 are used to derive the Burgers, Korteweg-de Vries, and Boussinesq equations and their fractional generalizations in the continuous limit. The conclusion is given in Sec. 7.

Transform operation
Let us consider a one-dimensional system of interacting oscillators that are described by the equations of motion, where s = 1, 2, and u n are displacements from the equilibrium. The terms F (u n ) characterize an interaction of the oscillators with the external on-site force. The termÎ n (u) is defined bŷ and it takes into account the interaction of the oscillators in the system.
For linear long-range interaction we have W (u n, u m ) = u n − u m, and the interaction term In this paper, we consider a wide class of interactions (3) that create a possibility of presenting the continuous medium equations with fractional derivatives. We also discuss the term (2) with W (u n, u m ) = f (u n ) − f (u m ) as nonlinear long-range interaction.
As the examples, we consider f (u) = u 2 and f (u) = u − gu 2 that gives the Burgers, Korteweg-de Vries and Boussinesq equations and their fractional generalizations in the continuous limit. Let us define the operation, which transforms equations (1) for u n (t) into continuous medium equation for u(x, t). We assume that u n (t) are Fourier coefficients of some functionû(k, t). Then we define the fieldû(k, t) on [−K/2, K/2] aŝ where x n = n∆x, ∆x = 2π/K is distance between oscillators, and u n (t) = 1 K These equations are the basis for the Fourier transform, which is obtained by transforming from a discrete variable to a continuous one in the limit ∆x → 0 (K → ∞). The Fourier transform can be derived from (4) and (5) in the limit as ∆x → 0. Replace the discrete u n (t) = (2π/K)u(x n, t) with continuous u(x, t) while letting x n = n∆x = 2πn/K → x. Then change the sum to an integral, and equations (4), (5) becomẽ Here,ũ (k, t) = Lû(k, t), and L denotes the passage to the limit ∆x → 0 (K → ∞). Note thatũ(k, t) is a Fourier transform of the field u(x, t), andû(k, t) is a Fourier series transform of u n (t), where we can use u n (t) = (2π/K)u(n∆x, t). The functionũ(k, t) can be derived fromû(k, t) in the limit ∆x → 0. The map of a discrete model into the continuous one can be defined by the transform operation.

Definition 1. Transform operationT is a combinationT
1) The Fourier series transform: 2) The passage to the limit ∆x → 0: 3) The inverse Fourier transform: The operationT = F −1 L F ∆ is called a transform operation, since it performs a transform of a discrete model of coupled oscillators into the continuous medium model.

From discrete to continuous equation
Let us consider the interparticle interaction that is described by (3), where J(n, m) satisfies the conditions Note that J(−n) = J(n).
Examples of functions J(n) for α-interactions can be summarized in the table of the Appendix.

Proposition 1. The transform operationT maps the discrete equations of motion
with noninteger α-interaction into the fractional continuous medium equations: where ∂ α /∂|x| α is the Riesz fractional derivative, and G α = g|∆x| α is a finite parameter.
Proof. To derive the equation for the fieldû(k, t), we multiply equation (17) by exp(−ikn∆x), and summing over n from −∞ to +∞. Then The left-hand side of (19) gives whereû(k, t) is defined by (4). The second term of the right-hand side of (19) is The first term on the right-hand side of (19) is Using (4) and (24), the first term in r.h.s. of (22) gives where we use (12) and J(m ′ + n, n) = J(m ′ ), and Note that where J(m, n ′ + m) = J(n ′ ) is used. As a result, equation (19) has the form where F ∆ {F (u n )} is an operator notation for the Fourier series transform of F (u n ).
The Fourier series transform F ∆ of (17) gives (26). We will be interested in the limit ∆x → 0. Using (15), equation (26) can be written as where we use finite parameter G α = g|∆x| α, and Note that R α satisfies the condition The expression forT α,∆ (k) can be considered as a Fourier transform of the operator (3). Note that g → ∞ for the limit ∆x → 0, if G α is a finite parameter. The passage to the limit ∆x → 0 for the third term of (27) gives Then where we use LF ∆ = F L. As a result, equation (27) in the limit ∆x → 0 obtains The inverse Fourier transform of (31) gives where T α (x) is an operator Here, we have used the connection between the Riesz fractional derivative and its Fourier transform [2]: |k| α ←→ −∂ α /∂|x| α. As a result, we obtain continuous medium equations (18).
Examples of the interaction terms J(n) that give the operators (33) in continuous medium equations are summarized in the following table.
4 Fractional three-dimensional lattice equation The generalization for the three-dimensional case can be easy realized. Let us consider the three-dimensional lattice that is described by the equations of motion where n = (n 1, n 2, n 3 ), and J(n, m) = J(n − m) = J(m − n). We suppose that u n (t) are Fourier coefficients of the functionû(k, t): where k = (k 1, k 2, k 3 ), and Here, a i are the translational vectors of the lattice. The continuous medium model can be derived in the limit |a i | → 0.
To derive the equation forû(k, t), we multiply (34) by exp(−ikr n ), and summing over n. Then, we obtain where F ∆ {F (u n )} is an operator notation for the Fourier series transform of F (u n ), and For the three-dimensional lattice, we define the α-interaction with α = (α 1, α 2, α 3 ), as an interaction that satisfies the conditions: where In the continuous limit (|a i | → 0), the α-interaction in the three-dimensional lattice gives the continuous medium equations with the derivatives ∂ α 1 /∂x α 1, ∂ α 2 /∂y α 2, and ∂ α 3 /∂z α 3 : This equation describes multifractional properties of continuous medium.

Linear power-law long-range interaction
Let us consider the chain with linear long-range interaction that is defined by the equation of motion where J(n, m) = J(|n − m|), and with positive integer number β.
The transition to the limit ∆x → 0 in Eq. (68) gives whereT The inverse Fourier transform to (71) is where As the result, we obtain the continuous medium equations (60) and (61).
For s = 1 and F (u) = 0, Eq. (60) is the fractional kinetic equation that describes the fractional superdiffusion [12,15,39]. If F (u) is a sum of linear and cubic terms, then Eq. (60) has the form of the fractional Ginzburg-Landau equation [40,41,42,43,44]. A remarkable property of the dynamics described by the equation with fractional space derivatives is that the solutions have power-like tails.

Nonlinear long-range interaction
Let us consider the discrete equations with nonlinear long-range interaction: where f (u) is a nonlinear function of u n (t), and J α (n, m) = J α (n−m) defines the α-interaction.
As an example of J α (n), we can use For α = 1, 2, 3, 4, the interactions with f (u) = u 2 and f (u) = u − gu 2 give the Burgers, Korteweg-de Vries and Boussinesq equations in the continuous limit. For fractional α in Eq.
(75), we can obtain the fractional generalization of these equations.

Proposition 4. The transform operation maps the equations of motion
where F is an external on-site force, and J α (n) defines the α-interaction, into the continuous medium equations where G α = g|∆x| α is a finite parameter.
