From finite-gap solutions of KdV in terms of theta functions to solitons and positons

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From finite-gap solutions of KdV in terms of theta functions to solitons and positons

Pierre Gaillard

To cite this version:

Pierre Gaillard. From finite-gap solutions of KdV in terms of theta functions to solitons and positons.

2010. �hal-00466159v2�

From finite-gap solutions of KdV in terms of theta functions to solitons

and positons

+

Pierre Gaillard,

Universit´e de Bourgogne, Dijon, France : e-mail: Pierre.Gaillard@u-bourgogne.fr,

March 18, 2010

Abstract

We degenerate the finite gap solutions of the KdV equation from the general formulation in terms of abelian functions when the gaps tends to points, to recover solutions of KdV equations given a few years ago in terms of wronskians called solitons or positons. For this we establish a link between Fredholm determinants and Wronskians.

1 The KdV equation and solutions in terms of theta functions

We consider the Riemann surface Γ of the algebraic curve defined by ω² =

Q2g+1

j=1 (z−E_j), with E_j 6= E_k, j 6= k. Let D be some divisor D = ^Pg_j=1P_j, P_j ∈Γ. The so-called finite gap solution of the KdV equation

u_t= 6uux−u_xxx (1)

can be expressed in the form [7]

u(x, t) =−2 d²

dx² lnθ(xg+tv+l) +C. (2) We recall briefly, the notations. In (2),θ is the Riemann function defined by

θ(z) = ^X

k∈Z^g

exp{πi(Bk|k) + 2πi(k|z)}, (3)

constructed from the matrix of the B-periods of the surface Γ, and the vectors g, v, l are defined by

g_j = 2ic_j1, (4)

vj = 8i(c_j1 2

2g+1

X

k=1

Ek+cj2), (5)

l_j =−

g

X

k=1

Z _Pk

∞ dU_j + j 2 − 1

2

g

X

k=1

B_kj, (6)

C=

2g+1

X

k=1

E_k−2

g

X

k=1

Z

ak

zdU_k, (7)

the coefficients cjk being relating with abelian differential dUj by dU_j =

Pg

k=1c_jkz^g−k

q Q2g+1

k=1 (z−E_k)

dz, (8)

and coefficients cjk can be obtained by solving the system of linear equations

Z

ak

dU_j =δ_jk, 1≤j ≤g, 1≤k≤g.

2 Degeneracy of solutions

We suppose thatE_j are real,E_m < E_j ifm < j and try to evaluate the limits of all objects in formula (2) when E2m, E2m+1 tends to −αm, −αm = κ²_m, κ_m > 0, for 1 ≤m ≤ g, and E₁ tends to 0 (these ideas were first presented by A. Its and V.B. Matveev, exposed for example in [1]).

2.0.1 Limit of P(z) = ^Q2g+1_j=1 (z−Ej)

The limit ofP(z) =^Q2g+1_j=1 (z−Ej) is evidently equal to ˜P(z) =z^Qg_j=1(z+αj)² 2.0.2 Limit of dUm =

Pg

k=1cmkz^g−k

qQ2g+1 k=1 (z−Ek)

dz

The limit ofdU_mis equal to ˜dU_m = ^√_z^Qϕgm(z)

j=1(z+αj)dz, whereϕ_m(z) =^Pg_k=1˜c_mkz^g−k. The normalization condition takes the form in the limit

Z

ak

dU_j → 2πiϕj(−α_k) κkQ

m6=k(αm−αk) =δ_kj, (9)

which proves that the numbers−αm,m 6=k are the zeros of the polynomials ϕ_k(z), and soϕ_k(z) can be written as ϕ_k(z) = ˜c_k1^Q_m6=k(z+α_m). By (9), we get in the limit

˜

ck1 = κk

2πi. So

dU˜_k= κ_k 2πi√

z(z+αk)dz 2.0.3 Limit of v_k and g_k

By identification of the powers of z^g−2 in (10)

˜ ϕk= ˜ck1

Y

l6=k

(z+αl) =

g

X

j=1

˜

ckjz^g−j, (10) we get in the limit

˜ c_k1

g

X

l=1

−α_l+ ˜c_k2 = κ³_k 2πi. So we have the limit values of vk and gk :

˜ vk = 4

πκ³_k and

˜ g_k= 1

πκ_k. 2.0.4 Limit of Uj(P) and Bmk

For λ0 = −αm = κ²_m, I = ^R_λ⁰₀dUk → ¹₂B˜mk. The integral I can be easily evaluate along the real axis on the upper sheet of surface Γ and we get

I → i 2π ln

κ_m+κ_k κm−κk

. So we have the limit values of matrix B :

B˜_mk = i π ln

κ_m+κ_k κm−κk

. So iB_kk tends to −∞. As previously, we have

Z P

∞ dU_j → − i 2πln

κ_j−√z_P κj+√zP

. (11)

2.0.5 Limit of argument of exponential in θ(p)

Let us denote A the argument of θ(p) =^P_k∈Z^gexp{πi(Bk|k) + 2πi(k|p)}. A can be rewritten in the form

A=πi

g

X

j=1

B_jjk_j(k_j−1) + 2πi^X

j>m

B_mjk_mk_j +

g

X

j=1

πi(2p_j +B_jj)k_j. (12) Using the inequality kj(kj −1) ≥ 0 for all k ∈ Z^g and the fact that iBkk

tends to −∞, we can reduce the limit ˜θ of θ(p) to a finite sum taken over vectors k ∈Z^g such that each k_j must be equal to 0 or 1.

So, A can be rewritten in the form A=πi

g

X

j=1

Bjjkj(kj −1) + 2πi ^X

j>m

Bmjkmkj +

g

X

j=1

kj[2πi(gjx+vjt)

−πi(−j+ 2

g

X

k=1

Z Pk

∞ dUj+ ^X

m6=j

Bmj)].

In other words A =πi

g

X

j=1

Bjjkj(kj −1) + 2πi^X

j>m

Bmjkmkj+

g

X

j=1

kjQj,

with

Qj = 2πi(gjx+vjt) +βj

and

β_j =−πi(−j+ 2

g

X

k=1

Z Pk

∞ dU_j + ^X

m6=j

B_mj).

The quantity β_j has a finite limit value ˜β_j independent from x and t.

2.0.6 Limit of θ(p)

By means of the inequality Kj(Kj −1)≥0 for all K ∈Z^g and the previous relation iB_kk tends to −∞, it turns out that the limit ˜θ of θ(xg +tv+l) reduce to a finite sum taken over vectorsk ∈Z^g with the property that each kj must be equal to 0 or 1.

θ˜= ^X

k∈Z^g, kj=0or1

exp{^X

m>j

2 ln

κ_m−κ_j κ_m+κ_j

k_mk_j+i(

g

X

j=1

2κjx+ 8κ³_jt+ 2κjx_j+πj

−^X

m6=j

iln

κm+κj

κ_m−κ_j

)kj}, with

xj =xj(κj) = 1 2iκj

g

X

k=1

ln

κ_j −√z_k κj+√zk

. It can be rewritten as

θ˜= ^X

J⊂{1,...,g}

Y

j∈J

Y

j,k∈J j<k

κ_j−κ_k κj +κk

2

exp 2i^X

j∈J

(κjx+ 4κ³_jt+κ_jx_j). ^Y

j∈J k6=j

κ_j +κ_k κj −κk

.(13)

Using the equality

Y

j,k∈J j<k

κj −κk

κ_j +κ_k

2

Y

j∈J k6=j

κj+κk

κ_j −κ_k

= ^Y

j∈J k /∈J

κj +κk

κ_j −κ_k

,

it can be reduced to θ˜= ^X

J⊂{1,...,g}

Y

j∈J

Y

j∈J k /∈J

κj+κk

κ_j−κ_k

exp 2i^X

j∈J

(κjx+ 4κ³_jt+κjxj). (14)

3 1-positon of order 1

A 1-positon of order 1 is given by (see [11]), u=−2∂_x²logW(φ, ∂Kφ), where W is the classical wronskian, and φ the function

φ=φ(x, K) = sin(K(x+x₁(K) + 4K²t)) In this case

W = 1

2(sin 2Θ−2Kγ), where Θ =K(x+x₁(K) + 4K²t) and γ =∂_KΘ.

We consider here a Riemann surface of genus 2.

Using the previous section, when we take the limit as E_j tend to K_j² the function θ tends to ˜θ which takes the form

θ˜= 1− ^K_K²₂^+K_−K¹₁ exp 2i(K1 + 4K₁³t+K1x1(K1)) + ^K_K^2+K1

2−K1 exp 2i(K2+ 4K₂³t+K2x2(K2))

−exp 2i((K₁+K₂) + 4(K₁³+K₂³)t+K₁x₁(K₁) +K₂x₂(K₂)).

It can be rewritten as

θ˜= 1 + (K2+K1)^{exp 2i(K2+4K23t+K2x2(K}_K^{2))−exp 2i(K1+4K13t+K1x1(K1))}

2−K1

−exp 2i((K1 +K2) + 4(K₁³+K₂³)t+K1x1(K1) +K2x2(K2)). (15) Now, it is clear that when K₂ tends toK₁ =K, we get

θ˜= 1 + 2Kiexp(2iΘ)2i∂KΘ−exp(4iΘ). (16) It can be reduced to

θ˜=−2iexp(2iΘ)(sin 2Θ−2Kγ) =−4iexp(2iΘ)×W. (17) Since Θ is a linear function of x, we recover exactly the 1-positon

u(x, t) = −2∂_x²ln ˜θ =−2∂_x²W(φ, , ∂_Kφ) = 16K²sin Θ(sin Θ−Kγcos Θ) (sin 2Θ−2Kγ)² .

4 1-positon of order 2

We define a 1-positon of order 2 by (see [11]),

u=−2∂_x²logW(φ, ∂Kφ, ∂_K² φ), where W is the classical wronskian, and φ the function

φ=φ(x, K) = sin(K(x+x₁(K) + 4K²t)) In this case

W = 4γ²K²cos Θ−2Kγsin Θ−48K³tsin Θ−sin 2Θ sin Θ, where Θ =K(x+x₁(K) + 4K²t) and γ =∂_KΘ.

Here we consider a Riemann surface of genus 3.

We can write the function θ using the previous section. When we take the limit as E_j tend to K_j² the function θ tends to ˜θ which takes the form

θ˜= 1−

K1+K2

K1−K2

K1+K3

K1−K3

exp 2i(K1+ 4K₁³t+K₁x₁(K)) +

K2+K1

K2−K1

K2+K3

K2−K3

exp 2i(K2+ 4K₂³t+K₂x₂(K))

−^KK³3^+K−K¹1

K3+K2

K3−K2

exp 2i(K3+ 4K₃³t+K₃x₃(K))

−

K1+K3

K1−K3

K2+K3

K2−K3

exp 2i(K₁+ 4K₁³t+K₁x₁(K) +K₂ + 4K₂³t+K₂x₂(K)) +^K_K^1+K2

1−K2

K3+K2

K3−K2

exp 2i(K1+ 4K₁³t+K1x1(K) +K3+ 4K₃³t+K3x3(K))

−

K2+K1

K2−K1

K3+K1

K3−K1

exp 2i(K2+ 4K₂³t+K2x2(K) +K3 + 4K₃³t+K3x3(K))

+ exp 2i(K1+ 4K₁³t+K1x1(K) +K2+ 4K₂³t+K2x21(K) +K3+ 4K₃³t+K3x3(K))

Now, it is clear that when K2 and K3 tends to K1 =K, we get

θ˜= 1−exp(2iΘ)−2K²(−4γ²+ 48iKγt) exp(2iΘ)−4iKγexp(2iΘ) + 16K²γ²exp(4iΘ)

−exp(4iΘ) + exp(6iΘ) + 4iKγexp(4iΘ) + 2K²(−4γ²+ 48iγKt) exp(4iΘ).

It can be reduced to

θ˜=−4 exp(3iΘ)(4γ²K²cos Θ−2Kγsin Θ−48K³tsin Θ−sin 2Θ sin Θ).(18) As Θ is linear in x, we recover exactly the 1-positon of order 2

u(x, t) =−2∂_x²ln ˜θ=−2∂_x²W(φ, ∂_Kφ, ∂_K² φ).

5 From Theta to Wronskian

5.1 From Theta to Fredholm

We consider the following matrix A= (a_jk)_1≤j,k≤N defined by ajk =^Y

l6=k

Kl+Kj

K_l−K_k

exp(i(Kjx+ 8K_j³t+ 2Kjxj), (19) where xj is an arbitrary parameter. Then det(I+A) has the following form det(I+A) = ^X

J⊂{1,...,N}

Y

j∈J

Y

j∈J k /∈J

Kj +Kk

K_j−K_k

exp(2i^X

j∈J

(K_jx+ 4K_j³t+K_jx_j).(20) By the previous section,

θ˜= ^X

J⊂{1,...,g}

Y

j∈J

Y

j∈J k /∈J

κ_j+κ_k κ_j−κ_k

exp 2i^X

j∈J

(κjx+ 4κ³_jt+κ_jx_j). (21) If we compare the expression (20) to (21), we have clearly the equality

θ˜= det(I+A). (22)

It remains to find the link between this Fredholm determinant and a certain wronskian.

5.2 From Fredholm to Wronskians

In this section, we consider the following functions

φj(x) = sin(Kjx+ 4K_j³t+Kjxj), (23) where K_j are real numbers such that K₁ ≤ . . . ≤ K_N, and x_j an arbitrary constant independent of x.

We use the following notations : θ_j =K_jx+ 4K_j³t+K_jx_j.

W =W(φj, . . . , φN) is the classical WronskianW = det[(∂_x^j−1φi)_{i, j∈[1,...,N]}].

We consider the matrix A= (ajk)j, k∈[1,...,N] defined by a_jk =^Y

l6=k

K_l+K_j K_l−K_k

exp(i(Kjx+ 8K_j³t+ 2Kjx_j). (24) Then we have the following statement

Theorem 5.1

det(I +A) = 2^Ni^N(N+5)2 exp(i^PN_j=1θj)

Q_N

j=2

Qj−1

i=1(Kj −Ki) W(φ1, . . . , φN) (25) Proof : We start to remove the factor (2i)⁻¹e^iθj in each row j in the Wronskian W for 1≤≤N.

Then

W =

N

Y

j=1

e^iθj(2i)^−N ×W1, (26) with

W₁ =

(1−e^−2iθ1) iK₁(1 +e^−2iθ1) . . . (iK₁)^N−1(1 + (−1)^Ne^−2iθ1) (1−e^−2iθ2) iK2(1 +e^−2iθ2) . . . (iK2)^N−1(1 + (−1)^Ne^−2iθ2)

... ... ... ...

(1−e^−2iθN) iK_N(1 +e^−2iθN) . . . (iKN)^N−1(1 + (−1)^Ne^−2iθ2N)

The determinant W1 can be written as

W₁ = det(αjke_j+β_jk),

where αjk = (−1)^k(iKj)^k−1, ej =e^−2iθj, and βjk = (iKj)^k−1.

Denoting U = (αij)i, j∈[1,...,N], V = (βij)i, j∈[1,...,N], the determinant of U is clearly equal to

det(U) = (−1)^N(N+1)2 (i)^{N(N2−1) Y}

N≥l>m≥1

(Kl−K_m). (27) Then we use the following Lemma

Lemma 5.1 Let A= (aij)i, j∈[1,...,N], B = (bij)i, j∈[1,...,N],

(Hij)i, j∈[1,...,N], the matrix formed by replacing the jth row ofA by the ith row of B

Then

det(aijxi+bij) = det(aij)×det(δijxi+det(Hij)

det(aij)) (28) Proof : For ˜A = (˜aij)i, j∈[1,...,N] the matrix of cofactors of A, we have the well known formula A×^tA˜= detA×I.

So it is clear that det( ˜A) = (det(A))^N−1.

The general term of the product (cij)i,j∈[1,..,N]= (aijxi+bij)_{i,j∈[1,..,N]}×(˜aij)i,j∈[1,..,N]

can be written as

c_ij =^PN_s=1(a_isx_i+b_is)טa_js

=xiPN

s=1ais˜ajs+^PN_s=1bis˜ajs

=δ_ijdet(A)xi+ det(Hij).

We get

det(cij) = det(aijxi+bij)×(det(A))^N−1 = (det(A))^N ×det(δijxi+ ^det(H_det(A)^ij)).

Thus det(aijx_i+b_ij) = det(A)×det(δijx_i+^det(H_det(A)^ij)).

2

Using the previous lemma (28), we get :

det(αije_i+β_ij) = det(αij)×det(δije_i+det(Hij) det(αij)),

where (Hij)i, j∈[1,...,N] is the matrix formed by replacing the jth row of U by the ith row of V defined previously.

We compute det(Hij) and we get

det(Hij) = (−1)^{N(N+1)2 +1}(i)^{N(N2−1) Y}

N≥l>m≥1, l6=j, m6=j

(Kl−K_m)^Y

l<j

(Kk−K_l)^Y

l>j

(Kk−K_l).(29)

We can simplify the quotient q= ^det(H_det(α^ij)

ij) : q=

Q

l6=k(K_l+K_k)

Q

l6=k(K_l−K_k). (30)

So det(δjke_j +^det(H_det(α^jk)

jk)) can be expressed as det(δjkej +det(Hjk)

det(α_jk)) =

N

Y

j=1

e^−2iθjdet(δjk+^Y

l6=k

Kl+Kk

K_l−K_k

e^2iθj), and therefore

det(δjkej+ det(Hjk) det(α_jk)) =

N

Y

j=1

e^−2iθjdet(I+A).

The Wronskian can be written as W(φ₁, . . . , φ_N) =

N

Y

j=1

e^iθj(2i)^−N(−1)^N(N+1)2 (i)^N(N2−1)

N

Y

j=2 j−1

Y

i=1

(K_j−K_i)

N

Y

j=1

e^−2iθjdet(I+A) It follows that

det(I+A) = e^iP

N j=1θj

(2)^N(i)^N(N+5)2

QN j=2

Qj−1

i=1(Kj −K_i) W(φ1, . . . , φ_N) (31) 2

6 Positons of arbitrary order

Now it is clear how to get multi-positons. It is the same strategy used for the preceding examples.

If we want to get the following general positon

u=−2∂_x²lnW(φ₁, . . . , φ^(k₁¹⁾, φ₂, . . . , φ^(k₂ ²⁾, . . . , φ_l, , . . . , φ^(k_l ^l)), (32) we consider a Riemann surface of genus g =^Pl_j=1k_j +l.

As defined in the first section, we suppose thatE_j are real, E_m ≤E_j ifm < j and we evaluate the limits of all objects in formula (2).

We consider K_i >0, for 1≤m≤g such thatK_m ≤K_j if m < j.

First, we choose the following limits : E₁ tends to 0.

for 1≤i≤k₁+ 1, E_2i, E_2i+1 tends toK_i²;

for 1≤i≤k2+ 1, E2(k1+1)+2i,E2(k1+1)+2i+1 tends toK_k²_1+1+i; we continue until;

for 1≤i≤k_l+1,E_{2(k1+...kl−1+l−1)+2i},E_{2(k1+...kl−1+l−1)+2i+1}tends toK_k²_{1+...+kl−1+l−1+i}. Then from the results (20), (22) and (25), we get

θ˜= det(I+A) = ^X

J⊂{1,...,g}

Y

j∈J

(−1)^{j Y}

j∈J k /∈J

K_j +K_k Kj −Kk

exp^X

j∈J

(i(2Kjx+8K_j³t+2Kjxj).

= 2^gi^g(g+5)2 exp(i^Pg_j=1θj)

Qg j=2

Qj−1

i=1(Kj−Ki) W(ϕ1, . . . , ϕl), with

θj =Kjx+ 4K_j³+Kjxj, and

ϕj = sin(θj).

We use the following notations : for 1≤i≤k1+ 1, ϕi =ϕ(Ki);

for 1≤i≤k₂+ 1, ϕ_k1+1+i =ϕ(Kk1+1+i);

and so on until;

for 1≤i≤kl+ 1, ϕk1+...+kl−1+l−1+i =ϕ(Kk1+...+kl−1+l−1+i).

We make here the following choice for K_j, 1≤j ≤g : for 1≤i≤k₁+ 1, K_i =κ₁+ (i−1)h;

for 1≤i≤k2+ 1, Kk1+1+i =κ2+ (i−1)h);

and so on until;

for 1≤i≤k_l+ 1, K_{k1+...+kl−1+l−1+i} =κ_l+ (i−1)h.

Then we consider the the classical difference derivative operator ∆h defined by the formula :

∆_hf(x) = f(x+h)−f(x)

h .

It is easy to prove that

∆^j_hf(x) = 1 h^j

j

X

k=0

(−1)^kC_j^kf(x+ (j−k)h), (33)

and it is obvious that for any function f(x)∈ C^j, (i.e. having j continuous derivatives),

hlim→0∆^j_hf(x) = f^(j)(x). (34) We consider W :=W(ϕ₁, . . . , ϕ₂, . . . , ϕ_l).

Combining the columns, W can be written as

W =

l

Y

i=1

h^ki(ki+1)2

×W(ϕ(κ₁),∆ϕ(κ₁), . . . ,∆^k1ϕ(κ₁), ϕ(κ₂),∆ϕ(κ₂), . . . ,∆^k2ϕ(κ₂), . . . , ϕ(κ^l),∆ϕ(κ^l), . . . ,∆^klϕ(κ^l)).

Then ˜θ can be expressed as

θ˜=c×W(ϕ(κ₁),∆ϕ(κ₁), . . . ,∆^k1ϕ(κ₁), ϕ(κ₂),∆ϕ(κ₂), . . . ,∆^k2ϕ(κ₂), . . . , ϕ(κ^l),∆ϕ(κ^l), . . . ,∆^k1ϕ(κ^l)),

with

c=c1×c2. The coefficient c₁ defined by

c1 = 2^gi^g(g+5)2 exp(i

g

X

j=1

θj) is such that θj is linear in x, and so verify 2∂_x²lnc1 = 0.

The coefficient c2 is defined by

c2= 1

Ql

j=12!. . . ki!Qk₂+1 m=1

Qk₁+1

i=1 (Kk1+1+m−Ki). . .Qkl+1 m=1

Qk₁+...+kl

−1+l−1

i=1 (Kk1+kl−1+l−1+m−Ki).

By definition of the terms K_j, the coefficient c₂ tends to a finite value inde- pendent of xwhen h tends to 0, and so verify 2∂_x²lnc2 = 0.

If we denote φ_j =ϕ(κj), then whenh tends to 0

W(ϕ(κ1),∆ϕ(κ1), . . . ,∆^k1ϕ(κ1), ϕ(κ2),∆ϕ(κ2), . . . ,∆^k2ϕ(κ2), . . . , ϕ(κl),∆ϕ(κl), . . . ,∆^k1ϕ(κl)) tends to

W(φ₁, . . . , φ^(k₁ ¹⁾, φ₂, . . . , φ^(k₂²⁾. . . , φ_l, , . . . , φ^(k_l ^l)).

So we get when h tends to 0

u(x, t) = −2∂_x²ln(˜θ) =−2∂_x²ln(W) =−2∂_x²lnW(φ₁, . . . , φ^(k₁ ¹⁾, φ₂, . . . , φ^(k₂²⁾. . . . , φ_l, , . . . , φ^(k_l ^l)).

We get clearly in the limit the positon defined by (32).

7 Conclusion

• This work takes its origin in the seminal paper of A. Its and V.B.

Matveev in 1975 [7], in the study of V.B. Matveev in 1976 of abelian functions and solitons [13] and other papers like [16], [12], [14], [15], [17].

This result is based on two remarks. First, it was essential to express the degenerate θ function into an explicit Fredholm determinant; this remark was initiated by the works of Kirillov and Van Diejen [22]. The second step was to get the transformation of the Fredholm determinant into a Wronskian.

• As a byproduct, we get the result for the case of a soliton; it is more simple.

If we want to get the following soliton

u= 2∂_x²lnW(φ₁, . . . , φ_l), (35) we consider a Riemann surface of genus g =l.

We choose Ej real such thatEm < Ej if m < j.

We consider K_i >0, for 1 ≤m ≤g. We choose the following limits : E₁ tends to 0.

for 1≤i≤l, E2i, E2i+1 tends to−αi =K_i²;

Then we get clearly in the limit the soliton defined by (35).

• We can also mentioned that we can get the famous DPT potential.

Let n and d be some non negative integers, m = n +d, If we choose the coefficients Kj defined by

Kp =p, if d6= 0, 1≤p≤m−n,

Kj =n−m+ 2j, if n 6= 0, m−n+ 1 ≤j ≤m,

Takingt = 0 andxj = 0, thenu= 2∂_x²lnW(φ₁, . . . , φ_m), is exactly the DPT potential

m(m+ 1)

sin²(x) + n(n+ 1) cos²(x) .

Its a consequence of a previous work. We refer the reader to the paper [4] for the details or to an another forthcoming publication containing a different approach.

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