Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids

1 Introduction

Korteweg–de Vries (KdV)-type equations, with the first-order nonlinear and dispersive terms being retained and in balance, have been used as approximate models governing weakly nonlinear long waves from different dynamical contexts [1–3]. For instance, they can describe the ion-acoustic waves in plasmas, shallow water waves in channels and oceans, and pulse waves in large arteries [4–6]. When the second-order nonlinear and dispersive terms are retained, the extend fifth-order KdV equation can be expressed as follows [7–16]:

(1)uτ + uξ + 6 α u uξ + α uξξξ + α2 c1 u2 uξ + α2 c2 uξ uξξ + α2 c3 u uξξξ + α2 c4 uξξξξξ = 0, (1)

where α<<1 is a non-dimensional measure of the small wave amplitude, u(ξ, τ) is the analytic function of the space variable ξ and the time variable τ, and c_i’s (i= 1, …, 4) are the coefficients of the higher order terms. Equation (1) can be used to describe the propagation of steeper waves of shorter wavelength than the KdV equation does [7]. As one member of the perturbed KdV equations, (1) also appears in the study on the free surface flow over an obstacle, and flow of a stratified fluid over an obstacle [8, 9]. By scaling both space variable ξ and time variable τ into x and t, (1) can be written in the compact form as [10–16]

(2)ut + 6 u ux + uxxx + α c1 u2 ux + α c2 ux uxx + α c3 u uxxx + α c4 uxxxxx = 0. (2)

In the context of surface water waves, (2) is useful for the progression from the shallow water waves to deep water waves [10–14]. Equation (2) can also be used to depict the internal waves of the moderate amplitude in a shallow and density-stratified fluid [10–14]. It has been found that the first-order solutions of (1) and (2) are the KdV soliton with O(1) amplitude [8–16], while the second-order nonlinear and dispersive terms affect the amplitude of the soliton even though these terms appear in (1) at O(α²) and in (2) at O(α) [8–16].

Equation (2) has been studied by some authors from various viewpoints. The asymptotic transformation from (2) to one higher order member of the integrable KdV hierarchy has been proposed [10]. Under the extended asymptotic transformation, (2) can be transformed to the KdV equation [11]. Via a local variant of the transformation, (2) has also been transformed into a member of the integrable KdV hierarchy and the asymptotic two-soliton solutions have been obtained [12]. Additionally, numerical solutions for (2) have been presented when the ratio α is small [13]. Lax pairs, conservation laws, bilinear forms, and N-soliton solutions for (1) and (2) under certain coefficient constraints have been derived, and soliton propagation and interaction have been illustrated graphically [14]. Via the simplified form of the bilinear method, multi-soliton solutions and multiple singular soliton solutions for (2) have been obtained, and the integrability has been investigated [15]. Besides, the bilinear forms with an auxiliary variable, Bäcklund transformation, superposition formula, and N-soliton solutions in terms of Wronski determinant for (2) under other coefficient constraints have been constructed [16].

Up to now, some methods for obtaining analytic solutions of nonlinear evolution equations (NLEEs) have been proposed, including the inverse scattering transformation, bilinear method, Bäcklund transformation, Darboux transformation (DT), and symmetry reduction [17–24]. Among these approaches, the DT can be employed to construct the soliton solutions for NLEEs through an algebraically iterative algorithm [20–30]. However, the elementary DT cannot be applicable directly to construct the rational solutions for NLEEs [25–30]. Therefore, the so-called generalised DT has been introduced to achieve the position solutions of the KdV equation [25]. In the case of the nonlinear Schrödinger equation, a generalised N-fold DT has been presented in terms of both a summation formula and determinants so that the rational solutions can be generated [26, 27].

Since DT and rational solutions for (2) under certain coefficient constraint have not been reported, we will apply the DT on this equation to search for the multi-soliton and rational solutions. In Section 2, with the aid of symbolic computation [31–34], based on the Lax pair and Darboux matrix method [20–30], we will derive the N-fold DT for (2), from which multi-soliton solutions will be derived. In Section 3, by means of the DT and Taylor expansion of those solutions for the Lax pair [20–30], the generalised DT for (2) will be given. Via iterations of the generalised DT, formulae of the first- and second-order rational solutions for (2) will be presented. Section 4 will be devoted to the conclusions.

2 Multi-Soliton Solutions for (2)

The Lax pair can assure the integrability of a nonlinear system, from which the initial problem of a given NLEE can be solved, and the integrable property such as conservation law, symmetry class, Hamiltonian structure, and DT can be derived [20–24]. Based on the Lax pair of (2) with the coefficient constraints

(3)c1 = 30 c4,  c2 = 20 c4,  c3 = 10 c4, (3)

the associated linear eigenvalue problem of the equation can be expressed as [14]

(4a)ϕx = U ϕ,  U = (λu−1−λ), (4a)

(4b)ϕt = V ϕ,  V = (V11V12V21−V11), (4b)

with

(5a)V11 = − 16 α λ5 − (8 α u + 4) λ3 − 4 α ux λ2 − (6 α u2 + 2 u + 2 α uxx) λ − 6 α u ux − ux − α uxxx, (5a)

(5b)V12 = − 16 α u λ4 − 8 α ux λ3 − (8 α u2 + 4 u + 4 α uxx) λ2 − (12 α u ux + 2 ux + 2 α uxxx) λ − 6 α u3 − 2 u2 − 6 α ux2 − 8 α u uxx − uxx − α uxxxx, (5b)

(5c)V21 = 16 α λ4 + (8 α u + 4) λ2 + 6α u2 + 2 u + 2 α uxx, (5c)

where ϕ= (ϕ₁, ϕ₂)^T is the vector eigenvalue, T represents the transpose of the matrix, and λ is the parameter independent of x and t. By direct calculation, the compatibility condition U_t – V_x + UV – VU=0 leads to (2) with constraints (3). Consider a DT for the eigenvalue problem (4) as

(6a)ϕ[1] = D[1] ϕ, (6a)

(6b)ϕ[1]x = U[1] ϕ[1], ϕ[1]t = V[1] ϕ[1], (6b)

where D is a nonsingular matrix; (U[1], V[1]) have the same forms as (U, V) by replacing u with u[1], and satisfy

(7)D[1]x + D[1] U = U[1] D[1], D[1]t + D[1] V = V[1] D[1]. (7)

In this way, DT (6) establishes the connection between the new potential u[1] in (U[1], V[1]) and the initial potential u in (U, V). Let matrix D[1] in (6a) and (7) be in the form of [20–30]

(8a)D[1] = (100−1)  [(λ00λ)−S], (8a)

(8b)S = H Λ H−1, (8b)

(8c)H = (ψ1ψ1 + 2λ0ψ2ψ2ψ2),  Λ = (λ000−λ0), (8c)

where ϕ₁=(ψ₁, ψ₂)^T is a special solution of the Lax pair (4) and (5) at u=u[0] and λ= λ₀. Therefore, under the operation of the Darboux matrix D[1] in (7), the elementary DT of (2) can be given as

(9a)u[1] = − u[0] − 2ψ1,1[0](ψ1,1[0] + 2λ0ψ1,2[0])ψ1,2[0]2,  (9a)

(9b)D[1] = (100−1)  [(λ00λ) − H[0] Λ[0] H[0]−1], (9b)

with

(10)H[0] = (ψ1,1[0]ψ1,1[0] + 2λ0ψ1,2[0]ψ1,2[0]ψ1,2[0]    ),  Λ[0] = (λ000−λ0), (10)

where ϕ₁[0]=(ψ_1,1[0], ψ_1,2[0])^T is equivalent to ϕ₁=(ψ₁, ψ₂)^T.

When N distinct basic solutions ϕ_j[0]=(ψ_j,1[0], ψ_j,2[0])^T of the Lax pair (4) and (5) at u= u[0] and λ=λ_j (j=1, 2, …, N) are given, the elementary DT can be repeated N times; then the N-fold DT can be obtained as

(11a)ϕN[N − 1] = D[N − 1] D[N − 2] ⋯ D[1] D[0] ϕN[0], (11a)

(11b)u[N] = − u[0] − ∑j = 1N2 ψj,1[j − 1](ψj,1[j − 1] + 2 λj ψj,2[j − 1])ψj,2[j − 1]2, (11b)

with

(12a)D[j] = (100−1)  [(λ00λ) − H[j − 1] Λ[j − 1] H[j − 1]−1], (12a)

(12b)H[j − 1] = (ψj,1[j − 1]ψj,1[j − 1] + 2λjψj,2[j − 1]ψj,2[j − 1]ψj,2[j − 1]), Λ[j − 1] = (λj−100−λj−1), (12b)

(12c)ϕj[j − 1] = (D[j − 1] D[j − 2] ⋯ D[0])|λ = λjϕj[0], j = 1, 2, …, N, (12c)

where ϕ_j[j– 1]=(ψ_j,1[j – 1], ψ_j,2[j – 1])^T, D[0]=I, and I is the identity matrix. According to the iterative algorithm of DT (11) and (12) based on the zero background, the multi-soliton solutions for (2) can be derived in a recursive manner. As a result, we take a trivial seed solution u=0 and solve the Lax pair (4) and (5) with λ= ξ₁, and can have the basic solutions ϕ₁ as

(13a)ϕ1 = (ψ1, ψ2)T, (13a)

(13b)ψ1 = Γ11 eξ1 x − 4ξ13 t − 16 α ξ15 t, (13b)

(13c)ψ2 = 2 Γ12 ξ1 e−ξ1 x + 4ξ13 t + 16 α ξ15 t − Γ11 eξ1 x − 4ξ13 t − 16 α ξ15 t2ξ1, (13c)

where Γ₁₁ and Γ₁₂ are constants. Through (9a), one-soliton solutions for (2) can be derived as

(14)u[1] = − 16 Γ11 Γ12 ξ13 e2 ξ1 x + 8 ξ13 t + 32 α ξ15 t(Γ11 e2 ξ1 x − 2 Γ12 ξ1 e8 ξ13 t + 32 α ξ15 t)2 = 2 ξ12 sech2(θ12), (14)

where

θ1 = 2 ξ1 x − 8 ξ13 t − 32 α ξ15 t + log(−Γ112 Γ12 ξ1).

From (14), one can find that the amplitude and width of the soliton are 2 ξ12 and 1ξ1, respectively. As illustrated in Figure 1, the velocity of the soliton is influenced by the coefficient α, while its amplitude and width are unchanged.

Figure 1:

Propagation of the one soliton via (14) with the parameters ξ₁=–0.75, Γ₁₁=Γ₁₂=1: (a) α=0.02; (b) α=0.2.

Considering another basic solution of the Lax pair (4) and (5) at λ=ξ₂, and using DT (11) and (12), one can obtain two-soliton solutions for (2) as

(15a)u[2] = − 8eθ1ξ12(1 + eθ1)2 + G2F2, (15a)

(15b)G2 = 8e−2θ1Γ212[− 4e2θ1(1 + eθ2)2ξ14 − 2eθ1(−1 + e2θ1)(−1 + e2θ2)ξ2ξ13 − eθ2(1 + eθ1)4ξ24+ 2eθ1 + (1 + eθ1)2(2eθ1 + eθ2 + 2eθ1 + θ2 + e2θ1 + θ2 + 2eθ1 + 2θ2)ξ22ξ12], (15b)

(15c)F2 = [(−1 + eθ1)(1 + eθ2)ξ1 − (1+eθ1)(−1 + eθ2)ξ2]2, (15c)

where θ2 = 2 ξ2 x − 8 ξ23 t − 32 α ξ25 t + log(−Γ212 Γ22 ξ2), while Γ₂₁ and Γ₂₂ are constants.

Through (15), the interaction between two solitons can be studied. From Figure 2, it can be found that the amplitude, velocity, and wave shape of each soltion remain unchanged after the interaction. The velocity of each soliton is influenced by the variation of α, while the amplitude and width are unvaried.

Figure 2:

Two-soliton interaction via (15) with the parameters ξ₁=–0.6, ξ₂=0.8, Γ₁₁=Γ₁₂=1, Γ₂₁=Γ₂₂=1: (a) α=0.02; (b) α=0.2.

By virtue of the iterative algorithm of DT (11) and (12) once again, three-soliton solutions can be derived as

(16a)u[3] = 8 (G3[1] + G3[2] + G3[3])F3,  (16a)

(16b)G3[1] = eθ1ξ16[(−1 + eθ2)(1 + eθ3)ξ2−(1 + eθ2)(−1 + eθ3)ξ3]2+(1 + eθ1)2ξ22ξ32(ξ22 − ξ32)[eθ2(−1 + eθ3)2ξ22 − eθ3(−1 + eθ2)2ξ32]−2(−1 + e2θ1)ξ13ξ3[eθ3(−1 + e2θ2)ξ3 − eθ2(−1 + e2θ3)ξ2](ξ23 − ξ2ξ32)+2(−1 + e2θ1)ξ1ξ3(ξ23 − ξ2ξ32)[eθ3(−1 + e2θ2)ξ33 − eθ2(−1 + e2θ3)ξ23], (16b)

(16c)G3[2] = − ξ14[(eθ1 + eθ2 − 4eθ1 + θ2 + e2θ1 + θ2 + eθ1 + 2θ2)(1 + eθ3)2ξ24−2eθ1(−1 + e2θ2)(−1 + e2θ3)ξ23ξ3 − (− 2eθ1 + eθ2 − 2eθ1 + θ2 + e2θ1 + θ2 − 2eθ1 + 2θ2 + eθ3−2eθ1 + θ3 + e2θ1 + θ3 + 4eθ2 + θ3 + 4e2θ1 + θ2 + θ3 + e2θ2 + θ3 + e2θ2 + θ3 − 2eθ1 + 2θ2 + θ3 + e2θ1 + 2θ2 + θ3−2eθ1 + 2θ3 + eθ2 + 2θ3ξ22ξ32 − 2eθ1 + θ2 + 2θ3 + e2θ1 + θ2 + 2θ3 − 2eθ1 + 2(θ2 + θ3)) + (1 + eθ2)2(eθ1 + eθ3−4eθ1 + θ3 + e2θ1 + θ3 + eθ1 + 2θ3)ξ34 − 2eθ1(−1 + e2θ2)(−1 + e2θ3)ξ2ξ33], (16c)

(16d)G3[3] = ξ12 [eθ2(−1 + eθ1)2(1 + eθ3)2ξ26 + (eθ1 − 2eθ2 − 2eθ1 + θ2 − 2e2θ1 + θ2 + eθ1 + 2θ2 + eθ3+4eθ1 + θ3 + e2θ1 + θ3 − 2eθ2 + θ3 − 2e2θ1 + θ2 + θ3 + e2θ2 + θ3 + 4eθ1 + 2θ2 + θ3 + e2θ1 + 2θ2 + θ3 + eθ1 + 2θ3−2eθ2 + 2θ3 − 2eθ1 + θ2 + 2θ3 − 2e2θ1 + θ2 + 2θ3 + eθ1 + 2(θ2 + θ3))ξ24ξ32 − 2eθ1(−1 + e2θ2)(−1 + e2θ3)ξ23ξ33+eθ3(−1 + eθ1)2(1 + eθ2)2ξ36 (eθ1 + eθ2 + 4eθ1 + θ2 + e2θ1 + θ2 + eθ1 + 2θ2 − 2eθ3 − 2eθ1 + θ3−2e2θ1 + θ3 − 2eθ2 + θ3− 2e2θ1 + θ2 + θ3 − 2e2θ2 + θ3 − 2eθ1 + 2θ2 + θ3 − 2e2θ1 + 2θ2 + θ3 + eθ1 + 2θ3+eθ2 + 2θ3 + 4eθ1 + θ2 + 2θ3 + e2θ1 + θ2 + 2θ3 + eθ1 + 2(θ2 + θ3))ξ22ξ34], (16d)

(16e)F3 = [(1 + eθ1)ξ12((−1 + eθ2)(1 + eθ3)ξ2 − (1 + eθ2)(−1 + eθ3)ξ3)+(1 + eθ1)((1 + eθ2)(−1 + eθ3)ξ2 − (−1 + eθ2)(1 + eθ3)ξ3) ξ2ξ3+(1 − eθ1)(1 + eθ2)(1 + eθ3)(ξ22 − ξ32)ξ1]2, (16e)

where θ3 = 2 ξ3 x − 8 ξ33 t − 32 α ξ35 t + log(−Γ312 Γ32 ξ3), while Γ₃₁ and Γ₃₂ are constants.

Via (16), the interaction among three solitons is depicted in Figure 3, from which it can be found that the three solitons exhibit the elastic interaction, during which they can preserve their properties after the interaction. Except for the velocity, the amplitude and width of each soliton are not influenced by the coefficient α.

Figure 3:

Three-soliton interaction via (16) with the parameters ξ₁=–0.5, ξ₂=0.7, ξ₃=–0.9, Γ₁₁=Γ₁₂=1, Γ₂₁=Γ₂₂=1, Γ₃₁=Γ₃₂=1: (a) α=0.02; (b) α=0.2.

3 Rational Solutions for (2) Through the Generalised DT

Rational solutions can be seen as the limiting cases of either periodic “Ma solitons” or “Akhmediev breathers” [35–42]. Ma solitons generate from initial conditions consisting of the background plane wave plus soliton, while Akhmediev breathers arise from modulation instability and may ultimately lead to the formation of the Peregrine soliton [35–42]. The study of rational solutions may help us comprehend further about the mechanism of the rogue waves (a kind of rational solution which is localised in both space and time) in the oceans, superfluids, Bose–Einstein condensations, and optics systems [35–43]. The rogue waves of (coupled) nonlinear Scrödinger-type equations have been constructed through different approaches such as the algebro-geometric method, bilinear method, and generalised DT [26–30, 44–46]. The rational solutions for KdV-type equations have also been reported [20–25]. Naturally, we are interested in whether the solutions of (2) can be expressed in the forms of a ratio of two polynomials. Motivated by this idea, we will derive the rational solutions for (2) by means of the generalised DT and Taylor expansion of those solutions for the associated Lax pair [26–30].

We assume that ψ= ϕ₁(λ + δ) is a special solution of the Lax pair (4) and (5) with δ a small parameter, and expand ψ into the Taylor series of δ [26–30],

(17)ψ = ϕ1[0] + ϕ1[1] δ + ϕ1[2] δ2+ ⋯ + ϕ1[N] δN, (17)

where

ϕ1[j] = (ϕ1[j], ϕ2[j])T = 1j ! ∂jϕ1(λ)∂λj|λ = λ0 (j = 1, 2, … ,N).

It can be testified that ϕ1[0] is the solutions for (4) and (5) with u= u[0]

and λ=λ₀. The one-fold generalised DT for (2) is constructed as

(18a)u[1] = − u[0] − 2ψ1[0](ψ1[0] + 2λ0ψ2[0])ψ2[0]2,  (18a)

(18b)D1[1] = (100−1)  [(λ00λ) − H[0] Λ[0] H[0]−1], (18b)

with

(19)H[0] = (ψ1[0]ψ1[0] + 2λ0ψ2[0]ψ2[0]ψ2[0] ),  Λ[0] = (λ000−λ0),  (ψ1[0], ψ2[0])T = ϕ1[0]. (19)

Taking the limit process

(20)limδ→0D1[1]|λ = λ0 + δψδ = limδ→0(δ I¯ + D1[1]|λ = λ0)(ϕ1[0] + ϕ1[1] δ)δ= I¯ ϕ1[0] + D1[1]|λ = λ0ϕ1[1] = (ψ1[1], ψ2[1])T, (20)

where I¯ = (100−1), we can obtain the solutions for the linear system (4) and (5) with u= u[1] and λ=λ₀. The process can be continued, and the two-fold generalised DT for (2) is derived as

(21a)u[2] = − u[1] − 2ψ1[1](ψ1[1] + 2λ0ψ2[1])ψ2[1]2,  (21a)

(21b)D1[2] = (100−1)  [(λ00λ)−H[1] Λ[1] H[1]−1], (21b)

with

(22)H[1] = (ψ1[1]ψ1[1] + 2λ0ψ2[1]ψ2[1]ψ2[1] ),  Λ[1] = (λ000−λ0).  (22)

In a similar manner, the following limit leads to the nontrivial solutions of the linear spectral problem with u=u[2] and λ=λ₀,

(23)limδ→0 (D1[2] D1[1])|λ = λ0 + δψδ2 = limδ→0(δ I¯ + D1[2]|λ = λ0) (δ I¯ + D1[1]|λ = λ0)(ϕ1[0] + ϕ1[1] δ + ϕ1[2] δ2)δ2=I¯2 ϕ1[0] + (I¯ D1[1]|λ = λ0 + D1[2]|λ = λ0 I¯)   ϕ1[1] + D1[2]|λ = λ0D1[1]|λ = λ0 ϕ1[2] = (ψ1[2], ψ2[2])T, (23)

and the three-fold generalised DT for (2) can be obtained as

(24a)u[3] = − u[2] − 2ψ1[2](ψ1[2] + 2λ0ψ2[2])ψ2[2]2, (24a)

(24b)D1[3] = (100−1) [(λ00λ)−H[2] Λ[2] H[2]−1], (24b)

with

(25)H[2] = (ψ1[2]ψ1[2] + 2λ0ψ2[2]ψ2[2]ψ2[2] ), Λ[2] = (λ000−λ0). (25)

Therefore, the N-fold generalised DT is

(26a)u[N] = − u[N − 1] − 2ψ1[N − 1](ψ1[N − 1] + 2λ0ψ2[N − 1])ψ2[N − 1]2, (26a)

(26b)D1[N] = (100−1) [(λ00λ)−H[N−1] Λ[N − 1] H[N − 1]−1], (26b)

with

(27)H[N − 1] = (ψ1[N − 1]ψ1[N − 1] + 2λ0ψ2[N − 1]ψ2[N − 1]ψ2[N − 1]), Λ[N − 1] = (λ000−λ0). (27)

By virtue of (17)–(27), we can construct the rational solutions for (2). The seed solution u=θ will be utilised, and the corresponding solutions for (2) at λ = ξ1 = θ + δ can be obtained as

(28a)ϕ1(ξ1) = (ψ1(ξ1), ψ2(ξ1))T, (28a)

(28b)ψ1(ξ1) = σ12ρ[ρ(eρx + e−ρx) − ξ1(eρx − e−ρx)] − σ2 θ2ρ(eρx − e−ρx), (28b)

(28c)ψ2(ξ1) = σ22ρ[ρ(eρx + e−ρx) − ξ1(eρx − e−ρx)] − σ12ρ(eρx − e−ρx), (28c)

with

(29a)σ1 = c12ρ[ρ(eχ + e−χ) − ξ1(eχ − e−χ)] − c2 θ2ρ(eχ − e−χ), (29a)

(29b)σ2 = c22ρ[ρ(eχ + e−χ) + ξ1(eχ − e−χ)] + c12ρ(eχ − e−χ), (29b)

where χ = 2 ρ[8 α ξ14 + (4 α θ + 2) ξ12 + (3 α θ + 1) θ] t, ρ = ξ12 − θ, c₁ and c₂ are constants, θ is a nonzero constant, while δ is a small parameter. Without loss of generality, we can set θ=1, c₁=1, c₂=1, and expand the solutions ψ₁(ξ₁) and ψ₂(ξ₁) in (28) at δ=0, which results in

(30a)ϕ1(ξ1) = ϕ1[0] + ϕ1[1] δ + ϕ1[2] δ2 + ⋯, (30a)

(30b)ϕ1[0] = (ψ1[0], ψ2[0])T = (2x − 12t(5α + 1) + 1−2x + 12t(5α + 1) + 1), (30b)

(30c)ϕ1[1] = (ψ1[1], ψ2[1])T=(2x33 + x2 + x − 1445αt + t)3 + 362x + 1)(5αt + t)2 − 2t[6(5α + 1)x2 + 65α + 1)x + 95α + 11]−2x33 + x2 − x + 1445αt + t)3 − 362x − 1)(5αt + t)2 + 2t[6(5α + 1)x2 − 65α + 1)x + 95α + 11] ), (30c)

(30d)ϕ1[2] = (ψ1[2], ψ2[2])T= (−25925(5αt + t)5 + 216(2x + 1)(5αt + t)4 + 130x2(2x3 + 5x2 + 20x + 15) + 6t2[4(5α + 1)2x3 + 4(475α2 + 150α + 11)x + 875α2 + 65αx + x)2 + 270α + 19] − 2t[(5α + 1)x4 + 25α + 1)x3 + 2(55α + 7)x2 + (95α + 11)x + 144α + 8]− 144t3(5α + 1)2[(5α + 1)x2 + 5αx + x + 45α + 5],25925(5αt + t)5 − 216(2x − 1)(5αt + t)4 + 130x2(−2x3 + 5x2 − 20x + 15) − 6t2[4(5α + 1)2x3 + 4(475α2 + 150α + 11)x − 875α2 − 6(5αx + x)2 − 270α − 19] + 2t[(5α + 1)x4 − 25α + 1)x3 + 2(55α + 7)x2 − (95α + 11)x + 144α + 8] + 144t3(5α + 1)2[(5α + 1)x2 − (5α + 1)x + 45α + 5])T. (30d)