Method of Multiple Scales and Travelling Wave Solutions for (2+1)-Dimensional KdV Type Nonlinear Evolution Equations

Burcu Ayhan; M. Naci Özer; Ahmet Bekir

doi:10.1515/zna-2016-0123

Article Publicly Available

Method of Multiple Scales and Travelling Wave Solutions for (2+1)-Dimensional KdV Type Nonlinear Evolution Equations

Burcu Ayhan , M. Naci Özer and Ahmet Bekir

Published/Copyright: June 24, 2016

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From the journal Zeitschrift für Naturforschung A Volume 71 Issue 8

Abstract

In this article, we applied the method of multiple scales for Korteweg–de Vries (KdV) type equations and we derived nonlinear Schrödinger (NLS) type equations. So we get a relation between KdV type equations and NLS type equations. In addition, exact solutions were found for KdV type equations. The (G′G)-expansion methods and the (G′G, 1G)-expansion methods were proposed to establish new exact solutions for KdV type differential equations. We obtained periodic and hyperbolic function solutions for these equations. These methods are very effective for getting travelling wave solutions of nonlinear evolution equations (NEEs).

Keywords: Exact Travelling Wave Solution; (G′G)$\left( {{{G'} \over G}} \right)$-Expansion Method; (G′G, 1G)$\left( {{{G'} \over G},{\rm{ }}{1 \over G}} \right)$-Expansion Method; Method of Multiple Scales

PACS: 02.30.Ik; 02.70.Wz; 05.45.Yv

1 Introduction

Nonlinear evolution equations (NEEs) are the mathematical models of problems that arise in many field of science. In recent years, NEEs have become an important field of study in applied mathematics [1], [2], [3]. Nonlinear phenomena come up in a variety of scientific applications such as optical fibres, fluid dynamics, plasma physics, chemical physics, and in engineering applications as well. Nonlinear equations that model these scientific phenomena have long been a major concern for research work [4]. The nonlinear Schrödinger (NLS) equation is an example of a universal nonlinear model that describes many physical nonlinear systems. The equation can be applied to hydrodynamics, nonlinear optics, nonlinear acoustics, quantum condensates, plasma physics, heat pulses in solids, and various other nonlinear instability phenomena [5]. It is well known that a multiple scales analysis of the KdV equation (and, indeed a wide variety of equations) leads to the NLS equation for the modulated amplitude [6], [7], [8], [9], [10]. In [6], Zakharov and Kuznetsov showed a much deeper correspondence between these integrable equations not only at the level of the equation but also at the level of the linear spectral problem by showing that a multiple scales analysis of the Schrödinger spectral problem leads to the Zakharov–Shabat problem for the NLS equation. Dag and Özer also showed the same relationship between the NLS equation and integrable fifth-order NEEs [11]. On the other hand, similar methods could also be used in [12], [13], [14], [15].

Travelling waves appear in many physical structures in solitary wave theory such as solitons, kinks, compactons, peakons, cuspons, and others. In recent years, lots of methods have been improved to find exact travelling wave solutions of NEEs [16], [17]. Through these methods, the (G′G)-expansion method and the (G′G, 1G)-expansion method are very simple, convenient, and straightforward to construct the travelling wave solutions of NEEs. Zhang et al. [18] and Aslan [19] used the (G′G)-expansion method to address some physically important NLDDEs. Zhang et al. [20] proposed a generalised (G′G)-expansion method to improve and extend the works of Wang et al. [21] and Tang et al. [22] for solving variable-coefficient equations and high-dimensional equations. More recently, the (G′G)-expansion method has been applied to fractional differential difference equation to get new exact travelling wave solutions in the sense of modified Riemann Liouville derivative [23]. On the other hand, (G′G, 1G)-expansion method was first improved by Li et al. [24] to find travelling wave solutions of NEEs and then the method has been applied to different type equations in [25], [26].

2 Description of the Method of Multiple Scales for NEEs

Multiple scales method was proposed by Zakharov and Kuznetsov [6] to reduce the KdV equation into the NLS equation and apply to a class of NEEs. They showed that using this method, conventionally employed in the theory of nonlinear waves, integrable systems are reduced to other integrable systems. If the initial system is nonintegrable, the result can be either integrable or nonintegrable. But if we treat an integrable system properly, we must always get an integrable system as the result of our analysis. This is the main purpose in the application of this method to integrable systems.

In this article, we consider the application of the multiple scales method to NEEs. Using the technique of Zakharov and Kuznetsov [6], getting the NLS type equations from KdV type equations was shown step by step.

Let consider the general evolution equation

(1)ut = K(u, ux, uy …) (1)

General NEEs K [u] are a function of u and its derivatives with respect to the x-spatial variables. The well known of this type equations is KdV equation.

L [∂_x, ∂_y]u is the linear part of K [u]. So, using K [u], we can reach the dispersion relation for the (1). Substituting the plane wave solution

(2)uk = Aei(kx + ry − ω(k,r)t)≡ Aeiθ (2)

into the linear part of (1)

(3)ut = L[∂x, ∂y]u (3)

we get dispersion relation

(4)ω(k, r) = iL[ik, ir] (4)

Then, dispersion relation (4) is substituted in (1). We assume the following series expansions for the solution of the (1):

u(x, y, t) = ∑n = 1∞ϵnUn(x, y, t, ξ, η, τ)

Based on this solution, we also define slow spaces ξ, η and multiple time variable τ with respect to the scaling parameter ϵ>0, respectively, as follows. A nonlinear equation modulates the amplitude of this plane wave solution in such a way that may consider it depend on the slow variables:

(5)ξ = ϵ(x − dω(k, r)dkt)η = ϵ(y − dω(k, r)drt)τ = −12ϵ2(d2ω(k, r)dk2 + d2ω(k, r)dr2)t (5)

If we choose the slow variables different forms, we can derive higher-order NLS equations. The multiple scales analysis starts with the assumption:

(6)u(x, y, t) = U(x, y, t, ξ, η, τ) (6)

and solution of U is in the form

(7)U(x, y, t, ξ, η, τ) = ϵU1 + ϵ2U2 + ϵ3U3 + … (7)

Then, considering transformation (6) and solution (7), using dispersion relation (4) and slow variables (5), we get u and its derivatives with respect to ϵ in (1). And we substitute these terms with (6) and (7) in (1). Collecting all terms with the same order of ϵ together, the left-hand side of (1) is converted into a polynomial in ϵ. Then setting each coefficient of this polynomial to zero, we get a set of algebraic eqautions. Using wave solution space (2) and dispersion relation (4), these equations can be solved by iteration and by use of Maple. So, we can get NLS type equations from (1). In addition, from this procedure, we can reach numerical solutions of KdV type equations.

3 Description of the (G′G)- Expansion Method for NEEs

(G′G)-expansion method has been described first by Wang et al. [27] and in ([23], [28], [29], [30], [31], [32]), this method has been applied to a lot of different type nonlinear partial differential equations. (G′G)-expansion method is very practicable and effective to find travelling wave solutions of NEEs. Using this method, hyperbolic and trigonometric solutions of NEEs are get.

We can describe the (G′G)-expansion method step by step as follows:

Let consider nonlinear partial differential equation in the form

(8)P(u, ut, ux, uy, uyy, utt, uxx, …) = 0, (8)

where P is a polynomial of u(x, y, t) and its derivatives.

Step 1: Let consider

(9)u(x, y, t) = U(ξ), ξ = kx + ry − ct (9)

travelling wave transformation for the travelling wave solutions of (8). With this wave transformation, (8) can be reduced to

(10)Q(U, −cU′, kU′, rU′, c2U″, r2U″, k2U″, …) = 0. (10)

ordinary differential equation where U=U(ξ) and prime denotes derivatives with respect to ξ.

Step 2: We predict the solution of (10) in the finite series form

(11)U(ξ) = ∑l = 0mαl(G′(ξ)G(ξ))l, αm ≠ 0 (11)

where m and α_l’s are constants to be determined later, G(ξ) satisfies a second-order linear ordinary differential equation:

(12)d2G(ξ)dξ2 + λdG(ξ)dξ + μG(ξ) = 0, (12)

where λ and μ are arbitrary constants. Using the general solutions of (12), we get following cases:

(13)G′(ξ)G(ξ) = {λ2 − 4μ2(C1sinh(λ2 − 4μ2ξ) + C2cosh(λ2 − 4μ2ξ)C1cosh(λ2 − 4μ2ξ) + C2sinh(λ2 − 4μ2ξ)) − λ2, λ2 − 4μ > 04μ − λ22(−C1sin(4μ − λ22ξ) + C2cos(4μ − λ22ξ)C1cos(4μ − λ22ξ) + C2sin(4μ − λ22ξ)) − λ2, λ2 − 4μ < 0 (13)

where C₁ and C₂ are arbitrary constants.

Step 3: We can easily determine the degree m of (11) using the homogeneous balance principle for the highest order nonlinear term(s) and the highest order partial derivative of U(ξ) in (10).

Step 4: As a final step, substituting (11) together with (12) into (10) and collecting all terms with the same order of (G′(ξ)G(ξ)), the left-hand side of (3) is converted into a polynomial in (G′(ξ)G(ξ)). For example, for m=1 in (12) is in the form

d2G(ξ)dξ2 + λdG(ξ)dξ + μG(ξ) = 0 ⇒ G″(ξ) = − λG′(ξ) − μG(ξ)

where

(14)U(ξ) = a0 + a1(G′(ξ)G(ξ)) (14)

so we get

(15)U′ = a1(−λG′G − μ − (G′G)2) (15)

and

(16)U″ = a1(λμ + (λ2 + 2μ)(G′G) + 3λ(G′G)2 + 2(G′G)3)⋮ (16)

as polynomials of (G′(ξ)G(ξ)). In addition, we can obtain these for higher order values of m. As a final step, substituting (11) and (14–16) together with (12) into (10) and collecting all terms with the same order of (G′(ξ)G(ξ)) together, the left-hand side of (10) is converted into a polynomial in (G′(ξ)G(ξ)). Equating each coefficient of (G′(ξ)G(ξ))l (l=0, 1, 2, …) to zero yields a set of algebraic equations for “α_l (l=0, 1, 2, …, N), k, c”. Solving these algebraic equations system, we can define “α_l (l=0, 1, 2, …, N), k, c”. Finally, we substitute these values into expression (11) and obtain various kinds of exact solutions to (8) by use of Maple.

4 Description of the (G′G, 1G)- Expansion Method for NEEs

(G′G, 1G)-Expansion method was first improved by Li et al. [24] to find travelling wave solutions of NEEs and it can be considered as a generalisation of the original (G′G)-expansion method. Then this method was applied to another nonlinear partial differential equations by a lot of scientists [25], [26].

The key idea of the (G′G)-expansion method is that the exact solutions of NEEs can be expressed by a polynomial in one variable (G′(ξ)G(ξ)) in which G=G(ξ) satisfies a second-order linear ODE

(17)G″(ξ) + λG′(ξ) + μG(ξ) = 0 (17)

where λ and μ are constants. The key idea of the (G′G, 1G)-expansion method is that the exact travelling wave solutions of NEEs can be expressed by a polynomial in the two variables (G′(ξ)G(ξ)) and (1G(ξ)), in which G=G(ξ) satisfies a second-order linear ODE

(18)G″(ξ) + λG(ξ) = μ (18)

where λ and μ are constants. Using homogeneous balance procedure between the highest order derivatives and nonlinear terms in the NEE, we can determine the degree of the polynomial. Finally, using the method, we can find the coefficients of the polynomial and we reach the exact travelling wave solutions of the NEEs.

Before describe the main steps of the method, we consider the following remarks:

Let

(19)ϕ = (G′(ξ)G(ξ)) and ψ = (1G(ξ)) (19)

Using (18) and (19), we get

(20)ϕ′ = G″G − G′G′G2= G″G − (G′G)2= μ − λGG − ϕ2= μψ − λ − ϕ2 (20)

and

(21)ψ′ = −G′G2 = −ϕψ (21)

Then, considering the general solution of linear ordinary differential equation (18), we get

(22){G(ξ) = C1sinh(−λξ) + C2cosh(−λξ) + μλ, λ < 0G(ξ) = C1sin(λξ) + C2cos(λξ) + μλ, λ > 0 (22)

where C₁ and C₂ are arbitrary constants. Thus, we get

(23){ψ2 = −λ(ϕ2 − 2μψ + λ)λ2σ + μ2, σ = C12 − C22 and λ < 0ψ2 = −λ(ϕ2 − 2μψ + λ)λ2σ − μ2, σ = C12 + C22 and λ > 0 (23)

Let consider nonlinear partial differential equation in the form

(24)F(u, ux, uy, ut, uxx, utt, uxt, …) = 0 (24)

where F is a polynomial of u and its partial derivatives.

Step 1: Using travelling wave transformation

u(x, y, t) = U(ξ), ξ = kx + ry − ct

nonlinear partial differential equation (24) can be reduced to ordinary differential equation

(25)P(U, U′, U″, …) = 0 (25)

where P is a polynomial of U and its total derivatives, while (′ = ddξ).

Step 2: Solution of the (25) can be expressed by a polynomial in the variables in ϕ ve ψ as follows:

(26)U(ξ) = ∑i = 0Naiϕi + ∑i = 1Nbiϕi − 1ψ (26)

where a_i (i=0, 1, 2, …, N) and b_i (i=0, 1, 2, …, N) are constants to be determined later.

Step 3: According to the homogeneous balance procedure, balancing the highest order derivatives, and the nonlinear terms in (25), we can find the positive integer N in (26).

Step 4: Substituting (26) into (25) along with (20), (21), and (23), the left-hand side of (25) convert to a polynomial in ϕ and ψ, in which the degree of ψ is not longer than 1. Then, equating the coefficients of this polynomial to zero, we get a system of algebraic equations.

Step 5: Solving the algebraic equation system with the aid of Maple, we find a_i, b_i, k, r, c, μ, C₁C₂ values for λ<0, λ>0. Finally substituting this values in (25), hyperbolic and trigonometric solutions can be found, respectively.

5 Applications

In this section, we apply (G′G)-expansion method and (G′G, 1G)-expansion method summarised in Section 2 to (2+1)-dimensional KdV4 equation.

5.1 (2+1)-Dimensional KdV4 Equation

Using symmetries of KdV equation, a new class of partial differential equations was obtained in (2+1) dimension by Gürses and Pekcan [33]. Thus, the authors presented a useful work and developed (2+1)-dimensional generalisation of the KdV equations. Then using this work, (2+1) dimensional KdV4 equation has been obtained by Wazwaz [34] as follows:

(27)uxy + uxxxt + uxxxx + 3(ux2)x + 4uxuxt + 2uxxut = 0 (27)

5.1.1 Method of Multiple Scales

(2+1)-Dimensional KdV4 equation is in the form (27). To find dispersion relation for (27), we consider the linear part of (27) in the form

(28)uxy + uxxxt + uxxxx = 0 (28)

and linear differential equation (28) satisfies the solution

(29)u(x, y, t) = eiθ, θ = kx + ry − ω(k, r)t (29)

Substituting the solution (29) into the linear differential equation (28), we get

⇒i2kr + i4k3(−ω) + i4k4 = 0⇒ − kr − ωk3 + k4 = 0

and from this we reach the

(30)ω(k, r) = k − rk2 (30)

dispersion relation. Thus, the solution of linear differential equation (28) is as follows:

(31)u(x, y, t) = ei(kx + ry − (k − rk2)t) (31)

Let the solution of (27) is in the form

(32)u(x, y, t) = U(x, y, t, ξ, η, τ), U(x, y, t, ξ, η, τ) = ϵU1 + ϵ2U2 + ϵ3U3 + … (32)

and slow variables are in the form

(33)ξ = ϵ(x − dω(k, r)dkt)= ϵ(x − (1 + 2rk3)t)η = ϵ(y − dω(k, r)drt)= ϵ(y + 1k2t)τ = −12ϵ2(d2ω(k, r)dk2t + d2ω(k, r)dr2t)= −3rϵ2k4t (33)

Then using (30)–(33), the total derivatives in (27) are obtained as follows:

(34)Dx = (∂x + ϵ∂ξ)Dxx = ∂xx + 2ϵ∂ξx + ϵ2∂ξξDxxx = ∂xxx + 3ϵ∂ξxx + 3ϵ2∂ξξx + ϵ3∂ξξξDxxxx = ∂xxxx + 4ϵ∂ξxxx + 6ϵ2∂ξξxx + 4ϵ3∂ξξξx + ϵ4∂ξξξξ (34)

(35)Dy = (∂y + ϵ∂η)Dt = (∂t − (1 + 2rk3)ϵ∂ξ + ϵk2∂η + 3rk4ϵ2∂τ)Dxy = (∂xy + ϵ∂xη + ϵ∂yξ + ϵ2∂ξη)Dxt = ∂xt − (1 + 2rk3)ϵ∂xξ + 3rk4ϵ2∂xτ + ϵk2∂xη + ϵ∂tξ −ϵ2(1 + 2rk3)∂ξξ + 3rk4ϵ3∂ξτ + ϵ2k2∂ξηDxxxt = ∂xxxt − (1 + 2rk3)ϵ∂xxxξ + 3rk4ϵ2∂xxxτ + ϵk2∂xxxη +3ϵ∂xxtξ − 3ϵ2(1 + 2rk3)∂xxξξ + 9rk4ϵ3∂xxξτ + 3ϵ2k2∂xxξη+ 3ϵ2∂xtξξ − 3ϵ3(1 + 2rk3)∂xξξξ + 9rk4ϵ4∂xξξτ + 3ϵ3k2∂xξξη+ ϵ3∂tξξξ − ϵ4(1 + 2rk3)∂ξξξξ + 3rk4ϵ5∂ξξξτ + ϵ4k2∂ξξξη (35)

Substituting (32), (34), and (35) into the (27), we get a polynomial in ϵ. Equating each coefficients of this polynomial to zero, we find

(36)ϵ : u1xxxx + u1xxxt + u1xy = 0 (36)

(37)ϵ2 : u2xxxx + u2xxxt + u2xy + u1xη + u1yξ + 4u1xxxξ + 3u1ξxxt + 6u1xu1xx + 4u1xu1xt + 2u1xxu1t − (1 + 2rk3)u1xxxξ + 1k2u1xxxη = 0 (37)

(38)ϵ3 : u3xxxx + u3xxxt + u3xy + 3rk4u1xxxτ − 3(1 + 2rk3)u1xxξξ+ 3k2u1ξxxη + u1ξη + u2yξ + 4u2ξxxx + 6u1ξξxx + 3u2ξxxt+ 6u1x(u2xx + 2u1ξx)6u1xx(u2x + u1ξ) + 4u1x(u2xt +1k2u1xη − (1 + 2rk3)u1ξx + u1ξt) + 4u1xt(u2x + u1ξ) + u2xη+ 2u1xx(u2t + 1k2u1η − (1 + 2rk3)u1ξ) + 3u1ξξxt + 2u1t(u2xx+ 2u1ξx) − (1 + 2rk3)u2ξxxx + 1k2u2xxxη = 0 (38)

(39)⋮ (39)

Then, we can find the solution of (36) as follows:

(40)u1(x, y, t, ξ, η, τ) = v1(ξ, η, τ)ei(kx + ry − (k − rk2)t) + c.c. (40)

where c.c. is complex conjugate of v₁. Substituting the solution (40) into (37), the solution of (37) is in the form

(41)u2(x, y, t, ξ, η, τ) = v2(ξ, η, τ)e2i(kx + ry − (k − rk2)t) + c.c.+ f(ξ, η, τ) (41)

where f (ξ, η, τ) is integration constant. Thus, we get

(42)v2(ξ, η, τ) = iv12(ξ, η, τ)2k, v−2(ξ, η, τ) = −iv−12(ξ, η, τ)2k (42)

where v₋₁ is the complex conjugate of v₁ and v₋₂ is the complex conjugate of v₂. Substituting solutions (40)–(42) into the (38), we find the solution of (38) in the form

(43)u3(x, y, t, ξ, η, τ) = v3(ξ, η, τ)e3i(kx + ry − (k − rk2)t) +f2(ξ, η, τ)e2i(kx + ry − (k − rk2)t) + c.c. (43)

where f₂ (ξ, η, τ) is integration constant. Then, we get

(44)v3(ξ, η, τ) = −v13(ξ, η, τ)4k2, v−3(ξ, η, τ) = −v−13(ξ, η, τ)4k2 (44)

and

(45)f2(ξ, η, τ) = −v1v1ξ2k2f3(ξ, η, τ) = −v−1v−1ξ2k2f(ξ, η, τ) = 0 (45)

where v₋₃ and f₃ are the complex conjugates of v₃ and f₂, respectively. Thus, the solutions of (36)–(38) are obtained as

(46)u1 = v1(ξ, η, τ)eiθ + c.c.u2 = i(2k)−1v12(ξ, η, τ)e2iθ + c.c.u3 = −12(k)−2v1(ξ, η, τ)v1ξ(ξ, η, τ)e2iθ −(2k)−2v13(ξ, η, τ)e3iθ + c.c (46)

where

(47)θ = (kx + ry − (k − rk2)t) (47)

Finally, substituting the solutions (46) into (38), we get

(48)⇒iv1τ = v1ξξ − 2v12v−1 − 2k3rv1ξη⇒iv1τ = v1ξξ − 2v1|v1|2 − 2k3rv1ξη (48)

and

(49)⇒−iv−1τ = v−1ξξ − 2v−12v1 − 2k3rv−1ξη⇒−iv−1τ = v−1ξξ − 2v−1|v−1|2 − 2k3rv−1ξη (49)

Describing as q=v₁, q^*=v₋₁ and (λ = 2k3r), from (48) and (49), we get the (2+1) dimensional NLS type equations in the form

(50)iqτ = qξξ − 2q|q|2 − λqξη (50)

and

(51)−iqτ∗ = qξξ∗ − 2q∗|q∗|2 − λqξη∗ (51)

In addition, numerical solution of the (2+1)-dimensional KdV4 equation (27) is found as

(52)u(x, y, t) = ϵkq(ξ, η, τ)eiθ + iϵ2(2k)−1q2(ξ, η, τ)e2iθ− 12k−2ϵ3q(ξ, η, τ)qξ(ξ, η, τ)e2iθ −(2k)−2ϵ3q13(ξ, η, τ)e3iθ + c.c. + … (52)

where q is solution of NLS equation [35].

5.1.2 (G′G)-Expansion Method

Using the travelling wave transformation,

u(x, y, t) = U(ξ), ξ = kx + ry − ct

(27) turns into

(53)krU″ − k3cU(4) + k4U(4) + 6k2(k − c)U′U″ = 0 (53)

ordinary differential equation. Suppose that the travelling wave solution of (53) is in the form

U(ξ) = ∑l = 0mαl(G′(ξ)G(ξ))l, αm ≠ 0

According the homogeneous balance procedure, in (53) balancing the highest order derivative term U⁽⁴⁾ and the highest order nonlinear term U′U″, we get

(54)U(4) ∼ U′U″ (54)

(55)⇒m + 4 = m + 1 + m + 2⇒m = 1 (55)

Thus substituting (55) into (11), the travelling wave solution of (53) is found as

(56)U(ξ) = α0 + α1(G′(ξ)G(ξ)), α1 ≠ 0 (56)

where α_l (l=0, 1) are constants to be determined later and G(ξ) satisfies second-order linear ordinary differential equation

(57)G″(ξ) + λG′(ξ) + μG(ξ) = 0, (57)

where λ and μ are arbitrary constants. Using (57) and (56), the derivatives in (53) are obtained as follows:

(58)U′ = −α1(λG′G + μ + G′2G)U″ = λ2G′G + α1λμ − μG′G + 3α1λG′2G + 3α1G′Gμ + 2α13G′G (58)

(59)U(4) = α1(15λ3G′2G + λ4G′G + λ3μ + 22λ2μG′G + 60λμG′2G+ 8λμ2 + 16μ2G′G + 50G′3Gλ2 + 40G′3Gμ + 60G′4Gλ+ 24G′5G) (59)

Substituting these terms together with (56) and (57), into (53), clearing the denominator and setting the coefficients of (G′(ξ)G(ξ))i to zero we have algebraic system for “α₀, α₁, k, r, c” in the form:

(60)(G′(ξ)G(ξ))0 : 6k2(c − k)μ2λα12 − kλμ(−8k3μ − k3λ2 +8k2cμ + k2cλ2 − r)α1(G′(ξ)G(ξ))1 : 12k2μ(λ2 + μ)(c − k)α12 − k(−k3λ4 −22k3μλ2 − 16k3μ2 + 22k2μcλ2 + 16k2μ2c+k2cλ4 − rλ2 − 2rμ)α1(G′(ξ)G(ξ))2 : 6k2λ(λ2 + 6μ)(c − k)α12 − 3kλ(−5k3λ2 −20k3μ + 5k2cλ2 + 20k2cμ − r)α1(G′(ξ)G(ξ))3 : 24k2(λ2 + μ)(c − k)α12 − 2k(−25k3λ2 −20k3μ + 25k2cλ2 + 20k2cμ − r)α1(G′(ξ)G(ξ))4 : 30k2(c − k)λα12 − 60k3(c − k)λα1(G′(ξ)G(ξ))5 : 12k2(c − k)α12 − 24k3(c − k)α1 (60)

Solving the algebraic equation system (60) for α₀, α₁, k, r, c by use of Maple, we get

(61)α0 = α0, α1 = 2k,c = c, k = k, r = k2(c − k)(λ2 − 4μ) (61)

Thus, hyperbolic and trigonometric solutions of (53) are found as follows [35]:

For λ²–4μ>0,
Substituting (61) and (13) into (56), we get hyperbolic function solutions of (53)
(62)U(ξ) = α0 + k(λ2 − 4μC1sinh(λ2 − 4μ2ξ) + C2cosh(λ2 − 4μ2ξ)C1cosh(λ2 − 4μ2ξ) + C2sinh(λ2 − 4μ2ξ) − λ) (62)
where C₁ and C₂ are arbitrary constants. From (62) using ξ=(kx+k²(c–k)(λ²–4μ)–ct), we obtain
u(x, y, t) = α0 + k(λ2 − 4μv(x, y, t) − λ)
where
v(x, y, t) = C1sinh(λ2 − 4μ2(kx + k2(c − k)(λ2 − 4μ) − ct)) + C2cosh(λ2 − 4μ2(kx + k2(c − k)(λ2 − 4μ) − ct))C1cosh(λ2 − 4μ2(kx + k2(c − k)(λ2 − 4μ) − ct)) + C2sinh(λ2 − 4μ2(kx + k2(c − k)(λ2 − 4μ) − ct))
For λ²–4μ<0,
Substituting (61) and (13) into (56), we find trigonometric function solutions of (53)

(63)U(ξ) = α0 + k(4μ − λ2−C1sin(4μ − λ22ξ) + C2cos(4μ − λ22ξ)C1cos(4μ − λ22ξ) + C2sin(4μ − λ22ξ) − λ) (63)

where C₁ and C₂ are arbitrary constants. From (63) using ξ=(kx+k²(k–c)(4μ–λ²)–ct), we obtain

u(x, y, t) = α0 + k(4μ − λ2 v(x, y, t) − λ)

where

v(x, y, t) = −C1sin(4μ − λ22(kx + k2(k − c)(4μ − λ2) − ct)) + C2cos(4μ − λ22(kx + k2(k − c)(4μ − λ2) − ct))C1cos(4μ − λ22(kx + k2(k − c)(4μ − λ2) − ct)) + C2sin(4μ − λ22(kx + k2(k − c)(4μ − λ2) − ct))

5.1.3 (G′G, 1G)-Expansion Method

Using the travelling wave transformation

(64)u(x, y, t) = U(ξ), ξ = kx + hy − ct (64)

(27) turns into

(65)khU″ + k3(k − c)U(4) + 6k2(k − c)U′U″ = 0 (65)

ordinary differential equation. Assume that the solution of (65) is in the form (26). Then using the homogeneous balance procedure like (54) and (55), we find

N = 1

Then the solution of (65) is found as

(66)U(ξ) = a0 + a1ϕ(ξ) + b1ψ(ξ) (66)

where “a₀, a₁, b₁” are constants to be determined later. Using (66) and

(67)ψ′ = −ϕψϕ′ = μψ − λ − ϕ2 (67)

we get the derivatives in (65) as follows:

(68)U′ = −a1ϕ2 + a1μψ − a1λ − b1ϕψU″ = 2a1ϕ3 − 3a1μϕψ + 2a1ϕλ + 2b1ϕ2ψ − b1ψ2μ + b1ψλU(4) = −60μϕ3ψ + 30a1μ2ϕψ2 − 36b1μϕ2ψ2 + 28b1λϕ2ψ −11b1μλψ2 − 45a1μλϕψ + 24b1ϕ4ψ + 40a1ϕ3λ +16a1ϕλ2 + 6b1μ2ψ3 + 5b1ψλ2 + 24a1ϕ5 (68)

Thus, new and different types hyperbolic and trigonometric function solutions for (2+1) dimensional KdV4 equation are obtained for λ<0 and λ>0 [35].

For λ<0;
Substituting (66), (68), and
(69)ψ2 = −λ(ϕ2 − 2μψ + λ)λ2ν + μ2, ν = C12 − C22 and λ < 0 (69)
into (65) and using (69), it can be seen that the powers of ψ are not higher than 1 in (65). Thus, the left-hand side of (65) becomes a polynomial in ϕ and ψ. Setting the each coefficient of equation to zero yields a system of algebraic equations:
(70)ϕ0: b1λ2μ(k2cλμ2 − k3λμ2 + hμ2 − 11k2cλ3ν + 11k3λ3ν+ 12λ3ka1ν(c − k) + hλ2ν)ψ: −b1λ(6λ5ka1ν2(k − c) + 5k2λ5ν2(c − k) − hλ4ν2 + 18λ3k2μ2ν(k − c) + 18λ3ka1μ2ν(c − k) + k2λμ4(c − k)+ k2λμ4(c − k) + hμ4)ϕ: 2λ(6ka12λ5ν2(c − k) + 8k2a1λ5ν2(k − c) + 3kb12λ4ν(k − c) + ha1λ4ν2 + 3ka12μ2λ3ν(c − k) + k2a1λ3μ2ν(k − c) + 2ha1μ2λ2ν + 3kb12λ2μ2(c − k) + 7k2a1λμ4(c − k) − 3kca12λμ4 + ha1μ4)ϕψ: −3μ(5k2a1λμ4(c − k) + 2ka12λμ4(k − c) + 8ka12λ3νμ2(c− k) + 2kb12λ2μ2(c − k) + 10ka12λ5ν2(c − k) + (k − c)10k2a1λ3μ2ν + 6kb12λ4ν(k − c) + 15k2a1λ5ν2(k − c)+ ha1(λ4ν2 + μ4)ϕ2: −b1μλ(35k2λμ2(c − k) + 36μ2λa1k(k − c) + 47k2λ3ν(c− k) + 48λ3kνa1(k − c) − hμ2 − hλ2ν)ϕ2ψ: 2b1(λ2ν + μ2)(24ka1λμ2(k − c) + 25k2λμ2(c − k) + 14k2λ3ν(k − c) + 15ka1λ3ν(c − k) + hμ2 + hλ2νϕ3: 18kb12λ4ν(k − c) + 6kb12λ2μ2(k − c) + 6ka12λμ4(c − k) + 2ha1μ4 + 10k2a1λμ4(k − c) + 40k2a1λ5ν2(k − c) − 24ka12λ5ν2(k − c) + 50k2a1λ3μ2ν(k − c) + 2ha1λ2ν(λ2ν+ 2μ2) − 30ka12μ2λ3ν(k − c)ϕ3ψ: 30k(λ2ν + μ2)μ(2μ2ka1 − a12(λ2ν + μ2) + 2ka1λ2ν + b12λ)(c − k)ϕ4: −36kb1(λ2ν+μ2)(−a1+k)(c−k)μλϕ4ψ: −24kb1(λ2ν + μ2)2(−a1 + k)(c − k)ϕ5: −12k(λ2ν + μ2)((λ2ν + μ2)(2ka1 − a12) + b12λ)(c − k) (70)
Solving the algebraic equations by Maple, we find
(71)a0 = a0, a1 = a1, b1 = ±−λ2ν + μ2λa1k = a1, h = (a1 − c)λa12, c = c (71)
From (71), we get
(72)ϕ(ξ) = −λ(A1cosh(−λξ) + A2sinh(−λξ))(A1sinh(−λξ) + A2cosh(−λξ)) + μλψ(ξ) = 1A1sinh(−λξ) + A2cosh(−λξ) + μλ (72)
where (ϕ, ψ) = (G′G, 1G) and
(73)G(ξ) = A1sinh(−λξ) + A2cosh(−λξ) + μλ, λ < 0 (73)
Finally substituting (71) and (72) into (66), the exact travelling wave solution of (65) is found as
U(ξ) = a0 + a1−λ(A1cosh(−λξ) + A2sinh(−λξ))(A1sinh(−λξ) + A2cosh(−λξ)) + μλ± −λ2ν + μ2λa11A1sinh(−λξ) + A2cosh(−λξ) + μλ
where
(74)ξ = a1x + (a1 − c)λa12y − ct (74)
And using (64) hyperbolic solution of (27) is obtained as follows:
u(x, y, t) = a0 + a1.v(x, y, t)
where
v(x, y, t) = (−λ(A1cosh(−λ(a1x + (a1 − c)λa12y − ct)) +A2sinh(−λ(a1x + (a1 − c)λa12y − ct))) ± −λ2ν + μ2λ)/.((A1sinh(−λ(a1x + (a1 − c)λa12y − ct)) +A2cosh(−λ(a1x + (a1 − c)λa12y − ct))) + μλ)
For λ>0;
Substituting (66), (68) and
(75)ψ2 = −λ(ϕ2 − 2μψ + λ)λ2σ − μ2, σ = C12 + C22 and λ > 0 (75)

into (65) and using (75), it can be seen that the powers of ψ are not higher than 1 in (65). Thus, the left-hand side of (65) becomes a polynomial in ϕ and ψ. Setting the each coefficient of equation to zero yields a system of algebraic equations:

ϕ0ψ0: b1λ2μ((23k2λμ2 − 24kλμ2a1 − 11k2λ3ν + 12λ3ka1ν)(c − k) + h(λ2ν − μ2))(λ2ν + μ2)2ψ: −b1λ((57k2λμ4 − 60λka1μ4 + 5k2λ5ν2 − 6ka1λ5ν2 − 38λ3k2μ2ν + 42λ3ka1μ2ν)(c − k) − h(3μ4 + λ4v2))(λ2ν + μ2)2ϕ: −2λ((8k2a1λ5ν2 − 6ka12λ5ν2 + 3kb12λ4ν − 31k2a1μ2λ3ν + 21ka12λ3μ2ν − 9kb12λ2μ2 + 23k2a1λμ4 − 15ka12μ4λ)(c − k)+ ha1λ2v(2μ2 − λ2v) − ha1μ4)(λ2ν + μ2)2ϕψ: 3μ((35k2a1λμ4 − 22ka12λμ4 − 50k2a1λ3μ2ν − 14kb12λ2μ2+ 32ka12λ3νμ2 − 10ka12λ5ν2 + 6kb12λ4ν + 15k2a1λ5ν2)(c − k)+ ha1λ2ν(2μ2 − λ2ν) − ha1μ4)(λ2ν + μ2)2ϕ2: b1μλ((59k2λμ2 − 60μ2λa1k + 48λ3kνa1 − 47k2λ3ν)(c − k)− hμ2 + hλ2ν)(λ2ν + μ2)2ϕ2ψ: −2b1(−λ2ν + μ2)((−54ka1λμ2 + 53k2λμ2 − 14k2λ3ν + 15ka1λ3ν)(c − k) − hμ2 + hλ2ν)(λ2ν + μ2)2ϕ3: −2((35k2a1λμ4 + 20k2a1λ5ν2 − 12ka12λ5ν2 − 21ka12λμ4 − 55k2a1λ3μ2ν + 33ka12μ2λ3ν + 9kb12λ4ν − 15kb12λ2μ2)(c − k)+ ha1λ2ν(2μ2 − λ2ν) − ha1μ4)(λ2ν + μ2)2ϕ3ψ: 30k(−λ2ν + μ2)μ(2μ2ka1 + a12(λ2ν − μ2) − 2ka1λ2ν− b12λ)(c − k)(λ2ν + μ2)2ϕ4: 36kb1(−λ2ν + μ2)(−a1 + k)(c − k)μλ(λ2ν + μ2)2ϕ4ψ: −24kb1(−λ2ν + μ2)2(−a1 + k)(c − k)(λ2ν + μ2)2ϕ5: −12k(−λ2ν + μ2)(−a1 + k)(c − k)(λ2ν + μ2)2((2k − a1)a1(−λ2ν + μ2) − b12λ)

Solving the algebraic equation systems by use of Maple, we get

(76)a0 = a0, a1 = a1, b1 = ±−λ2ν − μ2λa1k = a1, h = (a1 − c)λa12, c = c (76)

Then, we find

(77)ϕ(ξ) = λ(A1cos(λξ) − A2sin(λξ))(A1sin(λξ) + A2cos(λξ)) + μλψ(ξ) = 1(A1sin(λξ) + A2cos(λξ)) + μλ (77)

where (ϕ, ψ) = (G′G, 1G) and

(78)G(ξ) = A1sin(λξ) + A2cos(λξ) + μλ, λ > 0 (78)

Substituting (76) and (77) into (66), the exact travelling wave solution of (65) is found as

(79)U(ξ) = a0 + a1λ(A1cos(λξ) − A2sin(λξ))(A1sin(λξ) + A2cos(λξ)) + μλ± −λ2ν − μ2λa11(A1sin(λξ) + A2cos(λξ)) + μλ (79)

where

(80)ξ = a1x + (a1 − c)λa12y − ct (80)

Finally using (64), trigonometric solution of (27) is obtained in the form

u(x, y, t) = a0 + a1.v(x, y, t)

where

v(x, y, t) = (λ(A1cos(λ(a1x + (a1 − c)λa12y − ct)) −A2sin(λ(a1x + (a1 − c)λa12y − ct))) ± −λ2ν − μ2λ)/.((A1sin(λ(a1x + (a1 − c)λa12y − ct))+ A2cos(λ(a1x + (a1 − c)λa12y − ct))) + μλ)

6 Conclusion

The solutions of NEEs have many potential applications in physics and engineering. Through this type NEEs, we investigate the solutions of (2+1)-dimensional KdV4 equation and the relation between KdV4 equation and NLS equation. Applying the method of multiple scales to (2+1)-dimensional KdV4 equation, we get (2+1)-dimensional NLS type equations. Moreover, numerical solutions have been obtained for KdV4 equation by this method. In this article, we only studied on deriving NLS type equations from KdV type equations and their solutions by use of multiple scales method. In addition, we can seek the application of the method to the spectral problems and recursion operators of KdV4 equation in further works. By this way, we can get the spectral problems and recursion operators of NLS equation. At the same time if we choose the slow variables different forms, we can derive higher-order NLS type equations. Hovewer, this method can be applied not only this type NEEs but also nonlinear differential difference equations and higher-order NEEs as a future work. On the other hand, we have used the (G′G)-expansion method and (G′G, 1G)-expansion method to find the exact travelling wave solutions of KdV4 equation in this work. We concluded that the performances of these methods are reliable, simple, and these methods give many new trigonometric and hyperbolic type exact solutions. Finally, it is worth mentioning that the implementation of these proposed methods is very simple and straightforward, and it can also be applied to many other NEEs, differential difference equations, and fractional differential equations. The details about these methods and their applications to other NEEs are given in [35].

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Received: 2016-3-30

Accepted: 2016-6-2

Published Online: 2016-6-24

Published in Print: 2016-8-1

Articles in the same Issue

https://doi.org/10.1515/zna-2016-0123

Keywords for this article

Exact Travelling Wave Solution; (G′G)$\left( {{{G'} \over G}} \right)$-Expansion Method; (G′G, 1G)$\left( {{{G'} \over G},{\rm{ }}{1 \over G}} \right)$-Expansion Method; Method of Multiple Scales