Exact Solutions for N-Coupled Nonlinear Schrödinger Equations With Variable Coefficients

Bo Tang; Yingzhe Fan; Jixiu Wang; Shijun Chen

doi:10.1515/zna-2016-0128

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Exact Solutions for N-Coupled Nonlinear Schrödinger Equations With Variable Coefficients

Bo Tang , Yingzhe Fan , Jixiu Wang und Shijun Chen

Veröffentlicht/Copyright: 20. Juni 2016

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Abstract

In this paper, based on similarity transformation and auxiliary equation method, we construct many exact solutions of N-coupled nonlinear Schrödinger equations with variable coefficients, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions. These solutions may give insight into many considerable physical processes.

Keywords: Auxiliary Equation Method; Exact Solutions; N-Coupled Nonlinear Schrödinger Equations with Variable Coefficients; Similarity Transformation

1 Introduction

In recent years, the N-coupled nonlinear Schrödinger (NLS) equations which are used to describe the simultaneous propagation of N nonlinear waves in a uniform medium have received considerable interest due to their numerous applications in the areas of plasma physics [1], nonlinear optics and quantum electronics [2], Bose–Einstein condensates [3], and hydrodynamics [4, 5]. For better understanding the complicated nonlinear physical phenomena, the solution is much involved. In the past, various methods have been used to handle nonlinear partial differential nonlinear equations, such as the variational iteration method [6, 7], homotopy perturbation method [8, 9], the Bäcklund transformation method [10], the subsidiary ordinary differential equation method (sub-ODE for short) [11, 12], F-expansion method [13–15], sine-cosine method [16, 17], sech-tanh method [18, 19], Exp-function method [20, 21], Jacobi elliptic function method [22–24], residual power series method (RPSM for short) [25, 26], and homotopy analysis method [27, 28].

It is well known that the nonlinear partial differential equations with variable-coefficients are more realistic in various physical situations than their constant coefficients counterparts. So the aim of this paper is to construct exact solutions of the following N-coupled NLS equations with variable coefficients using similarity reduction and auxiliary equation method:

(1)i∂Ψk∂t = − ∂2Ψk∂x2 + vk(x, t)Ψk + ∑j = 1Ngkj(t)|Ψj|2Ψk + iγ(t)Ψk, (1)

where the physical field Ψ_k≡Ψ_k(x, t)(k=1, 2, …, N), the external potential v_k(x, t)(k=1, 2, …, N) is a real-valued function of time and spatial coordinates, and the nonlinear coefficient g_kj(t)(j, k=1, 2, …, N) and gain or loss coefficient γ(t) are real valued functions of time.

The rest of this paper is organised as follows: In Section 2, we study the similarity solutions by reducing (1) to the (1+1)-dimensional standard NLS equations with constant coefficients via the similarity transformation and auxiliary equation method. In Section 3, some conclusions are given.

2 Similarity Transformation and Solutions

In general, it is difficult to seek directly analytical solutions of (1). Here, we search for a similarity transformation connecting solutions of (1) with those of the (1+1)-dimensional standard NLS equations with constant coefficients

(2)i∂Φk(η, τ)∂τ = −∂2Φk(η, τ)∂η2 + ∑j = 1NGkj|Φj(η, τ)|2Φk(η, τ), (2)

where the physical field Φ_k(η, τ)(k=1, 2, …, N) are functions of two variables η≡η(x, t) and τ≡τ(t), which are to be determined later, and G_kj(j, k=1, 2, …, N) are constants.

To do this, we write the solution of (1) as

(3)Ψk(x, t) = ρ(t)eiφk(x, t)Φk[η(x, t), τ(t)] (3)

Substituting transformation (3) into (1) and after relatively simple algebra obtain the system of partial differential equations

(4)ηt + 2φkxηx = 0, ηxx = 0, τt − ηx2 = 0, (4)

(5)ρt + ρφkxx − γ(t)ρ = 0, (5)

(6)gkj(t)ρ2 − Gkjηx2 = 0, (6)

(7)vk(x, t) + φkx2 + φkt = 0. (7)

Solving (4), we can write the similarity variables η(x, t), τ(t), and the phase φ(x, t) in the following form

(8)η(x, t) = k(t)x + c(t), (8)

(9)τ(t) = = ∫0tk2(s)ds, (9)

(10)φk(x, t) = −k˙(t)4k(t)x2 − c˙(t)2k(t)x + wk(t), (10)

where k(t), c(t), w(t) are time-dependent functions and an overdot stands for the derivative with respect to time. Now, from (5) to (7), we obtain the functions ρ(t), v_k(x, t), and g_jk(t) as follows:

(11)ρ(t) = ρ0k(t)exp[∫0tγ(s)ds], (11)

(12)gkj = k(t)Gkjρ02exp[2∫0tγ(s)ds], (12)

(13)vk(x, t) = k¨(t)k(t) − 2k˙2(t)4k2(t)x2 + c¨(t)k(t) − 2k˙(t)c˙(t)2k2(t)x − w˙k(t) − c˙2(t)4k2(t), (13)

where ρ₀ is a constant.

To obtain the solutions of (2), we make the complex transformation

(14)Φk(η, τ) = Ak(η, τ)eiBk(η, τ), (14)

where A_k(η, τ) and B_k(η, τ)(k=1, 2, …, N) are real functions of η and τ.

Substituting Φ_k(η, τ)(k=1, 2, …, N) into (2) and setting the real and imaginary parts of the resulting equations to zero lead to the following set of PDEs:

(15)∂Ak∂τ + 2∂Ak∂η∂Bk∂η + Ak∂2Bk∂η2 = 0, (15)

(16)Ak∂Bk∂τ − ∂2Ak∂η2 + Ak(∂Bk∂η)2 + ∑j = 1NGkjAj2Ak = 0. (16)

The solution of (15) and (16) is chosen in the following form:

(17)Ak(η, τ) = ak + bkF(θ) + dkF−1(θ), (17)

(18)θ = pη + q(τ), (18)

(19)Bk(η, τ) = mkη + lk(τ), (19)

where a_k, b_k, d_k(k=1, 2, …, N), p, m_k, and q(τ), h(τ) to be determined later, and F′(θ) = dFdθ satisfies

(20)F′2(θ) = h0 + h2F2(θ) + h4F4(θ). (20)

With the aid of Mathematica, substituting (17)–(19) along with (20) into (15) and (16), setting the coefficients of monomials of F(θ) and η of the resulting system numerator to zero, we obtain a set of ODEs with respect to unknowns a_k, b_k, d_k, p, m_k, and q(τ), l_k(τ).

ak(l′k(τ) + mk2) = 0, q′(τ) + 2mkp = 0,∑j = 1NGkjbkbj2 = 2p2h4bk, ∑j = 1NGkjdkdj2 = 2p2h0dk,[l′k(τ) + (mk2 − p2h2)]bk + ∑j = 1NGkj(bj2dk + 2bkbjdj) = 0,[l′k(τ) + (mk2 − p2h2)]dk + ∑j = 1NGkj(dj2bk + 2dkdjbj) = 0.

Solving the system, we get the following solution set:

Case 1:

(21)ak = 0, q(τ) = −2mpτ + ξ0, lk(τ) = p2(h2 − 2(2 ± 1)h0h4)τ − m2τ + ξk,bk = ±p2h4DkD, dk = ±ϵp2h0DkD, p = p, mk = m, (21)

where ξ₀, ξ_k, p, and m are arbitrary constants, and the parameter ϵ can have the values ϵ=0, ±1.

Case 2:

(22)ak = ak, q(τ) = −2mpτ + ζ0, lk(τ) = −m2τ + ζk, h2 = 2(2 ± 1)h0h4, bk = ±p2h4DkD, dk = ±ϵp2h0DkD,p = p, mk = m, (22)

where ak, ζ₀, ζ_k, p, and m are arbitrary constants, and the parameter ϵ can have the values ϵ=0, ±1, the expression of D as following:

D = |G11G12⋯G1NG21G22⋯G2N⋮⋮⋱⋮GN1GN2⋯GNN|

and D_k is the determinant formed by replacing the k-th column of D by the unit column vector.

Therefore, we obtain exact traveling wave solutions of (2)

(23)Φk = ±p2DkD[h4F(θ) + ϵh01F(θ)]eiBk, (23)

where Bk = mk(t)x + p2(h2 − 2(2 ± 1)h0h4)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(24)Φk = ak ± p2DkD[h4F(θ) + ϵh01F(θ)]eiBk, (24)

where Bk = mk(t)x − m2∫0tk2(s)ds + mc(t) + ζk.

Substituting (21) and (24) along with (8)–(11) into (3), we get exact traveling wave solutions of (1) in the following form:

(25)Ψk = ±ρ0k(t)p2DkD[h4F(pk(t)x + pc(t) − 2mp∫0tk2(s)ds + ξ0) + ϵh0F(pk(t)x + pc(t) − 2mp∫0tk2(s)ds + ξ0)] exp[∫0tγ(s)ds+i(φk(x, t)+Bk)], (25)

where Bk = mk(t)x + p2(h2 − 2(2 ± 1)h0h4)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(26)Ψk = ρ0k(t){ak ± p2DkD[h4F(pk(t)x + pc(t) − 2mp∫0tk2(s)ds + ζ0)+ ϵh0F(pk(t)x + pc(t) − 2mp∫0tk2(s)ds + ζ0)]}exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (26)

where Bk = mk(t)x − m2∫0tk2(s)ds + mc(t) + ζk.

Remark 2.1. As we know the more solutions of (20) we find, the more exact solutions of (1) may be obtained. However, the general solutions are difficult to be listed because of the complexity of (20). Some special solutions [29–33] are listed in Tables 1–3.

Table 1:

Solutions of (20) m (0<m<1) denotes the modulus of the Jacobi elliptic function.

h₀	h₂	h₄	F(ξ)
0	>0	<0	−h2h4sech(h2ξ)
0	>0	>0	h2h4csch(h2ξ)
h224h4	<0	>0	−h22h4tanh(−h2ξ2)
0	<0	>0	−h2h4sec(−h2ξ)
0	<0	>0	−h2h4csc(−h2ξ)
h224h4	>0	>0	h22h4tan(h2ξ2)
1	–(1 +m²)	m²	snξ, cdξ = cnξdnξ
(1–m²)	2m²–1	–m²	cnξ
m²–1	2–m²	–1	dnξ
m²	(–1+m²)	1	nsξ = 1snξ, dcξ = dnξcnξ
14	1 − 2m22	14	nsξ±csξ
1 − m24	1 + m22	1 − m24	ncξ±scξ

k1 = 1 − m2,A, B, C(ABC≠0), and D are arbitrary constants.

Table 2:

Solutions of (20) (continued) m (0<m<1) denotes the modulus of the Jacobi elliptic function.

h₀	h₂	h₄	F(ξ)
–m²	2m²–1	1–m²	ncξ
–1	2–m²	m²–1	ndξ
1	2–m²	1–m²	scξ
1	2m²–1	–m²(1–m²)	sdξ
1–m²	2–m²	1	csξ
–m²(1–m²)	2m²–1	1	dsξ
m24	m2−22	14	nsξ±dsξ
m24	m2 − 22	m24	snξ±icnξ, dn(ξ)1 − m2sn(ξ) ± cn(ξ)
14	1 − 2m22	14	msnξ±idnξ, sn(ξ)1 ± cn(ξ)
14	m2 − 22	m24	sn(ξ)1 ± dn(ξ)
m2 − 14	m2 + 12	m2 − 14	dn(ξ)1 ± msn(ξ),msd(ξ)±nd(ξ)
1 − m24	m2 + 12	1 − m24	cn(ξ)1 ± sn(ξ), nc(ξ)±sc(ξ)
14	m2 + 12	(1 − m2)24	sn(ξ)dn(ξ) ± cn(ξ)
14	m2 − 22	m44	cn(ξ)1 − m2 ± dn(ξ)
−(1 − m2)24	m2 + 12	−14	mcn(ξ)±dn(ξ)

k1 = 1 − m2,A, B, C(ABC≠0), and D are arbitrary constants.

Table 3:

Solutions of (20) (continued) m (0<m<1) denotes the modulus of the Jacobi elliptic function.

h₀	h₂	h₄	F(ξ)
14		14	dn(ξ)mcn(ξ) ± i1 − m2
	1 − 2m22		sn(ξ)1 ± cn(ξ)
	1 − 2m22		cn(ξ)1 − m2sn(ξ) ± dn(ξ)
1	2 –4m²	1	sn(ξ)dn(ξ)cn(ξ)
(m − 1)24A2	1 + m2 + 6m2	A2(m − 1)24	dn(ξ)cn(ξ)A(1 + sn(ξ))(1 + msn(ξ))
(m + 1)24A2	1 + m2 − 6m2	A2(m + 1)24	dn(ξ)cn(ξ)A(1 + sn(ξ))(1 − msn(ξ))
–2m³+m⁴+m²	6m–m²–1	−4m	mdn(ξ)cn(ξ)1 + msn2(ξ)
2m³+m⁴+m²	–6m–m²–1	4m	mdn(ξ)cn(ξ)−1 + msn2(ξ)
2+2k₁–m²	6k₁–m²+2	4k₁	m2sn(ξ)cn(ξ)k1 − dn2(ξ)
2–2k₁–m²	–6k₁–m²+2	–4k₁	−m2sn(ξ)cn(ξ)k1 + dn2(ξ)
m2 − 14(C2m2 − B2)	m2 + 12	(C2m2 − B2)(m2 − 1)4	B2 − C2B2 − C2m2 + sn(ξ)Bcn(ξ) + Cdn(ξ)
m44(C2 + B2)	m22 − 1	(C2 + B2)4	B2 + C2 − C2m2B2 + C2 + dn(ξ)Bsn(ξ) + Ccn(ξ)
2m − m2 − 1B2	2m²+2	–B²m²–B²–2B²m	msn2(ξ) − 1B(msn2(ξ) + 1)
−2m + m2 + 1B2	2m²+2	–B²m²–B²–2B²m	msn2(ξ) + 1B(msn2(ξ) − 1)
0	1	12	±2 − 2tanh2(D − ξ)tanh(D − ξ)
0	0	>0	−1h4ξ

k1 = 1 − m2,A, B, C(ABC≠0), and D are arbitrary constants.

For example, if we choose ϵ=0, we can obtain the following solutions using (25) and Table 1:

Case 2.1 Soliton and soliton-like solutions

(27)Ψk = ±ρ0(t)k(t)p−2h2DkDsech(h2θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (27)

where Bk = mk(t)x + p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(28)Ψk = ±ρ0(t)k(t)p2h2DkDcsch(h2θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (28)

where Bk = mk(t)x + p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(29)Ψk = ±ρ0(t)k(t)ph2Dk−2Dtanh(−h2θ2)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (29)

where Bk = mk(t)x − m2∫0tk2(s)ds + mc(t) + ξk.

(30)Ψk = ±ρ0(t)k(t)ph2Dk−2Dtanh(−h2θ2)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (30)

where Bk = mk(t)x + 2p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

Case 2.2 Triangular periodic solutions

(31)Ψk = ±ρ0(t)k(t)p−2h2DkDsec(−h2θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (31)

where Bk = mk(t)x + p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(32)Ψk = ±ρ0(t)k(t)p−2h2DkDcsc(−h2θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (32)

where Bk = mk(t)x + p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(33)Ψk = ±ρ0(t)k(t)ph2Dk2Dtan(h2θ2)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (33)

where Bk = mk(t)x − m2∫0tk2(s)ds + mc(t) + ξk.

(34)Ψk = ±ρ0(t)k(t)ph2Dk2Dtan(h2θ2)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (34)

where Bk = mk(t)x + 2p2h2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

Case 2.3 Jacobi elliptic function solutions and combined Jacobi elliptic function solutions

(35)Ψk = ±mpρ0(t)k(t)2DkDsn(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (35)

where Bk = mk(t)x − p2(1 + m)2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(36)Ψk = ±mpρ0(t)k(t)2DkDsn(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (36)

where Bk = mk(t)x − p2(1 + 6m + m2)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(37)Ψk = ±mpρ0(t)k(t)2DkDcd(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (37)

where Bk = mk(t)x − p2(1 + m)2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(38)Ψk = ±mpρ0(t)k(t)2DkDcd(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (38)

where Bk = mk(t)x − p2(1 + 6m + m2)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(39)Ψk = ±impρ0(t)k(t)2DkDcn(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (39)

where Bk = mk(t)x + p2[2m2 − 2(2 ± 1)mm2 − 1 − 1]∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(40)Ψk = ±ipρ0(t)k(t)2DkDdn(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (40)

where Bk = mk(t)x + p2[2 − m2 − 2(2 ± 1)1 − m2]∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(41)Ψk = ±pρ0(t)k(t)2DkDns(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (41)

where Bk = mk(t)x − p2[1 + m2 + 2(2 ± 1)m]∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(42)Ψk = ±pρ0(t)k(t)2DkDdc(θ)exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (42)

where Bk = mk(t)x − p2[1 + m2 + 2(2 ± 1)m]∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(43)Ψk = ±pρ0(t)k(t)Dk2D[ns(θ) ± cs(θ)]exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (43)

where Bk = mk(t)x − p2m2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(44)Ψk = ±pρ0(t)k(t)Dk2D[ns(θ) ± cs(θ)]exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (44)

where Bk = mk(t)x − p2(1 + m2)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(45)Ψk = ±pρ0(t)(1 − m2)k(t)Dk2D[nc(θ) ± sc(θ)]exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (45)

where Bk = mk(t)x + p2m2∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

(46)Ψk = ±pρ0(t)(1 − m2)k(t)Dk2D[nc(θ) ± sc(θ)]exp[∫0tγ(s)ds + i(φk(x, t) + Bk)], (46)

where Bk = mk(t)x + p2(2m2 − 1)∫0tk2(s)ds − m2∫0tk2(s)ds + mc(t) + ξk.

Since k(t) and c(t) are arbitrary functions, the solution (27) has abundant properties. We show some properties of the bright soliton intensity of Eq. (27) in Figure 1. And the wave propagation of the dark soliton intensity of Eq. (29) is illustrated in Figure 2.

$Figure 1: Bright soliton intensity of (27) with k(t) = c(t) = t, h2 = p = 1, Dk = −D, m = −0.15, γ(t) = −2t, ρ0(t) = et 2.$k(t)\, = \,c(t)\, = \,t,{\rm{ }}{h_2}\, = \,p\, = \,1,{\rm{ }}{D_k}\, = \, - D,{\rm{ }}m\, = \, - 0.15,{\rm{ }}\gamma (t)\, = \, - 2t,{\rm{ }}{\rho _0}(t)\, = \,{e^{{t^2}}}.$$

Figure 1:

Bright soliton intensity of (27) with k(t) = c(t) = t, h2 = p = 1, Dk = −D, m = −0.15, γ(t) = −2t, ρ0(t) = et 2.

$Figure 2: Dark soliton intensity of (29) with k(t) = c(t) = t, h2 = −4, p = 1, Dk = 2D, m = −0.15, γ(t) = −2t, ρ0(t) = 1t + 2t2et2.$k(t)\, = \,c(t)\, = \,t,{\rm{ }}{h_2}\, = \, - 4,{\rm{ }}p\, = \,1,{\rm{ }}{D_k}\, = \,2D,{\rm{ }}m\, = \, - 0.15,{\rm{ }}\gamma (t)\, = \, - 2t,{\rm{ }}{\rho _0}(t)\, = \,{1 \over {\sqrt {t\, + \,2{t^2}} }}{e^{{t^2}}}.$$

Figure 2:

Dark soliton intensity of (29) with k(t) = c(t) = t, h2 = −4, p = 1, Dk = 2D, m = −0.15, γ(t) = −2t, ρ0(t) = 1t + 2t2et2.

Remark 2.2 We can obtain many other solutions using (25), (26), and Tables 1–3, but we omit them for simplicity.

Remark 2.3 There are some hyperbolic function solutions and trigonometric function solutions can be obtained at the limit case when m→1 and m→0, but we also omit them for simplicity.

3 Conclusion

In this paper, we have obtained many types of exact solutions for the N-coupled NLS equations with variable coefficients by means of the combination method of the similarity transformation and auxiliary equation method, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions. To the best of our knowledge, the solutions obtained in this letter have not been reported in the previous literature. These solutions may provide more information to further study the mechanisms of the complicated nonlinear physical phenomena.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11526088

Award Identifier / Grant number: 11501186

Funding source: Natural Science Foundation of Hubei Province

Award Identifier / Grant number: 2014CFB640

Funding statement: This work is supported by the National Natural Science Foundation of China (11526088, 11501186) and Natural Science Foundation of Hubei Province (2014CFB640).

Acknowledgments:

This work is supported by the National Natural Science Foundation of China (11526088, 11501186) and Natural Science Foundation of Hubei Province (2014CFB640).

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Received: 2016-4-1

Accepted: 2016-5-15

Published Online: 2016-6-20

Published in Print: 2016-7-1

Artikel in diesem Heft

https://doi.org/10.1515/zna-2016-0128

Schlagwörter für diesen Artikel

Auxiliary Equation Method; Exact Solutions; N-Coupled Nonlinear Schrödinger Equations with Variable Coefficients; Similarity Transformation