Role of Gilson-Pickering equation for the different types of soliton solutions: a nonlinear analysis

https://doi.org/10.1140/epjp/s13360-020-00646-8 R eg u l a r A r t i c l e

Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis

Asıf Yoku¸s¹, Hülya Durur², Kashif Ali Abro^3,4,a , Do˘gan Kaya⁵

1Department of Actuary, Faculty of Science, Firat University, Elazig 23100, Turkey

2Department of Computer Engineering, Faculty of Engineering, Ardahan University, Ardahan 75000, Turkey 3Faculty of Natural and Agricultural Sciences, Institute of Ground Water Studies, University of the Free

State, Bloemfontein, South Africa

4Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

5Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey

Received: 2 June 2020 / Accepted: 28 July 2020 / Published online: 16 August 2020

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract In this article, the soliton solutions of the Gilson–Pickering equation have been constructed using the sinh-Gordon function method (ShGFM) and (G^/G, 1/G)-expansion method, which are applied to obtain exact solutions of nonlinear partial differential equations.

A solution function different from the solution function in the classical (G^/G, 1/G)-expansion method has been considered which are based on complex trigonometric, hyperbolic, and ratio- nal solutions. By invoking ShGFM and (G^/G, 1/G)-expansion methods, different traveling wave solutions have been investigated. For the sake of avoiding the complex calculations, the ready package program has been tackled. The comparative analysis of sinh-Gordon function and (G^/G, 1/G)-expansion methods has shown several differences and similari- ties. A comparative analysis of ShGFM and (G^/G, 1/G)-expansion methods assures that the (G^/G, 1/G)-expansion method has been found to be more intensive, powerful, reliable and effective method for the Gilson–Pickering equation. The graphical illustrations of two-, three-dimensional, and contour graphs have been depicted as well.

1 Introduction

Analytical solutions of nonlinear partial differential equations play an active role in nonlinear sciences. Various techniques have been used to explore the analytical solutions of different types of nonlinear partial differential equations. Extensive studies of the literature provide some reliable contributions in this area. There are a variety of methods to solve nonlin- ear partial differential equations, such as modified exp(−Ω(ξ))-expansion function method [1–3], the residual power series method [4,5] (1/G^)-expansion method [6–8], F-expansion technique [9], (G^/G)-expansion method [10,11], collocation method [12,13], Adomian’s decomposition method [14,15], Laplace transform method [16], new sub-equation method [17,18], and so on [19–32].

ae-mail:kashif.abro@faculty.muet.edu.pk(corresponding author)

There are many studies related to the Gilson–Pickering equation; for instance, by apply- ing the qualitative theory of the polynomial differential system, the qualitative behavior and analytical solutions of the Gilson–Pickering equation were studied [33], different types of solutions of the Gilson–Pickering equation have been attained by applying the simplified G^/G expansion method [34], by using a hybrid scheme based on collocation and quintic B- spline functions, the Gilson–Pickering equation has been solved [35], complex soliton solu- tions of the Gilson–Pickering equation have been attained using the Bernoulli sub-equation function method [36], soliton solutions of the Gilson–Pickering equation have been attained using (1/G^)-expansion method [37], soliton solutions of the Gilson–Pickering equation have been attained using the solitary ansatz method and (G^/G)-expansion method [38], and the Gilson–Pickering equation has been solved using the numerical meshless method [39]. Ana- lytical solutions either from linear and nonlinear differential equations are of vital importance not only because they are the solutions for some fundamental application problems but also they serve as accuracy checks for experimental, asymptotic and numerical solutions [40–48].

Additionally, for the sake of deepness of analytical solutions, we invoke the recent stud- ies of fractional analytical solutions [49–53] as well as non-fractional analytical solutions [54–57] therein. In this study, we have been attained analytical solutions of the Gilson–Pick- ering equation by using ShGFM and (G^/G, 1/G)-expansion method. The Gilson–Pickering equation which is a model of the third order (NLPDE) is as follows [34]:

ut− εux xt+ 2kux− uux x x− αuux− βuxux x 0, (1) whereε, α, k, β are nonzero constants. By Gilson and Pickering, the Gilson–Pickering equa- tion was introduced in 1995 [58]. There are three types of special cases of the equation in the literature. Until some rescalings, these are, respectively, Eq. (1) is the Fornberg–Whitham (FW) equation forε  1, α  −1, k  0.5, β  3 [59–61], Eq. (1) is the Rosenau–Hyman (RH) equation forε  0, α  1, k  0, β  3 [62], and Eq. (1) is the Fuchssteiner—

Fokas–Camassa–Holm (CH) equation forε  1, α  −3, β  2 [63,64]. We see that the difference betweenα, β and ε values in Eq. (1) is important enough to change the name of the equation. In this study, the contributions of these constants to the solution will be discussed.

As is known, each method produces different types of solutions. Another aim of this article is to present different solutions by using different methods and to compare the methods. We will present a different perspective of the analytical solutions of the Gilson–Pickering equation with two effective and reliable methods of obtaining analytical solutions.

2 Sinh-Gordon function method (ShGFM)

For a given nonlinear partial differential equation in two variables t, x δ

u, uux, u²ut, uutt, . . .

 0, (2)

whereδ is polynomial of the function u. Consider sinh-Gordon equation [65],

u_xt  γ sinh(u), (3)

where u is the unknown function of the t, x andγ ∈{0} Using wave transmutation

u U(ξ), ξ  μ(x + ct), (4)

in Eq. (3), we get the following (NLODE), U^ γ

cμ² sinh(U), (5)

where U(ξ)  U, ξ is the width, μ is the height and c is the velocity of the traveling wave.

Integrating Eq. (5), we get the following

U 2

_2

 2γ cμ²sinh²

U 2

 + γ q

cμ², (6)

where q is the integration constant, resulting from integrating Eq. (5). Settingϕ  ^U₂ and σ  _cμ^2γ2  _cμ^{γ q}2, we get the following

ϕ^√

σ cosh(ϕ). (7)

Equation (7) is a variables separable equation, simplifying it, produces the following two significant equations;

cosh(ϕ)  tan√

σ(ξ + d)

, (8)

sinh(ϕ)  sec√

σ(ξ + d)

, (9)

where d is the integration constant. In order to produce new solutions of Eq. (2), as per the operation of the ShGFM, we consider the following two solution equations.

U(ϕ) 

n i1

coshⁱ⁻¹(ϕ)[Bisinh(ϕ) + Aicosh(ϕ)] + A, (10)

U(ξ) 

n i1

tanⁱ⁻¹√

σ(ξ + d)

Bisec√

σ(ξ + d)

+ Aitan√

σ(ξ + d)

+ A. (11)

The value of n Eqs. (10) and (11) are determined using the balancing technique so that the highest derivative and the highest power nonlinear term in the reduced NLODE are taken into account. Setting each sum of the coefficients of sinhⁱ(ϕ) cosh^j(ϕ), (0 ≤ i ≤ n, 0 ≤ j ≤ n) with the equal to zero power yields a group of algebraic equations. Simplifying this group of algebraic equations with the help of a computer package program gives the values of the coefficients A, Ai, Bi,μ, c, d and σ into Eq. (11) along with the value of n yields the new traveling wave solutions to Eq. (2).

3 (G^/G, 1/G)-expansion method

The form of NLPDEs containing two or more independent variables of which the solution will be explored by using this method is written as follows [66]

K

u, ux, ut, uy, uz, ux x, utt, . . .

 0. (12)

If u(x, t)  u  U(ξ), ξ  μ(x + ct) transmutation is used in Eq. (12) where c is the velocity of the traveling wave, Eq. (12) is converted into a NLODE and Eq. (12) can be written as:

P

U, U^, U^, U^, . . .

 0. (13)

Equation (13) can be integrated to reduce the operational complexity. This method G(ξ)  G provides the following second-order ODE as

G^(ξ) + λG(ξ)  τ. (14)

Also, it hasφ  φ(ξ)  G^

G andψ  ψ(ξ)  _G(ξ)¹ to provide operational esthetic. The derivatives of the functions described can be given as follows

φ^ −φ²+τψ − λ, ψ^ −φψ. (15)

We can offer the behavior of solution function Eq. (14) according to the state ofλ taking into account the equations given by Eq. (15) are:

(i) Ifλ <0,

G(ξ)  c1sinh√

−λξ

+ c2cosh√

−λξ +τ

λ, (16)

whereas c₁and c₂are reel numbers. By considering Eq. (16);

ψ² −λ λ²ρ + τ²

φ²− 2τψ + λ

, ρ  c²₁− c₂², (17) Equation (17) is written.

(ii) Ifλ >0,

G(ξ)  c1sin√ λξ

+ c2cos√ λξ

+τ

λ, (18)

here c₁and c₂are reel numbers. By considering Eq. (18), there is the following equation;

ψ² λ

λ²ρ − τ²

φ²− 2τψ + λ

, ρ  c²₁+ c²₂. (19) (iii) Ifλ  0,

G(ξ)  τ

2ξ²+ c1ξ + c2, (20)

c₁and c₂are reel numbers. By considering Eq. (17), there is the following equation:

ψ² 1

c²₁− 2τc2

φ²− 2τψ

. (21)

Forψ and φ polynomials, solution of Eq. (13) is U(ξ)  a0+

n i1

(aiφⁱ+ biψⁱ). (22)

In this study, we rearranged the solution function in the classical (G^/G, 1/G)-expansion method in the form of Eq. (22) with the logic that the solution function can be achieved with the 1/G^and the G^/G-expansion methods. This logic is considered in conjunction with the classical (G^/G, 1/G)-expansion method, and the method may be developed in the next studies, and different solutions can be offered.

Here, a_i, bi(i  1, . . . , n) are real numbers to be identified. n is a positive equilibrium term which may be attained by comparing maximum-order derivative with the maximum- order nonlinear term in Eq. (13), when Eq. (22) is written in Eq. (13) along with Eqs. (15), (17), (19) or (21), a polynomial function related toφ and ψ is written. Each coefficient ofφⁱψ^j (i  0, 1, . . . , n) ( j  0, 1, . . . , n) terms of the attained polynomial functions are equals to zero, and there is a system of algebraic equations for a_i, bi, c, τ, c1, c2, μ and λ (i  0, 1, . . . , n). The necessary coefficients are obtained solving algebraic equation with the help of computer package programs. Obtained coefficients are placed in Eq. (22) and U(ξ) solution function of the ODE given as in Eq. (13) is attained and ifξ  μ(x + ct) transmutation is employed, we will attain the exact solution u(x, t) of Eq. (12).

4 Solution of the Gilson–Pickering equation using ShGFM

We consider Eq. (1). Using u(x, t)  U(ξ), ξ  μ(x + ct), Eq. (1) is converted into an ODE

cμU^+ 2kμU^− αμUU^− βμ³U^U^− cεμ³U⁽³⁾− μ³U U⁽³⁾ 0. (23) Integrating Eq. (23) with respect toξ once yields

cμU + 2kμU −1

2αμU²+1

2μ³U^2−1

2βμ³U^2− cεμ³U^− μ³U U^+ K  0, (24) where K is an integration constant, if we make the necessary arrangement and simplify with

−¹₂μ,

αU²+μ²

(−1 + β)U^2+ 2cεU^

− 2U

c + 2k− μ²U^

+ W  0, (25)

where W ^−2K_μ ∈ R. In Eq. (25), we find the balance term n 2 and considering the series given in Eq. (10),

U(ϕ)  A + A1cosh[ϕ] + B1sinh[ϕ] + cosh[ϕ](A2cosh[ϕ] + B2sinh[ϕ]), (26) if the equation given by (26) is placed in Eq. (25) and the necessary arrangements are made, we can write the following equation system:

Const:− 2Ac − 4Ak + W + A²α − αB1² 0,

cosh[w] : −2cA1− 4k A1+ 2 Aα A1− 2Apμ²A1− 2cpεμ²A1− 2αB1B2 0,

cosh[w]²:α A²1− pμ²A²₁− pβμ²A²₁− 2cA2− 4k A2+ 2 Aα A2− 8Apμ²A2− 8cpεμ²A2+αB1²

− 4pμ²B₁²− αB2² 0,

cosh[w]³: 4 Apμ²A₁+ 4cpεμ²A₁+ 2α A1A₂− 6pμ²A₁A₂− 4pβμ²A₁A₂+ 2αB1B₂− 12pμ²B₁B₂

− 2pβμ²B₁B₂ 0,

cosh[w]⁴: 3 pμ²A²₁+ pβμ²A²₁+ 12 Apμ²A2+ 12cpεμ²A2+α A²2− 4pμ²A²₂− 4pβμ²A²₂+ 3 pμ²B₁² + pβμ²B₁²+αB2²− 8pμ²B₂²− 2pβμ²B₂² 0,

cosh[w]⁵: 12 pμ²A1A2+ 4 pβμ²A1A2+ 10 pμ²B1B2+ 4 pβμ²B1B2 0, cosh[w]⁶: 8 pμ²A²₂+ 4 pβμ²A²₂+ 7 pμ²B₂²+ 3 pβμ²B₂² 0,

sinh[w] : −2cB1− 4k B1+ 2 AαB1 0,

cosh[w] sinh[w] : 2α A1B₁− 2pμ²A₁B₁− 2cB2− 4k B2+ 2 AαB2− 2Apμ²B₂− 2cpεμ²B₂ 0,

cosh[w]²sinh[w] : 4Apμ²B1+ 4cpεμ²B1+ 2α A2B1− 8pμ²A2B1+ 2α A1B2− 2pμ²A1B2− 2pβμ²A1B2 0, cosh[w]³sinh[w] : 6pμ²A1B1+ 2 pβμ²A1B1+ 12 Apμ²B2+ 12cpεμ²B2+ 2α A2B2− 6pμ²A2B2

− 4pβμ²A₂B₂ 0,

cosh[w]⁴sinh[w] : 12pμ²A₂B₁+ 4 pβμ²A₂B₁+ 12 pμ²A₁B₂+ 4 pβμ²A₁B₂ 0, cosh[w]⁵sinh[w] : 16pμ²A₂B₂+ 8 pβμ²A₂B₂ 0,

cosh[w] sinh[w]⁴: 2 pμ²B₁B₂ 0,

cosh[w]²sinh[w]⁴: pμ²B₂²+ pβμ²B₂² 0. (27)

A, A₁, B₁, B₂, W andα, p, μ, β, c, ε, k constants are attained from Eq. (27) the system with the aid of a package program.

Case 1: If A1 −

√W− α A²

√α , B1  −

√W− α A²

√α , A2 0, B2  0,

Fig. 1 3D, 2D and contour graphics of Eq. (29) for A 1, α  1, W  2, d  4, σ  1, c  2

Fig. 2 3D, 2D and contour graphics of Eq. (31) for A₁ 2, A  1, μ  1.1, σ  2, d  3, c  1

μ  −

√α

√σ, β  −3, ε  −A

c, k  1

2(−c + α A), (28)

substituting values (28) into (26), we attain trigonometric soliton solution for Eq. (1) u₁(x, t)  A −

√W− A²α sec

d−^(ct+x)√_σ^√α √ σ

√α −

√W− A²α tan

d−^(ct+x)√_σ^√α √ σ

√α .

(29) The trigonometric soliton solution of Eq. (29) produced from the ShGFM is as in Fig.1.

Case 2: If

B1 A1, A2 0, B2 0, β  −3, ε  −A c, k 1

2

−c + Apμ²

, α  pμ², W  pμ² A²+ A²₁

,

(30) substituting values (30) into (26), we attain trigonometric soliton solution for Eq. (1)

u2(x, t)  A + sec

(d + (ct + x)μ)√ σ

A1+ A1tan

(d + (ct + x)μ)√ σ

. (31) The trigonometric soliton solution of Eq. (31) produced from the ShGFM is as in Fig.2.

Case 3: If

B1 −A1, A2 0, B2 0, β  −3, ε  − A

−2k + Apμ², c  −2k + Apμ², α  pμ², W  pμ²

A²+ A²₁

, (32)

substituting values (32) into (26), we attain trigonometric soliton solution for Eq. (1)

u₃(x, t)  A − sec √ σ

d +μ x + t

−2k + Aμ²σ

A₁+ A₁tan √ σ

d +μ x + t

−2k + Aμ²σ

. (33)

Fig. 3 3D, 2D and contour graphics of Eq. (33) for A₁ 2, A  3, μ  0.9, σ  3, d  1, k  2, t  1

Fig. 4 3D, 2D and contour graphics of the real and imaginary parts of Eq. (35) for A 2, α  0.5, W  3, d  4, σ  1, c  2

The trigonometric soliton solution of Eq. (33) produced from the ShGFM is as in Fig.3.

Case 4: If A1 −i√

3 A, B1 i√

3 A, A2 0, B2 0, β  −3, ε  −A

c, α  − W 2 A², k −2Ac − W

4 A , μ  − i√

√ W 2 A√

σ, (34)

substituting values (34) into (26), we attain complex trigonometric soliton solution for Eq. (1) u₄(x, t)  A + i√

3 A sec



d−i√

W(ct + x)

√2 A√σ

√ σ



− i√ 3 A tan



d−i√

W(ct + x)

√2 A√σ

√ σ

 . (35) The complex trigonometric soliton solution of Eq. (35) produced from the ShGFM is as in Fig.4.

5 Solution of the Gilson–Pickering equation using (G^/G, 1/G)-expansion method Considering Eq. (25), we get balancing term n 2 and in Eq. (22), and the following situation is obtained:

U(ξ)  a0+ a₁φ[ξ] + b1ψ[ξ] + a2φ[ξ]²+ b₂ψ[ξ]². (36) Equation (36) presented is different from the solution function in the classical (G^/G, 1/G)- expansion method. Therefore, the solutions to be obtained here will be different from the solutions in the classical method. If we substitute Eq. (36) in Eq. (25) and the coefficients of the algebraic equation are equal to zero, we can establish the following algebraic equation systems

Const: W− 2ca0− 4ka0+αa²0− λ²μ²a²₁+βλ²μ²a₁²− κ²λ²μ²a₁²

−μ²+λ²ρ+βκ²λ²μ²a₁²

−μ²+λ²ρ + 4cελ²μ²a₂+4cεκ²λ²μ²a₂

−μ²+λ²ρ + 4λ²μ²a₀a₂+4κ²λ²μ²a₀a₂

−μ²+λ²ρ − 2cεκλ²μ²b₁

−μ²+λ²ρ

−2κλ²μ²a₀b₁

−μ²+λ²ρ −8κλ³μ²a₂b₁

−μ²+λ²ρ + αλ²b₁²

−μ²+λ²ρ+ 2λ³μ²b²₁

−μ²+λ²ρ− 2cλ²b₂

−μ²+λ²ρ

− 4kλ²b₂

−μ²+λ²ρ+4cελ³μ²b₂

−μ²+λ²ρ + 2αλ²a₀b₂

−μ²+λ²ρ+4λ³μ²a₀b₂

−μ²+λ²ρ+4λ⁴μ²a₂b₂

−μ²+λ²ρ  0,

φ[ξ] : −2ca1− 4ka1+ 4cελμ²a₁+ 2αa0a₁+ 4λμ²a₀a₁+ 4βλ²μ²a₁a₂+4βκ²λ²μ²a₁a₂

−μ²+λ²ρ

−6κλ²μ²a₁b₁

−μ²+λ²ρ −2βκλ²μ²a₁b₁

−μ²+λ²ρ + 2αλ²a₁b₂

−μ²+λ²ρ+4λ³μ²a₁b₂

−μ²+λ²ρ+4βλ³μ²a₁b₂

−μ²+λ²ρ  0, (φ[ξ])²:αa²1+ 2λμ²a₁²+ 2βλμ²a₁²− κ²λμ²a₁²

−μ²+λ²ρ + βκ²λμ²a₁²

−μ²+λ²ρ− 2ca2− 4ka2+ 16cελμ²a₂ +4cεκ²λμ²a₂

−μ²+λ²ρ + 2αa0a₂+ 16λμ²a₀a₂+4κ²λμ²a₀a₂

−μ²+λ²ρ + 4βλ²μ²a²₂+4βκ²λ²μ²a₂²

−μ²+λ²ρ

−2cεκλμ²b₁

−μ²+λ²ρ −2κλμ²a₀b₁

−μ²+λ²ρ −26κλ²μ²a₂b₁

−μ²+λ²ρ −4βκλ²μ²a₂b₁

−μ²+λ²ρ + αλb²₁

−μ²+λ²ρ + 5λ²μ²b²₁

−μ²+λ²ρ+ βλ²μ²b₁²

−μ²+λ²ρ− 2cλb2

−μ²+λ²ρ− 4kλb2

−μ²+λ²ρ+16cελ²μ²b₂

−μ²+λ²ρ + 2αλa0b₂

−μ²+λ²ρ+16λ²μ²a₀b₂

−μ²+λ²ρ + 2αλ²a₂b₂

−μ²+λ²ρ+16λ³μ²a₂b₂

−μ²+λ²ρ +8βλ³μ²a₂b₂

−μ²+λ²ρ  0, (φ[ξ])³: 4cεμ²a₁+ 4μ²a₀a₁+ 2αa1a₂+ 12λμ²a₁a₂+ 8βλμ²a₁a₂+4βκ²λμ²a₁a₂

−μ²+λ²ρ

−6κλμ²a₁b₁

−μ²+λ²ρ −2βκλμ²a₁b₁

−μ²+λ²ρ + 2αλa1b₂

−μ²+λ²ρ+16λ²μ²a₁b₂

−μ²+λ²ρ +8βλ²μ²a₁b₂

−μ²+λ²ρ  0, (φ[ξ])⁴: 3μ²a²₁+βμ²a²₁+ 12cεμ²a₂+ 12μ²a₀a₂+αa2²+ 8λμ²a₂²+ 8βλμ²a₂²

+4βκ²λμ²a₂²

−μ²+λ²ρ −18κλμ²a₂b₁

−μ²+λ²ρ −4βκλμ²a₂b₁

−μ²+λ²ρ + 3λμ²b²₁

−μ²+λ²ρ+ βλμ²b₁²

−μ²+λ²ρ +12cελμ²b₂

−μ²+λ²ρ +12λμ²a₀b₂

−μ²+λ²ρ+ 2αλa2b₂

−μ²+λ²ρ+28λ²μ²a₂b₂

−μ²+λ²ρ +16βλ²μ²a₂b₂

−μ²+λ²ρ  0, (φ[ξ])⁵: 12μ²a₁a₂+ 4βμ²a₁a₂+12λμ²a₁b₂

−μ²+λ²ρ+4βλμ²a₁b₂

−μ²+λ²ρ  0,