Lagrangians and integrability for additive fourth-order difference equations
- PDF / 801,560 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 3 Downloads / 188 Views
Lagrangians and integrability for additive fourth-order difference equations Giorgio Gubbiottia School of Mathematics and Statistics F07, The University of Sydney, Sydney, NSW 2006, Australia Received: 12 February 2020 / Accepted: 11 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of the obtained family and we put it in relation with known classifications. Finally, we discuss the continuum limits of the integrable cases.
1 Introduction Discrete equations attracted the interest of many scientists during the past decades for several reason, spanning from philosophical to practical. For instance, several modern theory of physics led to hypothesis that the nature of space-time itself at very small scales, the socalled Planck length and Planck time, is discrete. From this assumption, it follows that discrete systems are actually at the very foundation of physical sciences [24]. On the other hand, discrete systems often appear in applied sciences as tools to investigate numerically equations whose closed-form solution is not available. In particular, discrete equations are related to finite difference methods for solving ordinary and partial differential equations [37]. These considerations greatly stimulated the theoretical study of discrete systems from different points of view and perspective, see [10,25]. In this paper, we will deal fourth-order difference equations, that is, functional equations for an unknown sequence {xn } where the xn+2 element is expressible in terms of the previous xn+i , i = −2, . . . , 1. That is a fourth-order difference equation is a relation of the form: xn+2 = F (xn+1 , xn , xn−1 , xn−2 ) .
(1.1)
Such kind of functional equations are also called recurrence relations of order four. A fourthorder difference equation is called invertible if it is possible to solve Eq. (1.1) in a unique way with respect to xn−2 . (xn+2 , xn+1 , xn , xn−1 ) . xn−2 = F
(1.2)
To be specific, using the solution of the inverse problem of calculus of variations we gave in [19], we will classify the variational additive fourth-order difference equations. We recall that a difference equation of order 2k
a e-mail: [email protected] (corresponding author)
0123456789().: V,-vol
123
853
Page 2 of 30
Eur. Phys. J. Plus
(2020) 135:853
xn+k = F (xn+k−1 , xn+k−2 , . . . , xn−k ) , k ≥ 1,
(1.3)
is said to be variational if there exists a discrete Lagrangian: L n = L n (xn+k , xn+k−1 , . . . , xn )
(1.4)
whose discrete Euler–Lagrange equation. k ∂ L n−l l=0
∂ xn
(xn+k−l , xn+k−1−l , . . . , xn−l ) = 0
(1.5)
coincide with Eq. (1.3). For a complete discussion on the variational formulation of difference equation, we refer to [2,19,28,34,40,41]. In the same way, we recall that a fourth-order d
Data Loading...
