Journal of Mathematical Physics 47 (2006) 092901
Map of Discrete System into Continuous
arXiv:0711.2612v1 [math-ph] 16 Nov 2007
Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia E-mail:
[email protected]
Abstract Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the continuous limit is discussed. The one-dimensional systems of coupled oscillators for this type of longrange interactions are considered. The discrete equations of motion are mapped into the continuum equation with the Riesz fractional derivative.
1
Introduction
Equations which involve derivatives or integrals of noninteger order [1, 2, 3, 4, 5] have found many applications in recent studies in mechanics and physics [6, 7, 8, 9, 10, 11]. Usually the fractional equations for dynamics or kinetics appear as some phenomenological models. Recently, the method to obtain fractional analogues of equations of motion was considered for sets of coupled particles with a long-range interaction [12, 13, 14]. Examples of systems with interacting oscillators, spins, or waves are used for numerous applications in physics, chemistry, biology [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Transfer from the equations of motion for discrete systems to the continuous medium equation with fractional derivatives is an approximate procedure. Different applications of the procedure have already been used to derive fractional sine-Gordon and fractional wave Hilbert equation [12, 14], to study synchronization of coupled oscillators [13], and for fractional Ginzburg-Landau equation [13]. Long-range interaction has been the subject of great interest for a long time. Thermodynamics of the model of classical spins with long-range interactions has been studied in Refs. [15, 16, 19, 17]. An infinite one-dimensional Ising model with long-range interactions was considered by Dyson [15]. The d-dimensional classical Heisenberg model
1
with long-range interaction is described in Refs. [16, 19], and their quantum generalization can be found in Ref. [17]. The long-range interactions have been widely studied in discrete systems on lattices as well as in their continuous analogues. Solitons in a one-dimensional lattice with the long-range Lennard-Jones-type interaction were considered in Ref. [27]. Kinks in the Frenkel-Kontorova model with long-range interparticle interactions were studied in Ref. [28]. The properties of time periodic spatially localized solutions (breathers) on discrete chains in the presence of algebraically decaying interactions were considered in Refs. [24, 25]. Energy and decay properties of discrete breathers in systems with long-range interactions have also been studied in the framework of the Klein-Gordon [22], and discrete nonlinear Schrodinger equations [29]. A remarkable property of the dynamics described by the equation with fractional space derivatives is that the solutions have power-like tails. Similar features were observed in the lattice models with power-like long-range interactions [23, 24, 25, 30, 31, 32, 14]. As it was shown in Ref. [13, 14], analysis of the equations with fractional derivatives can provide results for the space asymptotics of their solutions. The goal of this paper is to study a connection between the dynamics of system of particles with long-range interactions and the fractional continuous medium equations by using the transform operation. Here, we consider the one-dimensional lattice of coupled nonlinear oscillators. We make the transform to the continuous limit and derive the fractional equation which describes the dynamics of the oscillatory medium. We show how the continuous limit for the systems of oscillators with long-range interaction can be described by the corresponding fractional equation. In Sec. 2, the equations of motion for the system of oscillators with long-range interaction are considered. In Sec. 3, the transform operation that maps the discrete equations into continuous medium equation is defined. In Sec. 4, the Fourier series transform of the equations of a system with long-range interaction is realized. In Sec. 5, we consider a wide class of long-range interactions that can give the fractional equations in the continuous limit. In Sec. 6, the simple example of nearest-neighbor interaction is considered to demonstrate the application of the transform operation to the well-known case. In Sec. 7, the power-law long-range interactions with positive integer powers are considered. In Sec. 8, the power-law long-range interactions with noninteger powers and the correspondent continuous medium equations are discussed. In Sec. 9, the nonlinear long-range interactions for the discrete systems are used to derive the Burgers, Korteweg-de Vries and Boussinesq equations and their fractional generalizations in the continuous limit. In Sec. 10, the fractional equations are obtained from the dispersion law for three-dimensional discrete system. The conclusion is given in Sec. 11.
2
2
Equations of motion for interacting oscillators
Consider a one-dimensional system of interacting oscillators that are described by the equations of motion, ∂ 2 un = g Iˆn (u) + F (un ), (1) ∂t2 where un are displacements from the equilibrium. The terms F (un ) characterize an interaction of the oscillators with the external on-site force. The term Iˆn (u) is defined by +∞ X J(n, m) E(un , um ), (2) Iˆn (u) ≡ m=−∞ m6=n
and it takes into account the interaction of the oscillators in the system. Examples. 1) If J(n, m) = δn+1,m − δn,m , and E(un , um ) = um , then Iˆn (u) = un+1 − un = ∆un . 2) For J(n, m) = δn+1,m − 2δn,m + δn−1,m , and E(un , um ) = um , we get Iˆn (u) = un+1 − 2un + un−1 = ∆2 un . 3) We can consider the long-range interaction that is given by J(n) = |n|−(1+α) , where α is a positive real number. In this case, we have nonlocal coupling given by the power-law function. Constant α is a physical relevant parameter. Some integer values of α correspond to the well-known physical situations: Coulomb potential corresponds to α = 0, dipoledipole interaction corresponds to α = 2, and the limit α → ∞ is for the case of nearestneighbor interaction. For the term (2) with E(un , um ) = um , the translation invariance condition is +∞ X
J(n, m) = 0
(3)
m=−∞ m6=n
for all n. If (3) cannot be satisfied, we must define E(un , um) = un − um , and the interaction term (2) is +∞ X ˆ J(n, m) [un − um ]. (4) In (u) ≡ m=−∞ m6=n
This interaction term is translation invariant. Note that the noninvariant terms lead to the divergences in the continuous limit (see Appendix). In this paper, we consider the wide class of interactions (4) that create a possibility to present the continuous medium equations with fractional derivatives. We also discuss the term (2) with E(un , um ) = f (un ) − f (um ) as nonlinear long-range interaction. As the 3
examples, we consider f (u) = u2 and f (u) = u − gu2 that give the Burgers, Kortewegde Vries and Boussinesq equations and their fractional generalizations in the continuous limit.
3
Transform operation
In this section, we define the operation that transforms the system of equations for un (t) into continuous medium equation for u(x, t). To derive a continuous medium equation, we suppose that un (t) are Fourier coefficients of some function uˆ(k, t). We define the field uˆ(k, t) on [−K/2, K/2] as uˆ(k, t) =
+∞ X
un (t) e−ikxn = F∆ {un (t)},
(5)
n=−∞
where xn = n∆x, ∆x = 2π/K is distance between oscillators, and Z 1 +K/2 un (t) = dk uˆ(k, t) eikxn = F∆−1 {ˆ u(k, t)}. K −K/2
(6)
These equations are the basis for the Fourier transform, which is obtained by transforming from discrete variable to a continuous one in the limit ∆x → 0 (K → ∞). The Fourier transform can be derived from (5), (6) in the limit as ∆x → 0. Replace the discrete un (t) with continuous u(x, t) while letting xn = n∆x = 2πn/K → x. Then change the sum to an integral, and Eqs. (5) and (6) become Z +∞ u˜(k, t) = dx e−ikx u(x, t) = F {u(x, t)}, (7) −∞
1 u(x, t) = 2π where
Z
+∞
dk eikx u˜(k, t) = F −1 {˜ u(k, t)},
(8)
−∞
u˜(k, t) = Lˆ u(k, t),
(9)
and L denotes the passage to the limit ∆x → 0 (K → ∞). Note that u˜(k, t) is a Fourier transform of the field u(x, t), and uˆ(k, t) is a Fourier series transform of un (t). The function u˜(k, t) can be derived from uˆ(k, t) in the limit ∆x → 0. The procedure of the replacement of a discrete model by the continuous one is defined by the transform operation. Definition 1. Transform operation Tˆ is a combination Tˆ = F −1 L F∆ of the operations: 1) The Fourier series transform: F∆ :
un (t) → F∆ {un (t)} = uˆ(k, t). 4
(10)
2) The passage to the limit ∆x → 0: L:
uˆ(k, t) → L{ˆ u(k, t)} = u˜(k, t).
(11)
3) The inverse Fourier transform: F −1 :
u˜(k, t) → F −1 {˜ u(k, t)} = u(x, t).
(12)
The operation Tˆ = 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. Proposition 1. The transform operation Tˆ maps the function F (un ) into the function F (u(x, t)), i.e., Tˆ F (un (t)) = F (u(x, t)), (13) where u(x, t) = Tˆ un (t), if the function F satisfies LF (un ) = F (Lun ). Proof. The Fourier series transform leads to F∆ :
F (un ) → F∆ F (un ).
(14)
Note that F∆ F (un ) 6= F (F∆ un ) = F (ˆ u(k, t)). The passage to the limit ∆x → 0 gives L:
F∆ F (un ) → LF∆ F (un ).
(15)
Then LF∆ {F (un )} = F {LF (un)} = F {F (Lun)} = F {F (u(x, t))},
(16)
where we use LF∆ = F L. The inverse Fourier transform get F −1 :
F {F (u(x, t))} → F −1 {F {F (u(x, t))}} = F (u(x, t)).
(17)
As the result, we prove (13).
4
Equations for momentum space
Let us consider a system of infinite numbers of oscillators with interparticle interaction that is described by (4). We suppose that J(n, m) satisfies the condition J(n, m) = J(n − m),
∞ X n=1
where J(−n) = J(n).
5
|J(n)|2 < ∞,
(18)
Proposition 2. The Fourier series transform F∆ maps the equations of motion +∞ X ∂ 2 un (t) J(n, m)[un − um ] + F (un ), = g ∂t2 m=−∞
(19)
m6=n
where un is the position of the nth oscillator, and F is an external on-site force, into the equation ∂ 2 uˆ(k, t) = g[Jˆα (0) − Jˆα (k∆x)]ˆ u(k, t) + F∆ {F (un )}, (20) ∂t2 where uˆ(k, t) = F∆ {un (t)}, Jˆα (k∆x) = F∆ {J(n)}, and F∆ {F (un )} is an operator notation for the Fourier series transform of F (un ). Proof. To derive the equation for the field uˆ(k, t), we multiply Eq. (19) by exp(−ikn∆x), and summing over n from −∞ to +∞. Then +∞ X
−ikn∆x
e
n=−∞
+∞ +∞ +∞ X X X ∂2 −ikn∆x e J(n, m) [un − um ] + e−ikn∆x F (un ). un (t) = g ∂t2 n=−∞ n=−∞ m=−∞ m6=n
(21) From uˆ(k, t) =
+∞ X
e−ikn∆x un (t),
(22)
n=−∞
the left-hand side of (21) gives +∞ X
−ikn∆x ∂
e
n=−∞
+∞ ∂ 2 X −ikn∆x ∂ 2 uˆ(k, t) un (t) = e u (t) = . n ∂t2 ∂t2 n=−∞ ∂t2
2
(23)
The second term of the right-hand side of (21) is +∞ X
e−ikn∆x F (un ) = F∆ {F (un )}.
(24)
n=−∞
The first term of the right-hand side of (21) is defined by the function J(n, m). Let us introduce the notation Jˆα (k∆x) =
+∞ X
e−ikn∆x J(n).
(25)
n=−∞ n6=0
Using J(−n) = J(n), the function (25) can be presented by Jˆα (k∆x) =
+∞ X n=1
+∞ X J(n) e−ikn∆x + eikn∆x = 2 J(n) cos (k∆x) . n=1
6
(26)
From (26) it follows that Jˆα (k∆x + 2πm) = Jˆα (k∆x),
(27)
where m is an integer. The interaction term in (21) is +∞ X
+∞ X
e−ikn∆x J(n, m)[un − um ] =
n=−∞ m=−∞ m6=n
=
+∞ X
+∞ X
e−ikn∆x J(n, m)un −
+∞ X
+∞ X
e−ikn∆x J(n, m)um .
(28)
n=−∞ m=−∞ m6=n
n=−∞ m=−∞ m6=n
Using (22) and (25), the first term on the r.h.s. of (28) gives +∞ X
+∞ X
−ikn∆x
e
+∞ X
J(n, m)un =
n=−∞ m=−∞ m6=n
−ikn∆x
e
un
n=−∞
+∞ X
J(m′ ) = uˆ(k, t)Jˆα (0),
(29)
m′ =−∞ m′ 6=0
where we use (18) and J(m′ + n, n) = J(m′ ), and Jˆα (0) =
+∞ X
J(n) = 2
n=−∞ n6=0
∞ X
J(n).
(30)
n=1
For the second term on the r.h.s. of (28): +∞ X
+∞ X
e−ikn∆x J(n, m)um =
n=−∞ m=−∞ m6=n
=
+∞ X
m=−∞
+∞ X
m=−∞
−ikm∆x
um e
+∞ X
um
+∞ X
e−ikn∆x J(n, m) =
n=−∞ n6=m
′ e−ikn ∆x J(n′ ) = uˆ(k, t)Jˆα (k∆x),
(31)
n′ =−∞ n′ 6=0
where we use J(m, n′ + m) = J(n′ ). As a result, Eq. (21) has the form ∂ 2 uˆ(k, t) = g[Jˆα (0) − Jˆα (k∆x)]ˆ u(k, t) + F∆ {F (un )}, ∂t2 where F∆ {F (un )} is an operator notation for the Fourier series transform of F (un ).
7
(32)
5
Alpha-interaction
Let us consider the interaction term Iˆn (u) ≡
+∞ X
J(n, m) [un − um ].
(33)
m=−∞ m6=n
where J(n, m) = J(n − m) = J(m − n),
∞ X
|J(n)|2 < ∞.
(34)
n=1
In Sec. 4., we prove that the Fourier series transform F∆ of (33) gives F∆ {Iˆn (u)} = g[Jˆα (0) − Jˆα (k∆x)]ˆ u(k, t),
(35)
where uˆ(k, t) = F∆ {un (t)}, and Jˆα (k) =
+∞ X
e−ikn J(n) = 2
n=−∞ n6=0
∞ X
J(n)cos(kn).
(36)
n=1
Definition 2. The interaction term (33) in equation of motion (1) is called α-interaction if the function (36) satisfies the condition [Jˆα (k) − Jˆα (0)] = Aα , k→0 |k|α lim
(37)
where α > 0 and 0 < |Aα | < ∞. If the function Jˆα (k) is given, then J(n) can be defined by Z 1 π ˆ Jα (k)cos(nk) dk. J(n) = π 0
(38)
The condition (37) means that Jˆα (k) − Jˆα (0) = O(|k|α), i.e. Jˆα (k) − Jˆα (0) = Aα |k|α + Rα (k),
(39)
lim Rα (k)/|k|α = 0.
(40)
for k → 0, where k→0
Examples. 1) The first example of the α-interaction is Jˆα (k) = Aα |k|α . 8
Using (38), we obtain J(n) = Aα
(−1)n π α+1 (−1)n π 1/2 − L1 (α + 3/2, 1/2, πn) , α+1 (α + 1)|n|α+1/2
(41)
where L1 (µ, ν, z) is the Lommel function [45]. 2) The second example of the α-interaction is (−1)n . n2
J(n) =
(42)
Using (Ref. [46], Sec. 5.4.2.12) ∞ X (−1)n n=1
n2
1 cos(nk) = 4
π2 k − 3 2
,
|k| ≤ π,
we get Jˆα (k) = 2
+∞ X (−1)n
n2
n=1
Then we have α = 2, and
1 π2 cos(kn) = k 2 − , 2 6
|k| ≤ π.
Jˆα (k) − Jˆα (0) = (1/2) k 2 .
(43)
(44)
The inverse Fourier transform of this expression gives the coordinate derivatives of second order 1 ∂2 . F −1 {Jˆα (k) − Jˆα (0)} = − 2 ∂x2 3) For the interaction (42), we have α = 2 and the inverse Fourier transform of (44) gives the second-order derivative. At the same time, the interaction J(n) =
1 n2
gives α = 1 and then the first-order coordinate derivative. It can be proved by using (Ref. [46], Sec. 5.4.2.12) Jˆα (k) = 2
+∞ X cos(kn) n=1
n2
1 = [3k 2 − 6πk + 2π 2 ], 6
(0 ≤ k ≤ 2π),
(45)
and Jˆα (k) − Jˆα (0) ≈ −πk for k → 0. Therefore, the inverse Fourier transform leads to the derivative of first order. 4) For noninteger and odd numbers s, J(n) = |n|−(s+1) , 9
s>0
(46)
is an α-interaction. For 0 < s < 2 (s 6= 1), we get Jˆα (k) − Jˆα (0) = 2Γ(−s) cos(πs/2) |k|s .
(47)
Jˆα (k) − Jˆα (0) = −(π/2) k.
(48)
Jˆα (k) − Jˆα (0) = −ζ(α − 1) k 2 ,
(49)
For s = 1, For noninteger s > 2, where ζ(z) is the Riemann zeta-function. The interaction (46) is considered in Sec 7. 5) The other example is (−1)n J(n) = . Γ(1 + α/2 + n)Γ(1 + α/2 − n)
(50)
Using the series (Ref. [46], Sec.5.4.8.12) ∞ X n=1
(−1)n 22β−1 cos(nk) = sin2β Γ(β + 1 + n)Γ(β + 1 − n) Γ(2β + 1)
1 k − 2 , 2 2Γ (β + 1)
(51)
where β > −1/2 and 0 < k < 2π, we get Jˆα (k) − Jˆα (0) =
2α sinα Γ(α + 1)
k . 2
(52)
In the limit k → 0, we obtain Jˆα (k) − Jˆα (0) ≈
1 |k|α . Γ(α + 1)
(53)
For noninteger α, the inverse Fourier transform of (53) gives the fractional Riesz derivative [1] of order α. 6) The α-interaction J(n) = gives ˆ = J(k)
(−1)n , a2 − n2
1 π cos(ak) − 2 . a sin(πa) a
(54)
aπ k2. 2 sin(aπ)
(55)
For k → 0, we obtain Jˆα (k) − Jˆα (0) ≈
10
The inverse Fourier transform of (55) leads to the coordinate derivative of second order. 7) For J(n) = 1/n!, we use ∞ X cos(kn) n=1
n!
= ecos k cos(sin k),
|k| < ∞.
(56)
The passage to the limit k → 0 gives Jˆα (k) − Jˆα (0) ≈ −4e k.
(57)
Then α = 1, and we get the derivative of first order. Proposition 3. The transform operation Tˆ maps the discrete equations of motion +∞ X ∂ 2 un J(n, m) [un − um ] + F (un ) =g ∂t2 m=−∞
(58)
m6=n
with noninteger α-interaction into the fractional continuous medium equation: ∂α ∂2 u(x, t) − G A u(x, t) − F (u(x, t)) = 0, α α ∂t2 ∂|x|α
(59)
where ∂ α /∂|x|α is the Riesz fractional derivative, and Gα = g|∆x|α
(60)
is a finite parameter. Proof. The Fourier series transform F∆ of (58) gives (20). We will be interested in the limit ∆x → 0. Then Eq. (20) can be written as ∂2 uˆ(k, t) − Gα Tˆα,∆ (k) uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2
(61)
where we use finite parameter (60), and Tˆα,∆ (k) = −Aα |k|α − Rα (k∆x)|∆x|−α .
(62)
Note that Rα satisfies the condition Rα (k∆x) = 0. ∆x→0 |∆x|α lim
The expression for Tˆα,∆ (k) can be considered as a Fourier transform of the operator (4). Note that g → ∞ for the limit ∆x → 0, if Gα is a finite parameter. 11
In the limit ∆x → 0, Eq. (61) gets ∂2 u˜(k, t) − Gα Tˆα (k) u˜(k, t) − F {F (u(x, t))} = 0, ∂t2
(63)
where u˜(k, t) = Lˆ u(k, t),
Tˆα (k) = LTˆα,∆ (k) = −Aα |k|α .
The inverse Fourier transform of (63) gives ∂2 u(x, t) − Gα Tα (x) u(x, t) − F (u(x, t)) = 0, ∂t2
(64)
where Tα (x) is an operator ∂α Tα (x) = F −1 {Tˆα (k)} = Aα . ∂|x|α
(65)
Here, we have used the connection between the Riesz fractional derivative and its Fourier transform [1]: ∂α |k|α ←→ − . (66) ∂|x|α The properties of the Riesz derivative can be found in Refs. [1, 2, 3, 4]. Note that the Riesz derivative could be represented as 1 ∂α α α D u(x, t) + D u(x, t) , u(x, t) = − + − ∂|x|α 2 cos(πα/2)
(67)
α where α 6= 0, 1, 3, 5..., and D± are Riemann-Liouville left and right fractional derivatives defined by [1, 2, 3, 4] Z x u(ξ, t)dξ ∂m 1 α , D+ u(x, t) = m Γ(m − α) ∂x −∞ (x − ξ)α−m+1 Z ∞ (−1)m ∂ m u(ξ, t)dξ α D− u(x, t) = , (68) Γ(m − α) ∂xm x (ξ − x)α−m+1
where m − 1 < α < m. As the result, we obtain continuous medium equations (59) from (64) and (65).
6
Simple example of nearest-neighbor interaction
In this section, we demonstrate the application of transform operation to the well-known case: J(n, m) = δn+1,m − 2δn,m + δn−1,m , (69) 12
where δn,m is the Kronecker symbol. Then the interaction term (2) has the form Iˆn (u) = (un+1 − un ) − (un − un−1 ),
(70)
and describes the nearest-neighbor interaction. As the result, equations of motion (19) have the form ∂ 2 un = g[un+1 − 2un + un−1 ] + F (un ). (71) ∂t2 The well-known result is the following. Proposition 4. The transform operation Tˆ maps the equation of motion (71) into the continuous medium equation ∂ 2 u(x, t) ∂2 = G u(x, t) + F (u), 2 ∂t2 ∂x2
(72)
G2 = g(∆x)2
(73)
where is a finite parameter. Proof. To derive the equation for the field uˆ(k, t), we multiply Eq. (71) by exp(−ikn∆x), and summing over n from −∞ to +∞. Then +∞ X
−ikn∆x
e
n=−∞
+∞ +∞ X X ∂2 −ikn∆x un (t) = g e [un+1 − 2un + un−1] + e−ikn∆x F (un ). (74) ∂t2 n=−∞ n=−∞
The first term on the r.h.s. of (74) is +∞ X
−ikn∆x
e
J(n, m)um =
=
−ikn∆x
e
+∞ X
un+1 − 2
n=−∞
=
e−ikn∆x [un+1 − 2un + un−1] =
n=−∞
n=−∞ +∞ X
+∞ X
−ikn∆x
e
n=−∞
+∞ X
−ik(m−1)∆x
e
=e
+∞ X
+∞ X
um − 2ˆ u(k, t) +
=e
e−ik(s+1)∆x us =
s=−∞ −ikm∆x
e
−ik∆x
um − 2ˆ u(k, t) + e
+∞ X
e−iks∆x us =
s=−∞
m′ =−∞ ik∆x
e−ikn∆x un−1 =
n=−∞
m′ =−∞ ik∆x
un +
+∞ X
−ik∆x
uˆ(k, t) − 2ˆ u(k, t) + e
ik∆x
uˆ(k, t) = [e
+ e−ik∆x − 2]ˆ u(k, t) =
= 2[cos (k∆x) − 1]ˆ u(k, t) = −4 sin2 (k∆x) uˆ(k, t). 13
(75)
As the result, we obtain ∂ 2 uˆ(k, t) = g Jˆα (k∆x) uˆ(k, t) + F∆ {F (un (t))}, ∂t2
(76)
Jˆα (k∆x) = −4 sin2 (k∆x) .
(77)
where For ∆x → 0, the asymptotics of the sine is ∞ X 1 (−1)m+1 2m+1 sin(z) = z ≈ z − z3, (2m + 1)! 6 m=0
and (77) can be presented by 1 Jˆα (k∆x) ≈ − (k∆x)2 + (k∆x)4 . 12
(78)
Using the finite parameter (73), the transition to the limit ∆x → 0 in Eq. (76) gives ∂ 2 u˜(k, t) = −G2 k 2 u˜(k, t) + F −1 {F (u)}, ∂t2
(79)
where we use 0 < |G2 | < ∞. As the result, the inverse Fourier transform of (79) leads to the continuous medium equation (72).
7
Integer power-law interaction
Let us consider the power-law interaction (4) with J(n) = |n|−(s+1)
(80)
with positive integer number s. Proposition 5. The power-law interaction (80) for the odd number s is α-interaction with α = 1 for s = 1, and α = 2 for s = 3, 5, 7.... For even numbers s, (80) is not αinteraction. For odd number s, the transform operation Tˆ maps the equations of motion with the interaction (80) into the continuous medium equation with derivatives of first order for s = 1, and the second order for other odd s. Proof. From (20), we get the equation for uˆ(k, t) in the form ∂ 2 uˆ(k, t) + g [Jˆα (k∆x) − Jˆα (0)] uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2 14
(81)
where Jˆα (k∆x) =
+∞ X
e−ikn∆x |n|−(1+s) .
(82)
n=−∞ n6=0
The function (82) can be presented by Jˆα (k∆x) =
+∞ +∞ X X 1 1 −ikn∆x ikn∆x cos (kn∆x) . e +e =2 1+s 1+s n n n=1 n=1
(83)
Then we can use (Ref. [46] Sec. 5.4.2.12 and Sec. 5.4.2.7) the relations ∞ X cos(nk) n=1
n2
∞ X cos(nk) n=1
n2m
=
1 3k 2 − 6πk + 2π 2 , 12
(−1)m−1 (2π)2m = B2m 2(2m)!
k 2π
(0 ≤ k ≤ 2π),
,
(0 ≤ k ≤ 2π),
(84)
(85)
where m = 1, 2, 3, ..., and B2m (z) are the Bernulli polynomials [44]. These polynomials are defined by n X Bn (k) = Cns Bs k n−s , (86) s=0
where Bs are the Bernoulli numbers from ∞
X zs z = Bs , ez − 1 s! s=0
(|z| < 2π).
(87)
For example, B2 (k) = k 2 − k + 1/6,
B4 (k) = k 4 − 2k 3 + k 2 − 1/30.
(88)
Note B2m−1 = 0 for m = 2, 3, 4... [44]. For s = 1, we have 1 Jˆα (k∆x) − Jˆα (0) = (k∆x)2 − πk∆x ≈ −πk∆x. 2
(89)
For s = 2m − 1 (m = 2, 3, ...), we have (−1)m−1 Jˆα (k) = (2π)2m B2m (2m)! Then
k 2π
(0 ≤ k ≤ 2π).
(90)
(−1)m−1 (2π)2m−2 B2m−2 (k∆x)2 . Jˆα (k∆x) − Jˆα (0) ≈ 4(2m − 2)!
(91)
15
,
For example, the interaction (80) with s = 3 gives 1 4 π2 Jˆα (k) − Jˆα (0) = − k − 4πk 3 + 4π 2 k 2 ≈ − k 2 . 48 12
(92)
For s = 0, we have (Ref. [46], Sec. 5.4.2.9) the relation ∞ X cos(nk) n=1
n
= − ln [2 sin(k/2)] .
(93)
Then, the limit ∆x → 0 gives
For even numbers s,
Jˆα (k∆x) ≈ − ln(k∆x) → ∞.
(94)
ˆ s ˆ J (k∆x) − J (0) / |k∆x| → ∞ α α
(95)
since the expression has the logarithmic poles. The transition to the limit ∆x → 0 in Eq. (81) with s = 1 gives ∂ 2 u˜(k, t) − G1 k u˜(k, t) − F {F (u(x, t))} = 0, ∂t2
(96)
where G1 = πg∆x is a finite parameter. The inverse Fourier transform of (96) leads to the continuous medium equation with coordinate derivative of first order: ∂2 ∂ u(x, t) − iG u(x, t) − F (u(x, t)) = 0. 1 ∂t2 ∂x
(97)
This equation can be considered as the nonlinear Schroedinger equation. The limit ∆x → 0 in Eq. (81) with s = 2m − 1 (m = 2, 3, ...) gives ∂ 2 u˜(k, t) + G2 k 2 u˜(k, t) − F {F (u(x, t))} = 0, ∂t2 where G2 =
(98)
(−1)m−1 (2π)2m−2 B2m−2 g(∆x)2 4(2m − 2)!
is a finite parameter. The inverse Fourier transform of (98) leads to the partial differential equation of second order: ∂2 ∂2 u(x, t) − G u(x, t) − F (u(x, t)) = 0. 2 ∂t2 ∂x2 This equation can be considered as a nonlinear wave equation.
16
(99)
8
Noninteger power-law interaction
Let us consider the power-law interaction with J(n) = |n|−(s+1) ,
(100)
where s is a positive noninteger number. Proposition 6. The power-law interaction (100) with noninteger s is α-interaction with α = s for 0 < s < 2, and α = 2 for s > 2. For 0 < s < 2 (s 6= 1), the transform operation Tˆ maps the discrete equations with the interaction (100) into the continuous medium equation with fractional Riesz derivatives of order α. For α > 2 (α 6= 3, 4, 5, ...), the continuous medium equation has the coordinate derivatives of second order. Proof. From Eq. (20), we obtain the equation for uˆ(k, t) in the form ∂ 2 uˆ(k, t) + g [Jˆα (k∆x) − Jˆα (0)] uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2 where Jˆα (k∆x) =
+∞ X
e−ikn∆x
n=−∞ n6=0
1 . |n|1+α
(101)
(102)
For fractional positive α, the function (102) can be presented by Jˆα (k∆x) =
+∞ X n=1
1 n1+α
e−ikn∆x + eikn∆x = Li1+α (eik∆x ) + Li1+α (e−ik∆x ),
(103)
where Liβ (z) is a polylogarithm function. Using the series representation of the polylogarithm [43]: z
Liβ (e ) = Γ(1 − β)(−z)
β−1
+
∞ X ζ(β − n) n=0
n!
zn ,
|z| < 2π, β 6= 1, 2, 3...,
(104)
we obtain Jˆα (k∆x) = Aα |∆x|α |k|α + 2
∞ X ζ(1 + α − 2n) n=0
(2n)!
(∆x)2n (−k 2 )n ,
where ζ(z) is the Riemann zeta-function, |k∆x| < 2π, and πα . Aα = 2 Γ(−α) cos 2 From (105), we have
Jα (0) = 2ζ(1 + α). 17
α 6= 0, 1, 2, 3..., (105)
(106)
Then Jˆα (k∆x) − Jˆα (0) = Aα |∆x|α |k|α + 2
∞ X ζ(1 + α − 2n)
(∆x)2n (−k 2 )n ,
(107)
(∆x)2n (−k 2 )n uˆ(k, t) − F∆ {F (un (t))} = 0.
(108)
(2n)!
n=1
where α 6= 0, 1, 2, 3..., and |k∆x| < 2π. Substitution of (107) into Eq. (101) gives ∂ 2 uˆ(k, t) + g Aα |∆x|α |k|α uˆ(k, t)+ ∂t2 + 2g
∞ X ζ(α + 1 − 2n)
(2n)!
n=1
We will be interested in the limit ∆x → 0. Then Eq. (108) can be written in a simple form ∂2 uˆ(k, t) + Gα Tˆα,∆ (k) uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2
α 6= 0, 1, 2, ...,
(109)
where we use the finite parameter Gα = g|∆x|min{α;2} , and Tˆα,∆ (k) =
(110)
( Aα |k|α − |∆x|2−α ζ(α − 1)k 2 , 0 < α < 2, |∆x|
α−2
α
2
Aα |k| − ζ(α − 1)k , α > 2,
(α 6= 1);
(α 6= 3, 4, ...).
(111)
The expression for Tˆα,∆ (k) can be considered as a Fourier transform of the interaction operator (2). From (110), we see that g → ∞ for the limit ∆x → 0, and finite value of Gα . Note that (111) has a scale k0 : k0 = |Aα /ζ(α − 1)|1/(2−α) |∆x|−1
(112)
such that the nontrivial expression Tˆα,∆ (k) ∼ |k|α appears only for 0 < α < 2, (α 6= 1), k ≪ k0 . The transition to the limit ∆x → 0 in Eq. (109) gives ∂2 u˜(k, t) + Gα Tˆα (k)˜ u(k, t) − F −1 {F (u(x, t))} = 0 (α 6= 0, 1, 2, ...), 2 ∂t where Tˆα (k) =
( Aα |k|α ,
0 < α < 2, 2
−ζ(α − 1) k , 2 < α, 18
α 6= 1;
α 6= 3, 4, ....
(113)
(114)
The inverse Fourier transform to (113) is ∂2 u(x, t) + Gα Tα (x) u(x, t) − F (u(x, t)) = 0 α 6= 0, 1, 2, ..., ∂t2
(115)
where Tα (x) = F −1 {Tˆα (k)} =
(
−Aα ∂ α /∂|x|α , 2
(0 < α < 2, 2
ζ(α − 1) ∂ /∂|x| , (α > 2,
α 6= 1);
α 6= 3, 4, ...).
Here, we have used the connection between the Riesz fractional derivative and its Fourier transform [1]: ∂α ∂2 α 2 |k| ←→ − , k ←→ − . (116) ∂|x|α ∂|x|2 The properties of the Riesz derivative can be found in [1, 2, 3, 4]. As the result, we obtain the continuous medium equations ∂2 ∂α u(x, t) − G A u(x, t) = F (u(x, t)) , α α ∂t2 ∂|x|α
0 < α < 2,
(α 6= 1),
(117)
∂2 ∂2 u(x, t) + G ζ(α − 1) u(x, t) = F (u(x, t)) , α ∂t2 ∂|x|2
α > 2,
(α 6= 3, 4, ...).
(118)
and
Analogously, the continuous limit for the system +∞ X ∂un |n − m|−α−1 [un − um ] + F (un ) =g ∂t m=−∞
(119)
m6=n
gives the partial differential equations ∂ ∂α u(x, t) − Gα Aα u(x, t) = F (u(x, t)) , ∂t ∂|x|α
0 < α < 2,
(α 6= 1),
(120)
(α 6= 3, 4, ...).
(121)
and ∂2 ∂ u(x, t) + Gα ζ(α − 1) u(x, t) = F (u(x, t)) , ∂t ∂|x|2
α > 2,
For F (u) = 0, Eq. (120) is the fractional kinetic equation that describes the fractional superdiffusion [33, 34, 35]. If F (u) is a sum of linear and cubic terms, then Eq. (120) has the form of the fractional Ginzburg-Landau equation [36, 37, 38, 39, 40]. A remarkable property of the dynamics described by the equation with fractional space derivatives is that the solutions have power-like tails. 19
9
Nonlinear long-range interaction
In this section, we consider the discrete equations with nonlinear long-range interaction: Iˆn (u) =
+∞ X
Jα (n, m)[f (un ) − f (um )],
(122)
m=−∞ m6=n
where f (u) is a nonlinear function of un (t), and Jα (n, m) defines the α-interaction. As the example of Jα (n, m) = Jα (n − m), we can use the functions Jα (n) =
(−1)n . Γ(1 + α/2 + n)Γ(1 + α/2 − n)
(123)
We consider the interaction with f (u) = u2 and f (u) = u − gu2 that give the Burgers, Korteweg-de Vries and Boussinesq equations in the continuous limit for α = 1, 2, 3, 4. If we use the fractional α in Eq. (123), we can obtain the fractional generalization of these equations. Proposition 7. The Fourier series transform F∆ maps the equations of motion +∞ X ∂ 2 un (t) Jα (n − m)[f (un ) − f (um)] + F (un ), = g ∂t2 m=−∞
(124)
m6=n
where F is an external on-site force, into the equation ∂ 2 uˆ(k, t) = g[Jˆα(0) − Jˆα (k∆x)]F∆ {f (un)} + F∆ {F (un )}, ∂t2
(125)
where uˆ(k, t) = F∆ {un (t)}, and Jˆα (k∆x) = F∆ {J(n)}. If Jα (n) defines the α-interaction, then the continuous limit ∆x → 0 and the inverse Fourier transform give the equation ∂ 2 u(x, t) ∂α = G A f (u(x, t)) + F (u(x, t)), α α ∂t2 ∂|x|α
(126)
where Gα = g|∆x|α is a finite parameter. Proof. The Fourier series transform of the interaction term (122) can be presented as +∞ X
n=−∞
e−ikn∆x Iˆn (u) =
+∞ X
+∞ X
e−ikn∆x J(n, m)[f (un ) − f (um )] =
n=−∞ m=−∞ m6=n
20
=
+∞ X
+∞ X
−ikn∆x
e
J(n, m)f (un ) −
+∞ X
+∞ X
e−ikn∆x J(n, m)f (um ).
(127)
n=−∞ m=−∞ m6=n
n=−∞ m=−∞ m6=n
For the first term on the r.h.s. of (127): +∞ X
+∞ X
−ikn∆x
e
J(n, m)f (un ) =
n=−∞ m=−∞ m6=n
+∞ X
−ikn∆x
e
f (un )
n=−∞
+∞ X
J(m′ ) = F∆ {f (un )} Jˆα (0),
m′ =−∞ m′ 6=0
(128) ′
′
where we use J(m + n, n) = J(m ). For the second term on the r.h.s. of (127): +∞ X
+∞ X
−ikn∆x
e
J(n, m)f (um ) =
n=−∞ m=−∞ m6=n
=
+∞ X
+∞ X
f (um )
m=−∞
−ikm∆x
f (um )e
m=−∞
+∞ X
+∞ X
e−ikn∆x J(n, m) =
n=−∞ n6=m
′ e−ikn ∆x J(n′ ) = F∆ {f (un )} Jˆα (k∆x),
(129)
n′ =−∞ n′ 6=0
where we use J(m, n′ + m) = J(n′ ). As the result, we obtain Eq. (125). For the limit ∆x → 0, Eq. (125) can be written as ∂2 uˆ(k, t) − Gα Tˆα,∆ (k) uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2
(130)
where we use finite parameter Gα = g|∆x|α , and Tˆα,∆ (k) = −Aα |k|α − Rα (k∆x)|∆x|−α .
(131)
Here, the function Rα satisfies the condition Rα (k∆x) = 0. ∆x→0 |∆x|α lim
In the limit ∆x → 0, we get ∂2 u˜(k, t) − Gα Tˆα (k) F {f (u(x, t))} − F {F (u(x, t))} = 0, ∂t2
(132)
where u˜(k, t) = Lˆ u(k, t),
Tˆα (k) = LTˆα,∆ (k) = −Aα |k|α .
The inverse Fourier transform of (132) gives ∂2 u(x, t) − Gα Tα (x) f (u(x, t)) − F (u(x, t)) = 0, ∂t2 21
(133)
where Tα (x) is an operator ∂α Tα (x) = F −1 {Tˆα (k)} = Aα . ∂|x|α
(134)
As the result, we obtain the continuous medium equation (126). Let us consider examples of quadratic-nonlinear long-range interactions. 1) The continuous limit of the lattice equations +∞ +∞ X X ∂un (t) 2 2 J2 (n, m)[un − um ], J1 (n, m)[un − um ] + g2 = g1 ∂t m=−∞ m=−∞
(135)
m6=n
m6=n
where Ji (n) (i = 1, 2) define the αi -interactions with α1 = 1 and α2 = 2, gives the Burgers equation [47] that is a nonlinear partial differential equation of second order: ∂ ∂ ∂2 u(x, t) + G1 u(x, t) u(x, t) − G2 2 u(x, t) = 0. ∂t ∂x ∂x
(136)
It is used in fluid dynamics as a simplified model for turbulence, boundary layer behavior, shock wave formation, and mass transport. If we consider J2 (n, m) with fractional α2 = α, then we get the fractional Burgers equation that is suggested in Ref. [48]. 2) The continuous limit of the system of equations +∞ +∞ X X ∂un (t) 2 2 J3 (n, m)[un − um ], J1 (n, m)[un − um ] + g3 = g1 ∂t m=−∞ m=−∞
(137)
m6=n
m6=n
where Ji (n) (i = 1, 3) define the αi -interactions with α1 = 1 and α3 = 3, gives Korteweg-de Vries equation ∂ ∂ ∂3 u(x, t) − G1 u(x, t) u(x, t) + G3 3 u(x, t) = 0. (138) ∂t ∂x ∂x First formulated as part of an analysis of shallow-water waves in canals, it has subsequently been found to be involved in a wide range of physics phenomena, especially those exhibiting shock waves, traveling waves, and solitons. Certain theoretical physics phenomena in the quantum mechanics domain are explained by means of a KdV model. It is used in fluid dynamics, aerodynamics, and continuum mechanics as a model for shock wave formation, solitons, turbulence, boundary layer behavior, and mass transport. If we use noninteger αi -interactions for Ji (n), then we get the fractional generalization of the KdV equation [49, 50].
22
3) The continuous limit of +∞ +∞ X X ∂ 2 un (t) J4 (n, m)[un − um ], J2 (n, m)[f (un ) − f (um )] + g4 = g2 ∂t2 m=−∞ m=−∞
(139)
m6=n
m6=n
where f (u) = u − gu2, and Ji (n) define the αi -interactions with α2 = 2 and α4 = 4, gives the Boussinesq equation that is a nonlinear partial differential equation of fourth order ∂2 ∂2 ∂2 2 ∂4 u(x, t) − G2 2 u(x, t) + gG2 2 u (x, t) + G4 4 u(x, t) = 0. ∂t2 ∂x ∂x ∂x
(140)
This equation was formulated as part of an analysis of long waves in shallow water. It was subsequently applied to problems in the percolation of water in porous subsurface strata. It also crops up in the analysis of many other physical processes.
10
Fractional derivatives from dispersion law
Let us consider the three-dimensional lattice that is described by the equations of motion +∞ X ∂un J(n, m) [un − um ] + F (un), =g ∂t m=−∞
(141)
m6=n
where n = (n1 , n2 , n3 ), and J(n, m) = J(n − m) = J(m − n). We suppose that un (t) are Fourier coefficients of the function uˆ(k, t): uˆ(k, t) =
+∞ X
un (t) e−ikrn = F∆ {un (t)},
(142)
n=−∞
where k = (k1 , k2 , k3 ), and rn =
3 X
ni ai .
i=1
Here, ai are translational vectors of the lattice. The continuous medium model can be derived in the limit |ai | → 0. To derive the equation for uˆ(k, t), we multiply (141) by exp(−ikrn ), and summing over n. Then, we obtain h i ∂ uˆ(k, t) = g Jˆα (0) − Jˆα (ka) uˆ(k, t) + F∆ {F (un)}, ∂t 23
(143)
where F∆ {F (un )} is an operator notation for the Fourier series transform of F (un), and Jˆα (ka) =
+∞ X
e−ikrn J(n).
(144)
n=−∞
For the three-dimensional lattice, we define the α-interaction with α = (α1 , α2 , α3 ), as an interaction that satisfies the conditions: [Jˆα (k) − Jˆα (0)] = Aαi , k→0 |ki |αi lim
(i = 1, 2, 3),
(145)
where 0 < |Aαi | < ∞. Conditions (145) mean that Jˆα (0) − Jˆα (k) =
3 X
αi
Aα1 |ki| +
i=1
3 X
Rαi (k),
(146)
i=1
where lim Rαi (k)/|ki|αi = 0.
(147)
ki →0
In the continuous limit (|ai | → 0), the α-interaction in the three-dimensional lattice gives the continuous medium equations with the derivatives ∂ α1 /∂xα1 , ∂ α2 /∂y α2 , and ∂ α3 /∂z α3 . Let us recall the appearance of the nonlinear parabolic equation [51, 52, 53, 54]. Consider wave propagation in some media and present the wave vector k in the form k = k0 + κ = k0 + κk + κ⊥ ,
(148)
where k0 is the unperturbed wave vector and subscripts (k, ⊥) are taken respectively to the direction of k0 . A symmetric dispersion law ω(k) = ω(k) = Jˆα (ka) − Jˆα (0)
(149)
for κ = |k − k0 | ≪ k0 = |k0 | can be written as 1 ω(k) = ω(|k|) = ω(k0 + [|k| − k0 ]) ≈ ω(k0) + vg (|k| − k0 ) + vg′ (|k| − k0 )2 , 2 where vg =
∂ω ∂k
and |k| = |k0 + κ| =
q
,
vg′
k=k0
=
∂2ω ∂k 2
,
(150)
(151)
k=k0
(k0 + κk )2 + κ2⊥ ≈ k0 + κk +
1 2 κ . 2k0 ⊥
(152)
Substitution of (152) into (150) gives vg′ 2 vg 2 ω(k) ≈ ω0 + vg κk + κ + κk , 2k0 ⊥ 2 24
(153)
where ω0 = ω(k0 ). Expressions (143) and (153) in the dual space (”momentum representation”) correspond to the following equation for u = u(r, t) in the coordinate space i
vg′ ∂u vg ∂u = ω0 u − ivg − ∆⊥ u − ∆k u + F (u) ∂t ∂x 2k0 2
(154)
with respect to the field u = u(t, x, y, z), where x is along k0 , and we use the operator correspondence between the dual space and usual space-time: ω(k) ←→ i (κ⊥ )2 ←→ −∆⊥ = −
∂ , ∂t
κk ←→ −i
∂2 ∂2 − , ∂y 2 ∂z 2
∂ , ∂x
(κk )2 ←→ −∆k = −
∂2 . ∂x2
(155)
Equation (154) is known as the nonlinear parabolic equation [51, 52, 53, 54]. The change of variables from (t, x, y, z) to (t, x − vg t, y, z) gives −i
vg′ vg ∂u ∆⊥ u + ∆k u − ω0 u − F (u), = ∂t 2k0 2
(156)
which is also known as the nonlinear Schr¨odinger equation. Wave propagation in oscillatory medium with long-range interaction of oscillators can be easily generalized by rewriting the dispersion law (153), in the following way: ω(k) = ω0 + vg κk + Gα (κ2⊥ )α/2 + Gβ (κ2k )β/2 ,
(1 < α, β < 2)
(157)
with new finite constants Gα , and Gβ . Using the connection between Riesz fractional derivative and its Fourier transform [1] (−∆⊥ )α/2 ←→ (κ2⊥ )α/2 ,
(−∆k )β/2 ←→ (κ2k )β/2 ,
(158)
we obtain from (157) i
∂u ∂u = −ivg + Gα (−∆⊥ )α/2 u + Gβ (−∆k )β/2 u + ω0 u + F (u), ∂t ∂x
(159)
where u = u(t, x, y, z). By changing the variables from (t, x, y, z) to (t, ξ, y, z), ξ = x−vg t, and using ∂β ∂β = , (160) (−∆k )β/2 = ∂|x|β ∂|ξ|β we obtain from (159) i
∂u = Gα (−∆⊥ )α/2 u + Gβ (−∆k )β/2 u + ω0 u + F (u), ∂t
(161)
which can be called the fractional nonlinear parabolic equation. For Gβ = 0 and F (u) = b|u|2 u, we get the fractional Ginzburg-Landau equation [36, 37, 38, 39, 40]. 25
We may consider one-dimensional simplifications of Eq. (161), i.e., i
∂u ∂β u = Gβ + ω0 u + F (u), ∂t ∂|ξ|β
(162)
where u = u(t, ξ), ξ = x − vg t, or i
∂u ∂αu = Gα + ω0 u + F (u), ∂t ∂|z|α
(163)
where u = u(t, z). Let us comment on the physical structure of (161). The first and second terms on the right-hand side are related to wave propagation in oscillatory medium with long-range interaction of oscillators. The term with F (u) on the right-hand side of Eqs. (159), and (161) correspond to wave interaction due to the nonlinear properties of the media. Thus, Eq. (161) can describe fractal processes of self-focusing and related issues.
11
Conclusion
One-dimensional system of long-range interacting oscillators serves as a model for numerous applications in physics, chemistry, biology, etc. Long-range interactions are important type of interactions for complex media. An interesting situation arises when we consider the wide class of α-interactions, where α is noninteger. A remarkable feature of these interactions is the existence of a transform operation that replaces the set of coupled individual oscillator equations by the continuous medium equation with the space derivative of noninteger order α. Such transform operation is an approximation that appears in the continuous limit. This limit allows us to consider different models in a unified way by applying tools of fractional calculus. Periodic space-localized oscillations, which arise in discrete systems, have been widely studied for short-range interactions. In the paper, the systems with long-range interactions were considered. The method to map the discrete equations of motion into the continuous fractional order differential equation is developed by the transform operation. It is known that the properties of a system with long-range interaction are very different from short-range one. The method of fractional calculus can be a new tool for the analysis of different lattice systems.
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Appendix: Divergence of non-invatiant interaction term Noninvariant interaction term leads to the infinity in the continuous medium equation. To demonstrate this property, we prove the following proposition. Proposition 8. The α-interaction term g
+∞ X
J(n, m) um ,
(164)
m=−∞ m6=n
where J(n, m) = |n − m|−(α+1) is not translation-invatiant. The transform operation Tˆ of the term (164) leads to the divergence of order |∆x|−α in the continuous medium equations. Let us prove this proposition for 0 < α < 2 (α 6= 1), and the following equations of motion +∞ X ∂ 2 un J(n, m) um − F (un ) = 0. (165) +g ∂t2 m=−∞ m6=n
Since
+∞ X
J(n, m) =
m=−∞ m6=n
+∞ X
|n − m|−(α+1) 6= 0,
m=−∞ m6=n
then the interparticle interaction term in (165) is noninvariant with respect to translations. To derive the equation for uˆ(k, t), we multiply Eq. (165) by exp(−ikn∆x), and summing over n. Then, we obtain ∂ 2 uˆ(k, t) + g Jˆα (k∆x) uˆ(k, t) − F∆ {F (un (t))} = 0, ∂t2
(166)
where Jˆα (k) is defined by (102). Using (105), we present Eq. (166) in the form ∂ 2 uˆ(k, t) + g Aα |∆x|α |k|α uˆ(k, t) + 2gζ(α + 1)ˆ u(k, t)+ 2 ∂t + 2g
∞ X ζ(α + 1 − 2n) n=1
(2n)!
(∆x)2n (−k 2 )n uˆ(k, t) − F∆ {F (un (t))} = 0,
(167)
where ζ is the Riemann zeta-function and Aα is defined by (106). For the limit ∆x → 0 and 0 < α < 2 (α 6= 1), Eq. (167) can be written as ∂2 uˆ(k, t) + Gα Aα |k|α uˆ(k, t) + 2gζ(α + 1)ˆ u(k, t) − F∆ {F (un (t))} = 0, ∂t2 31
(168)
where 0 < α < 2, α 6= 1, and Gα = g|∆x|α is a finite parameter. Note that g → ∞ for ∆x → 0, if Gα is a finite. Therefore, the transition to the limit ∆x → 0 in Eq. (168) gives the divergence term lim gζ(α + 1)ˆ u(k, t) = ζ(α + 1)Gα u˜(k, t) lim |∆x|−α → ∞.
∆x→0
∆x→0
(169)
To have the continuous model equations without divergences, we must consider [um (t) − un (t)] instead of um (t) in the interaction terms (164).
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