICE EQUATIONS WITH TWO CONTINUUM LIMITS R. N. Garifullin and R. I. Yamilov
UDC 517.547
Abstract. We study a new example of a lattice equation, which is one of the key equations of a generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits, which are the well-known fifth-order partial-differential equations, namely, the Sawada–Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an L-A pair and a hierarchy of conservation laws. Keywords and phrases: differential-difference equation, integrability, Lax pair, conservation law. AMS Subject Classification: 37K10, 35G50, 39A10
1.
Introduction. We consider the differential-difference equation un+2 un (un+1 + 1)2 un−2 un (un−1 + 1)2 un,t = (un + 1) − + (2un + 1)(un+1 − un−1 ) , un+1 un−1
(1)
where n ∈ Z, un (t) is an unknown function of one discrete variable n and one continuous variable t, and the index t in the notation un,t denotes the time derivative. Equation (1) is obtained in the generalized symmetry classification of five-point differential-difference equations of the form un,t = F (un+2 , un+1 , un , un−1 , un−2 ),
(2)
performed in [8–10]. Equation (1) coincides with Eq. (E16) in [9]; it was obtained earlier in [2]. Equations of the form (2) play an important role in the study of four-point discrete equations on square lattices, which are very relevant today (see e.g., [1, 5, 6, 16]). At the present time, there is very little information on Eq. (1). It was proved in [9] that Eq. (1) possesses a nine-point generalized symmetry of the form un,θ = G(un+4 , un+3 , . . . , un−4 ). As for relations to other known integrable equations of the form (2), nothing useful from the viewpoint of constructing solutions is known (see details in the next section). However, this equation occupies a special place in the classification (see [8–10]). In particular, it possesses a remarkable property discovered in [7]: this equation possesses two different continuum limits, which are the well-known Kaup–Kupershmidt and Sawada–Kotera equations. For this reason, Eq. (1) deserves a more detailed study. In Sec. 2, we discuss the known properties of Eq. (1). In order to justify the integrability of (1), we construct an L-A pair in Sec. 3 and show that it provides an infinite hierarchy of conservation laws in Sec. 4. 2. Special place of Eq. (1) in the classification of [8–10]. In two lists of integrable equations of the form (2) presented in [9, 10], the following four equations occupy a special place: they are Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018. c 2021 Springer Science+Business Media, LLC 1072–3374/21/2522–0283
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Eq. (1) and
un,t
un,t = (u2n − 1) un+2 u2n+1 − 1 − un−2 u2n−1 − 1 , un,t = u2n un+2 un+1 − un−1 un−2 − un un+1 − un−1 , = un+1 u3n un−1 un+2 un+1 − un−1 un−2 − u2n un+1 − un−1 .
(3) (4) (5)
Equations (3)–(5)
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