The Derivation and Study of the Nonlinear Schrodinger ...internonlinearscience.org/upload/papers/20110308101224985.pdf · moving with a ﬁxed velocity. Of particular interest are

ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.9(2010) No.4,pp.430-443

The Derivation and Study of the Nonlinear Schrodinger Equation for Long Wavesin Shallow Water Using the Reductive Perturbation and Complex Ansatz Methods

A. M. Abourabia1 ∗, K. M. Hassan2, E. S. Selima31,2,3Department of Mathematics, Faculty of Science, Menoufiya University, Shebin El-koom 32511, Egypt

(Received 13 November 2009, accepted 10 June 2010)

Abstract: In this paper, the water wave flow problem for an incompressible and inviscid fluid of constantdepth is studied under the influence of acceleration of gravity and surface tension. The nonlinear Schrodinger(NLS) equation and the dispersion relation are derived from the nonlinear shallow water equations by usingthe reductive perturbation technique, which differs from the derivations of the same problem illustrated inprevious works. The complex ansatz method is presented for constructing exact traveling wave solutions ofNLS equation and from them the physical variables of the water wave problem are obtained. Depending onthe Ursell parameter, the diagrams are drawn to illustrate the behavior of the solutions of NLS equation, freesurface elevation and velocity of the model. The results indicate that the solutions of this problem are in theform of envelope traveling solitary waves where the Ursell parameter affects the wave profile significantly. Itis concluded the chosen methods of solutions are sufficiently accurate to demonstrate that the conservationof power is satisfied except at very few spots where it fluctuates about zero by the orders of 10−14 − 10−18

depending on the values of the Ursell parameter.

Keywords: shallow water equations; reductive perturbation method; nonlinear Schrodinger equation; com-plex ansatz method

1 IntroductionMany phenomena in physics and other applied fields are described by nonlinear partial differential equations (PDEs).To understand the physical situation of these phenomena in nature, we need to find the exact solutions of the PDEs,which have become one of the most important topics in mathematical physics. In the study of equations modeling wavephenomena, one of the fundamental objects of study is the traveling wave solution, meaning a solution of constant formmoving with a fixed velocity. Of particular interest are three types of traveling waves: solitary waves, periodic waves andkink waves [1-9].

The problem of nonlinear waves propagating in plasma and fluid can be described by the KdV, nonlinear Schrodinger,Davy-Stewartson equations and others. These last equations are derived by multiple scale, reductive perturbation methodsand other asymptotic methods [10-19].

Due to its central importance to the theory of quantum mechanics, the nonlinear equation of Schrodinger type has agreat interest. They arise in many physical problems, including nonlinear water waves, ocean waves, waves in plasma,propagation of heat pulses in a solid self trapping phenomenon in nonlinear optics, nonlinear waves in a fluid-filled vis-coelastic tube, and various nonlinear instability phenomena in fluids and plasma, and are of importance in the developmentof solitons and inverse scatting transform theory [1-4].

In the past the exact solution of Schrodinger type was obtained by converting these equations into real forms throughsome transformations and then using some methods such as Jacobi elliptic expansion, tanh-function method, Cole-Hopftransformation, Hirota bilinear method, inverse scattering method and so on. Recently, direct methods are proposed toobtain the exact wave solutions of these equations such as complex tanh-function method, complex hyperbolic-functionmethod, sub-ODE method, complex ansatz method, complex Jacobi elliptic method and others [1, 2, 20-30].

∗Corresponding author. E-mail address: am [email protected]. Tel.: +2010-1382826; fax: +2 048-2235689.dr [email protected](K. Hassan), [email protected](E. Selima).

Copyright c⃝World Academic Press, World Academic UnionIJNS.2010.06.30/370

A. Abourabia, K. Hassan, E. Selima: The Derivation and Study of the Nonlinear Schrodinger Equation for Long Waves ⋅ ⋅ ⋅ 431

In this paper we study the irrotational incompressible flow of a shallow layer of inviscid fluid moving under theinfluence of gravity as well as surface tension. Previously Dullin et al studied this case without surface tension, whichin the shallow water approximation leads to the Camassa-Holm (CH) equation [7]. Also Dullin et al derived the CHas a shallow water wave equation in an asymptotic expansion that extends one order beyond the Korteweg-de Vriesequation (KdV) and they showed that CH is asymptotically equivalent to the fifth-order integrable equation in the KdVhierarchy by using the Kodama transformation [8, 9]. Lizuka and Wadati applied the reductive perturbation method tothe two-dimensional Rayleigh-Tayler problem for the interface between light and heavy inviscid incompressible fluids byincluding the effect of surface tension and derived stable and unstable NLS equations and nonlinear diffusion equation[10]. Abourabia et al. studied the problem of water waves that propagating at the interface between two inviscid fluids byusing the multiple scale method to obtain the NLS equation [19].

The paper consists of the following. In section 2, we describe and formulate the water wave problem, and then study itsgoverning equations. In section 3, the NLS equation and dispersion properties for the nonlinear shallow water equationsare derived by the reductive perturbation technique. In section 4, the complex ansatz method is applied to the obtainedNLS equation to find the solutions of the problem. Section 5 is devoted for the conclusions.

2 Theoretical formulation of the basic equations for the problemLet us consider a shallow water wave propagating in a finite depth water that has a free surface. So we assume anincompressible, inviscid fluid of constant undisturbed depth ℎ0 and constant density 𝜌 with acceleration gravity 𝑔 andsurface tension . We assume also that 𝑥− 𝑦 plane is the undisturbed free surface with the 𝑧-axis positive upward. The freesurface elevation above the undisturbed depth is 𝑧 = 𝜂(𝑥, 𝑦, 𝑡), so that the free surface is at 𝑧 = ℎ0+𝜂 and the horizontalflat bottom is at 𝑧 = 0. Denote by 𝑢 and 𝑣 the horizontal and vertical velocity components, respectively.

The velocity potential 𝜑(𝑥, 𝑦, 𝑧, 𝑡) and the free surface elevation 𝜂(𝑥, 𝑦, 𝑡) are governed by the following Laplaceequation and boundary conditions:

∇2𝜑 = 𝜑𝑥, 𝑥 + 𝜑𝑦, 𝑦 + 𝜑𝑧, 𝑧 = 0, 0 < 𝑧 < ℎ0 + 𝜂, −∞ < 𝑥, 𝑦 < ∞. (1)

The dynamic and kinematic conditions at the free surface are

𝜑𝑡 + 𝑔 𝜂 +1

2(∇𝜑)

2 − Γ

𝜌(𝜂𝑥, 𝑥 + 𝜂𝑦, 𝑦) = 0, on 𝑧 = ℎ0 + 𝜂, (2)

𝜂𝑡 + 𝜂𝑥 𝜑𝑥 + 𝜂𝑦 𝜑𝑦 − 𝜑𝑧 = 0, on 𝑧 = ℎ𝑜 + 𝜂. (3)

The boundary condition at the flat bottom is

𝜑𝑧 = 0, on 𝑧 = 0, (4)

here and hereafter, the subscripts 𝑥, 𝑦, 𝑧 and 𝑡 denote the partial derivatives.These equations for a fluid will be written in a nondimensional form by introducing the following flow variables which

is based on different length scales, a typical horizontal wave length 𝜆0 and a typical vertical length ℎ0 [1, 2, 8, 9, 31]

(𝑥, 𝑦) = 𝜆0 (��, 𝑦), 𝑧 = ℎ0 𝑧, 𝑡 =𝜆0

𝑐0𝑡, 𝜂 = 𝑎 𝜂, 𝜑 =

𝑎𝜆0 𝑐0ℎ0

𝜑, (5)

where 𝑎 is the surface wave amplitude, 𝑐0 =√𝑔 ℎ0 is a typical horizontal velocity (shallow water wave speed), and the

bars refer to the nondimensional variables.We also introduce two fundamental parameters to characterize the nonlinear shallow water waves namely 𝜀 = 𝑎/ℎ0

as a measure of nonlinearity and 𝛿 = ℎ20

/𝜆20 as a measure of dispersion (shallowness) [1].

Introducing (5) into (1)-(4), the governing equations can be expressed in nondimensional form, dropping the bars, asfollows

𝛿 (𝜑𝑥, 𝑥 + 𝜑𝑦, 𝑦) + 𝜑𝑧, 𝑧 = 0, 0 < 𝑧 < 1 + 𝜀 𝜂, (6)

𝜑𝑡 + 𝜂 +𝜀

2

(𝜑2𝑥 + 𝜑2

𝑦

)+

𝜀

2 𝛿𝜑2𝑧 − 𝜎 𝛿 (𝜂𝑥, 𝑥 + 𝜂𝑦, 𝑦) = 0, on 𝑧 = 1 + 𝜀 𝜂, (7)

𝛿 (𝜂𝑡 + 𝜀 (𝜂𝑥 𝜑𝑥 + 𝜂𝑦 𝜑𝑦))− 𝜑𝑧 = 0, on 𝑧 = 1 + 𝜀 𝜂, (8)

𝜑𝑧 = 0, on 𝑧 = 0. (9)

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432 International Journal of NonlinearScience,Vol.9(2010),No.4,pp. 430-443

where 𝜎 = Γ/ℎ0 𝜌 𝑐

20 is the Bond number, which is the ratio of surface tension force Γ to the body force, [8, 9, 19]. A

low Bond number indicates that the system is relatively unaffected by surface tension effects, while a high Bond numberindicates that surface tension dominates. Intermediate Bond numbers indicate a non-trivial balance between the twoeffects.

The basic equations (6)-(9) are too complicated to handle in order to obtain a compact evolution equation for onephysical variable without approximations. So, we can begin the assumption that 𝛿 is small, which might be interpretedas the characteristic feature of the shallow water theory. It follows that 𝜙 could be expanded in terms of 𝛿 without anyassumption yet about 𝜀 and write

𝜑 = 𝜑0 + 𝛿 𝜑1 + 𝛿2 𝜑2 + ⋅ ⋅ ⋅ (10)

Substituting (10) into (6), we find that at𝑂(𝛿0) : 𝜑0 𝑧, 𝑧 = 0, (11)

𝑂(𝛿1) : 𝜑1 𝑧, 𝑧 + 𝜑0 𝑦, 𝑦 + 𝜑0 𝑥, 𝑥 = 0, (12)

𝑂(𝛿2) : 𝜑2 𝑧, 𝑧 + 𝜑1 𝑦, 𝑦 + 𝜑1 𝑥, 𝑥 = 0. (13)

Integrating (11) twice w. r. t. 𝑧 and by using (9), we get 𝜑0 = 𝑐1(𝑥, 𝑦, 𝑡), which indicates that the horizontal velocitycomponents are independent of the vertical coordinate [1].

In this problem of water waves, the motion may be taken to be irrotationals, which physically means that the individualfluid particles don’t rotate. Mathematically, this implies that the vorticity vanishes, 𝑐𝑢𝑟𝑙 �� = 0. So that there exists a singlevalued velocity potential, �� = ∇𝜑, or equivalently

𝜑0 𝑥 = 𝑢, 𝜑0 𝑦 = 𝑣. (14)

Hence, integrating (12) 𝑤.𝑟.𝑡.𝑧 to obtain

𝜑1 𝑧 = − 𝑧 (𝑢𝑥 + 𝑣𝑦) + 𝑐2(𝑥, 𝑦, 𝑡). (15)

Using Eq. (9) the integration constant 𝑐2(𝑥, 𝑦, 𝑡) equals zero, and integrating the resulting equation we obtain

𝜑1 =−𝑧2

2(𝑢𝑥 + 𝑣𝑦), (16)

where the integration constant is assumed equal to zero.Substituting (16) into (13) and integrating the resulting equation twice 𝑤.𝑟.𝑡.𝑧 we get

𝜑2 =−𝑧4

24

((∇2𝑢)𝑥+

(∇2𝑣)𝑦

)(17)

Using Eqs. (15)-(17), we substitute (10) into the free surface boundary conditions (7)-(8) considering that theseconditions retain all terms up to order 𝛿, 𝜀 in Eq. (7) and 𝛿, 𝛿2, 𝛿 𝜀 in (8). It turns out that

𝜑0 𝑡 + 𝜂 +𝜀

2

(𝑢2 + 𝑣2

)− 𝛿

2(𝑢𝑡, 𝑥 + 𝑣𝑡, 𝑦)− 𝜎 𝛿 (𝜂𝑥, 𝑥 + 𝜂𝑦, 𝑦) = 0, (18)

𝛿 (𝜂𝑡 + (1 + 𝜀 𝜂) (𝑢𝑥 + 𝑣𝑦) + 𝜀 (𝑢 𝜂𝑥 + 𝑣 𝜂𝑦))− 𝛿2

6

((∇2𝑢)𝑥+

(∇2𝑣)𝑦

)= 0. (19)

Using (14), differentiation of (18) first 𝑤.𝑟.𝑡.𝑥 and then 𝑤.𝑟.𝑡.𝑦 gives the following two equations

𝑢𝑡 + 𝜂𝑥 + 𝜀 (𝑢𝑢𝑥 + 𝑣 𝑣𝑥)− 𝛿

2(𝑢𝑥, 𝑥, 𝑡 + 𝑣𝑥, 𝑦, 𝑡)− 𝜎 𝛿 (𝜂𝑥, 𝑥, 𝑥 + 𝜂𝑥, 𝑦, 𝑦,) = 0, (20)

𝑣𝑡 + 𝜂𝑦 + 𝜀 (𝑢𝑢𝑦 + 𝑣 𝑣𝑦)− 𝛿

2(𝑢𝑦, 𝑥, 𝑡 + 𝑣𝑦, 𝑦, 𝑡)− 𝜎 𝛿 (𝜂𝑥, 𝑥, 𝑦 + 𝜂𝑦, 𝑦, 𝑦) = 0. (21)

Using the fact that 𝜑0 is irrotational, that is 𝑢𝑦 = 𝑣𝑥, we obtain the nondimensional shallow water equations

𝑢𝑡 + 𝜂𝑥 + 𝜀 (𝑢𝑢𝑥 + 𝑣 𝑢𝑥)− 𝛿

2

(∇2𝑢)𝑡− 𝜎 𝛿

(∇2𝜂)𝑥= 0, (22)

𝑣𝑡 + 𝜂𝑦 + 𝜀 (𝑢 𝑣𝑥 + 𝑣 𝑣𝑦)− 𝛿

2

(∇2𝑣)𝑡− 𝜎 𝛿

(∇2𝜂)𝑦= 0, (23)

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𝜂𝑡 + (𝑢 (1 + 𝜀 𝜂))𝑥 + (𝑣 (1 + 𝜀 𝜂))𝑦 −𝛿

6

((∇2𝑢)𝑥+

(∇2𝑣)𝑦

)= 0. (24)

In the following study we consider the one-dimensional case of the above equations and the nonlinear parameter 𝜀tends to zero for 𝛿 fixed, so that Eqs. (22)-(24) reduce to

𝛿−1 (𝑢𝑡 + 𝜂𝑥) + 𝑈𝑟 𝑢𝑢𝑥 − 1

2𝑢𝑥, 𝑥, 𝑡 − 𝜎 𝜂𝑥, 𝑥, 𝑥 = 0, (25)

𝛿−1 (𝜂𝑡 + 𝑢𝑥) + 𝑈𝑟 (𝑢 𝜂)𝑥 − 1

6𝑢𝑥, 𝑥, 𝑥 = 0, (26)

We take these two equations as the starting point to apply the reductive perturbation method, and taking that

𝑈𝑟 =𝜀

𝛿=

𝑎𝜆20

ℎ30

, (27)

defined as the Ursell parameter [32], which and is derived from the Stokes’ perturbation series for nonlinear periodic wavesin the long-wave limit of shallow water. This parameter characterizes the relative role of nonlinearity and dispersion:small values of 𝑈𝑟 correspond to the almost linear dispersive waves; while large values of 𝑈𝑟 correspond to the nonlinearnondispersive waves [33].

3 Derivation of NLS Equation Using Reductive Perturbation TechniqueHere, we carry out the standard reductive perturbation analysis [10, 14, 15] for the nonlinear shallow water equations(25) and (26) to obtain the NLS equation which governs the behavior of the one dimensional case of shallow water wavesthrough the constant water depth.

In this technique, the independent variables are scaled according to the Gardner-Morikawa transformation [10]

𝜉 = 𝜀 (𝑥− 𝑣𝑔 𝑡), 𝜏 = 𝜀2 𝑡, (28)

where 𝜀 is a small (0 < 𝜀 < 1) expansion parameter defined before and 𝑣𝑔 is the group velocity of the water wave thatwill be determined later.

The original equations (25) and (26), are transformed according to

∂

∂ 𝑡⇒ ∂

∂ 𝑡− 𝜀 𝑣𝑔

∂

∂ 𝜉+ 𝜀2

∂

∂ 𝜏,

∂

∂ 𝑥⇒ ∂

∂ 𝑥+ 𝜀

∂

∂ 𝜉, (29)

to yield

𝛿−1 (𝑢𝑡 − 𝜀 𝑣𝑔𝑢𝜉 + 𝜀2 𝑢𝜏 + 𝜂𝑥 + 𝜀 𝜂𝜉) + 𝑈𝑟 𝑢 (𝑢𝑥 + 𝜀 𝑢𝜉)− 12 (𝑢𝑡, 𝑥, 𝑥 + 𝜀 (2𝑢𝑡, 𝑥, 𝜉 − 𝑣𝑔𝑢𝑥, 𝑥, 𝜉)

+𝜀2 (𝑢𝑡, 𝜉, 𝜉 − 2 𝑣𝑔 𝑢𝑥, 𝜉, 𝜉 + 𝑢𝜏, 𝑥, 𝑥) + 𝜀3 (2𝑢𝜏, 𝑥, 𝜉 − 𝑣𝑔 𝑢𝜉, 𝜉, 𝜉) + 𝜀4 𝑢𝜏, 𝜉, 𝜉)− 𝜎 (𝜂𝑥, 𝑥, 𝑥+3 𝜀 𝜂𝑥, 𝑥, 𝜉 + 3 𝜀2 𝜂𝑥, 𝜉, 𝜉 + 𝜀3 𝜂𝜉, 𝜉, 𝜉) = 0,

(30)

𝛿−1 (𝜂𝑡 − 𝜀 𝑣𝑔𝜂𝜉 + 𝜀2 𝜂𝜏 + 𝑢𝑥 + 𝜀 𝑢𝜉) + 𝑈𝑟 (𝑢 (𝜂𝑥 + 𝜀𝜂𝜉) + 𝜂 (𝑢𝑥 + 𝜀 𝑢𝜉))− 1

6 (𝑢𝑥, 𝑥, 𝑥 + 3 𝜀 𝑢𝑥, 𝑥, 𝜉 + 3 𝜀2 𝑢𝑥, 𝜉, 𝜉 + 𝜀3 𝑢𝜉, 𝜉, 𝜉) = 0,(31)

thus the functions 𝑢(𝑥, 𝑡) and 𝜂(𝑥, 𝑡) are now treated as functions of the variables (𝜉, 𝜏 ).Using the small perturbation parameter 𝜀, the dependent physical variables can be expressed as

𝜂 =∞∑

𝑛=1

𝜀𝑛∞∑

𝑚=−∞𝑒𝑚𝐸 𝜂𝑚𝑛 (𝜉, 𝜏) , (32)

𝑢 =

∞∑𝑛=1

𝜀𝑛∞∑

𝑚=−∞𝑒𝑚𝐸 𝑢𝑚𝑛 (𝜉, 𝜏) , (33)

where 𝐸 = 𝑖 (𝑘 𝑥− 𝜔 𝑡), 𝑘 denotes the wave number and 𝜔 the angular frequency. Since the physical quantities 𝜂 and 𝑢are real, the relations

𝜂(−𝑚)𝑛 = 𝜂∗𝑚𝑛, 𝑢(−𝑚)𝑛 = 𝑢∗𝑚𝑛, (34)



are satisfied, where the asterisk denotes the complex conjugate.Applying the above series expansion to Eqs. (30)-(31) and equating each coefficient of different powers of 𝜀 to zero,

we get, for the first order of 𝜀 and 𝑛 = 1, the first order quantity 𝜂11 in terms of 𝑢11 as

𝜂11 =𝑘 (6 + 𝑘2 𝛿)

6𝜔𝑢11, (35)

and the dispersion relation

𝜔2 =𝑘2 (6 + 𝑘2 𝛿) (1 + 𝑘2 𝛿 𝜎)

6 + 3 𝑘2 𝛿, (36)

for the shallow water equations, from it we can obtain the phase velocity

𝑣𝑝 =𝜔

𝑘=

√(6 + 𝑘2 𝛿) (1 + 𝑘2 𝛿 𝜎)

6 + 3 𝑘2 𝛿. (37)

For 𝑛 = 1 and 𝑚 = 2, 3, we get𝜂21 = 𝜂31 = 𝑢21 = 𝑢31 = 0. (38)

For the order of 𝜀2 with 𝑚 = 0, 1, 2, we obtain the following equations

𝜂01 = 𝑢01 = 0, (39)

𝜂12 =𝑘 (6 + 𝑘2 𝛿)

6𝜔𝑢12 +

𝑖 (𝑘 𝑣𝑔(6 + 𝑘2 𝛿)− 3 (2 + 𝑘2 𝛿)𝜔)

6𝜔2𝑢11𝜉 , (40)

𝑣𝑔 =𝑘 (12 + 𝑘2 𝛿 (4 + 24𝜎 + 𝑘2 𝛿 (1 + 2 (6 + 𝑘2 𝛿)𝜎)))

3 (2 + 𝑘2 𝛿)2 𝜔≡ ∂ 𝜔

∂ 𝑘, (41)

𝜂22 = − (2 + 𝑘2 𝛿) (9 + 𝑘2 𝛿 (15 + 2 𝑘2 𝛿))𝑈𝑟

6 𝑘2 (−2 + (6 + 𝑘2 𝛿 (5 + 2 𝑘2 𝛿))𝜎)𝜔(𝑢11)2, (42)

𝑢22 = − (6 + 𝑘2 𝛿) (3 + 𝑘2 𝛿 (1 + (9 + 4 𝑘2 𝛿)𝜎))𝑈𝑟

6 𝑘 (−2 + (6 + 𝑘2 𝛿 (5 + 2 𝑘2 𝛿))𝜎)𝜔(𝑢11)2. (43)

where the compatibility condition (41) is interpreted as the group velocity of the shallow water wave equations.The graphs of this paper will be drawn to describe the physical situation of the model when we use suitable values for

the parameters as ℎ = 1𝑚, 𝜌 = 1000 𝑘𝑔/𝑚3, Γ = 0.07197𝑁/𝑚, 𝜎 = 7.34×10−6, 𝑔 = 9.8𝑚/𝑠2. The dependenceof 𝜔, 𝑣𝑝 and 𝑣𝑔 on the wave number 𝑘 are shown in Fig. 1.

Figure 1: The graphs of the dispersion properties against 𝑘 at 𝜀 = 𝛿 = 0.03.

In Fig. 1 we found that the phase velocity represented by the dashed line is greater than the group velocity representedby the solid line (𝑣𝑝 > 𝑣𝑔) which is satisfied for gravity waves, while the opposite is true for capillary waves, see [3].This means that the dispersion relation for the shallow water equations is normal. We also noticed that the phase and groupvelocities decrease as the wave number k increases which is consistent with long waves that travel faster than shorter ones.

For the order of 𝜀3 with the zeroth order harmonic mode of carrier wave (𝑚 = 0), we obtain

𝜂02 =𝛿 (𝑘 𝑣𝑔(6 + 𝑘2 𝛿) + 3𝜔)𝑈𝑟

3 (−1 + 𝑣2𝑔)𝜔(𝑢11)2, (44)



𝑢02 =𝛿 (𝑘 (6 + 𝑘2 𝛿) + 3 𝑣𝑔 𝜔)𝑈𝑟

3 (−1 + 𝑣2𝑔)𝜔(𝑢11)2. (45)

Now, by using the above derived equations for the order of 𝜀3 with the first harmonic mode (𝑚 = 1), we get a closedevolution equation for 𝑢11:

𝑖 𝑢11𝜏 + 𝑝 𝑢11

𝜉, 𝜉 + 𝑞∣∣𝑢11

∣∣2 𝑢11 = 0, (46)

which is the well-known envelope nonlinear Schrodinger equation; it describes the evolution of the envelope of the mod-ulated wave group. According to the stability criterion established in [1, 2, 14, 19], the wave solutions of this equation arestable if 𝑝𝑞 < 0 or unstable if 𝑝𝑞 > 0. To investigate these effects in more detail, we plot the product 𝑝𝑞 with 𝑘 and 𝛿 asin Fig. 2. The exact solutions of Eq. (46) will be discussed in the following section.

The dispersion coefficient 𝑝 satisfies the relationship

𝑝 =1

2

∂2𝜔

∂ 𝑘2, (47)

and the nonlinearity coefficient 𝑞 reads

𝑞 =(

−16(𝑣2

𝑔−1)(−2+(6+𝑘2𝛿(5+2𝑘2𝛿))𝜎)𝜔(𝑘2(6+𝑘2𝛿)(1+𝑘2𝛿 𝜎)+3(2+𝑘2𝛿)𝜔2))

)× (

𝛿(4𝑘14𝛿6𝜎2 + 2𝑘12𝛿5𝜎(2 + (31− 2𝑣2𝑔)𝜎) + 𝑘10𝛿4𝜎(59 + 333𝜎 − 𝑣2𝑔(5 + 57𝜎))+36𝑘11𝑣𝑔𝛿

5𝜎2𝜔 + 54𝑘7𝑣𝑔𝛿3𝜎(5 + 12𝜎)𝜔 + 18𝑘9𝑣𝑔𝛿

4𝜎 (2 + 17𝜎)𝜔 − 216(𝑣2𝑔 − 1)𝜔2

+72𝑘𝑣𝑔𝛿(3𝜎 − 1)𝜔3 + 36𝑘2(3− 3𝑣2𝑔 + 2𝛿(1 + 15𝜎 − 2𝑣2𝑔(2 + 3𝜎))𝜔2) + 36𝑘5𝑣𝑔𝛿2𝜔(−1

+2𝜎(6 + 9𝜎 + 𝛿𝜔2)) + 36𝑘3𝑣𝑔𝛿𝜔(−6 + 𝜎(18 + 5𝛿𝜔2)) + 12𝑘4𝛿(−6 + 72𝜎 + 𝛿(8+9𝜎(9 + 2𝜎))𝜔2 − 2𝑣2𝑔(3 + 18𝜎 + 𝛿(5 + 18𝜎)𝜔2)) + 3𝑘6𝛿2(−11 + 228𝜎 + 252𝜎2

+2𝛿(2 + 69𝜎 + 30𝜎2)𝜔2 − 𝑣2𝑔(5 + 4𝛿𝜔2 + 2𝜎(54 + 54𝜎 + 23𝛿𝜔2))) + 𝑘8𝛿3(−𝑣2𝑔(1+ 12𝜎(6 + 21𝜎 + 𝛿𝜔2)) + 3(−1 + 4𝜎(25 + 63𝜎 + 3𝛿(1 + 2𝜎)𝜔2))))𝑈2

𝑟

).

(48)

In the following figures we study the stability of the NLS equation:

Figure 2: The graphs for the variation of 𝑝 𝑞 at 𝜀 = 0.03.

We can see from Fig. 2(a) that the wave solution of the NLS equation is stable at all values of the wave number 𝑘 for𝛿 = 0.03, but in Fig. 2(b) the parameter space 𝑘-𝛿 is divided into two regions, the wave solution is unstable for very smallvalues of 𝑘 and 𝛿

(0 < 𝑘 ≤ 10−2, 0 < 𝛿 ≤ 10−3

), while at larger values, the wave solution is stable. It means that there

are two types of the soliton solutions of the NLS equation.From the above results (35)-(43) we can obtain the physical solutions of the shallow water wave problem for first and

second harmonics as

𝜂 = 𝜀𝑘(6+𝑘2𝛿)(𝑒−𝐸+𝑒𝐸)𝑢11

6𝜔

+𝜀2𝑈𝑟(𝑢11)2

((2+𝑘2𝛿)(9+𝑘2𝛿(15+2𝑘2𝛿))(𝑒−2𝐸+𝑒2𝐸)

−6𝑘2(−2+(6+𝑘2𝛿(5+2𝑘2𝛿))𝜎) +𝛿(6𝑘𝑣𝑔+𝑘3𝑣𝑔𝛿+3𝜔)

3(𝑣2𝑔−1)𝜔

),

(49)

𝑢 = 𝜀(𝑒−𝐸 + 𝑒𝐸)𝑢11

+𝜀2𝑈𝑟(𝑢11)2

((6+𝑘2𝛿)(3+𝑘2𝛿(1+(9+4𝑘2𝛿))(𝑒−2𝐸+𝑒2𝐸)

−6𝑘(−2+(6+𝑘2𝛿(5+2𝑘2𝛿))𝜎)𝜔 +𝛿(6𝑘+𝑘3𝛿+3𝑣𝑔𝜔)

3(𝑣2𝑔−1)𝜔

).

(50)

Note that these expansions have been obtained symbolically by using the formal computation software ”Mathematica”.



3.1 Conservation law for NLS equationZakharov and Shabat [1] proved that the NLS equation has infinite number of Polynomial conservation laws. These lawsare somewhat similar to those already proved for the KdV equation [1, 2, 19].

Using the relation (35), we replace 𝑢11 in Eq. (46) by 𝜂11. Then we multiply the resulting equation by 𝜂11 ∗ and itscomplex conjugate by 𝜂11, and hence subtracting the latter from the former to obtain the conservation of potential energywith respect to the elevation (amplitude) of the water wave which satisfy the continuity equation as follows:

𝑊 ≡ 𝑖 𝜌𝜏 + 𝐽𝜉 = 0, (51)

where 𝜌 = 𝜂11 ∗ 𝜂11 is the energy density and 𝐽 = 𝑝(𝜂11 ∗ 𝜂11𝜉 − 𝜂11 ∗

𝜉 𝜂11)

is the energy current density. We willdiscuss the continuity equation 𝑊 in the following section.

4 The exact solutions of NLS equation using the complex Ansatz methodIn this section, without transforming to real and imaginary parts, a complex ansatz method is presented to derive the exactcomplex traveling wave solutions of NLS equation [30].

For a given partial differential equations (PDEs)

𝑓(𝑢, 𝑢𝑥, 𝑢𝑡, 𝑢𝑥, 𝑥, ⋅ ⋅ ⋅ ) = 0, (52)

Using the complex transformation 𝜁 = 𝑖 (𝐾 𝜉 − Ω 𝜏), Eq. (52) becomes an ordinary differential equation (ODEs)

𝑔(𝑢, 𝑖𝐾 𝑢𝜁 , −𝑖Ω𝑢𝜁 , −𝐾2 𝑢𝜁, 𝜁 , ⋅ ⋅ ⋅ ) = 0, (53)

the proposed solutions could be taken in the form

𝑢(𝜁) =𝑙∑

𝑖=0

𝑎𝑖 𝐹𝑖(𝜁) +

𝑙∑𝑖=1

𝑏𝑖 𝐹𝑖−1(𝜁)𝐺𝑖(𝜁), (54)

where 𝑎𝑖, 𝑏𝑖 are constants to be determined and the integer 𝑙 is determined by balancing the linear term of highest orderwith the nonlinear term.

The coupled Riccati equations,

𝐹𝜁(𝜁) = −𝐹 (𝜁) 𝐺(𝜁) and 𝐺𝜁(𝜁) = 1− 𝐺2(𝜁), (55)

admits two types of solutions as

𝐹 (𝜁) = ± secℎ(𝜁), 𝐺(𝜁) = tanh(𝜁) with the relation 𝐺2 = 1− 𝐹 2, (56)

and𝐹 (𝜁) = ± cscℎ(𝜁), 𝐺(𝜁) = coth(𝜁) with the relation 𝐺2 = 1 + 𝐹 2, (57)

Then substituting (54) into (53), using (55) repeatedly with (56) or (57), and setting each coefficient of 𝐹 𝑖 and 𝐺𝐹 𝑖 tozero yields a set of algebraic equations for 𝑎𝑖, 𝑏𝑖, 𝐾, Ω. Solving them gives the exact solutions for Eq. (55), accordinglythe exact solutions of NLS equation (46) can be obtained. The ODE form of this equation is

Ω𝑢𝜁11 − 𝑝𝐾2𝑢𝜁𝜁

11 + 𝑞𝑢112𝑢11∗ = 0, (58)

and the balancing between 𝑢𝜁, 𝜁11 and 𝑢112 (𝑢11)

∗ yields 𝑙 = 1. This suggests the following form of solution (56)

𝑢 = 𝑎0 + 𝑎1 𝐹 + 𝑏1 𝐺. (59)

Using the relation in Eq. (56) with 𝐹 ∗ = 𝐹, 𝐺∗ = −𝐺, substituting (59) into (58) and equating all coefficients of 𝐹 𝑖

and 𝐺𝐹 𝑖 to zero, we obtain

𝑞 𝑎30 − 𝑞 𝑎0 𝑏21 = 0, −Ω 𝑎1 + 2 𝑞 𝑎0 𝑎1 𝑏1 = 0, 3 𝑞 𝑎0 𝑎

21 +Ω 𝑏1 + 𝑞 𝑎0 𝑏

21 = 0,−𝐾2 𝑝 𝑎1 + 3 𝑞 𝑎20 𝑎1

−𝑞 𝑎1 𝑏21 = 0, 2𝐾2 𝑝 𝑎1 + 𝑞 𝑎31 + 𝑞 𝑎1 𝑏

21 = 0, 𝑞 𝑎20 𝑏1 − 𝑞 𝑏31 = 0, 2𝐾2 𝑝 𝑏1 + 𝑞 𝑎21 𝑏1 + 𝑞 𝑏31 = 0.

(60)



Solving these algebraic equations gives the following casesCase 1, 2:

Ω = −2𝐾2𝑝, 𝑎0 = ∓𝑖𝐾

√2 𝑝

𝑞, 𝑎1 = 0, 𝑏1 = −𝑎0, (61)

Case 3, 4:

Ω = 2𝐾2𝑝, 𝑎0 = ∓𝑖𝐾

√2 𝑝

𝑞, 𝑎1 = 0, 𝑏1 = 𝑎0. (62)

Therefore, we obtain the following family of complex exact traveling wave solutions for NLS equation as

𝑢111,2 = 𝑖𝐾

√2 𝑝

𝑞(∓1± tanh (𝑖𝐾(𝜉 + 2𝐾 𝑝 𝜏))), (63)

𝑢113,4 = ∓𝑖𝐾

√2 𝑝

𝑞(1 + tanh (𝑖𝐾(𝜉 − 2𝐾 𝑝 𝜏))). (64)

In terms of these solutions of 𝑢11 we can obtain the physical variables of the water wave problem in Eqs. (49) and(50).

To show the properties of the exact solutions for the NLS equation and the shallow water wave equations, we takesome of these solutions as illustrative examples and draw their graphs as in Figs. 3, 4, 5 and 6.

(a) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (b) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 3: The graphs for the solutions of NLS equation (u111 ) at different values of Ursell parameter.

(c) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (d) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86



(e) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (f) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 4: The graphs for the elevation of the free surface (𝜂1) at different values of Ursell parameter.

(g) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (h) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

(i) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (j) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 5: The graphs for the elevation of the free surface (u1) at different values of Ursell parameter.

From the above figures we notice that the imaginary parts of solutions are in the form of bright and dark solitons, seeFigs. 3(a), 4(e) and 5(i). Their amplitudes decrease as the Ursell parameter increases, see Figs. 3(a, b), 4(e, f) and 5(i,j), while the real parts of solutions are in the form of bright solitons and their amplitudes increase as the Ursell parameterincreases, see Figs. 4(c, d) and 5(g, h).

Similarly when we use the relation in Eq. (57) with 𝐹 ∗ = −𝐹, 𝐺∗ = −𝐺, and by means of the same steps, we obtainthe following algebraic equations

𝑞 𝑎30 − 𝑞 𝑎0 𝑏21 = 0, −Ω 𝑎1 − 2 𝑞 𝑎0 𝑎1 𝑏1 = 0, 𝑞 𝑎0 𝑎

21 +Ω 𝑏1 + 𝑞 𝑎0 𝑏

21 = 0,−𝐾2 𝑝 𝑎1 + 𝑞 𝑎20 𝑎1

−3𝑞 𝑎1 𝑏21 = 0,− 2𝐾2 𝑝 𝑎1 − 𝑞 𝑎31 − 3 𝑞 𝑎1 𝑏

21 = 0, 𝑞 𝑎20 𝑏1 − 𝑞 𝑏31 = 0, 2𝐾2 𝑝 𝑏1 + 3 𝑞 𝑎21 𝑏1 + 𝑞 𝑏31 = 0.

(65)

Solving them gives the following cases



(k) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (l) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 6: The graphs for the solutions of NLS equation (W1) at different values of Ursell parameter.

Case 1, 2:

Ω = −𝐾2𝑝, 𝑎0 = −𝑖𝐾

√𝑝

2 𝑞, 𝑎1 = ± 𝑎0, 𝑏1 = −𝑎0, (66)

Case 3, 4:

Ω = −𝐾2𝑝, 𝑎0 = 𝑖𝐾

√𝑝

2 𝑞, 𝑎1 = ∓ 𝑎0, 𝑏1 = −𝑎0, (67)

Case 5, 6:

Ω = 𝐾2𝑝, 𝑎0 = −𝑖𝐾

√𝑝

2 𝑞, 𝑎1 = ± 𝑎0, 𝑏1 = 𝑎0, (68)

Case 7,8:

Ω = 𝐾2𝑝, 𝑎0 = 𝑖𝐾

√𝑝

2 𝑞, 𝑎1 = ∓ 𝑎0, 𝑏1 = 𝑎0, (69)

Case 9,10:

Ω = 2𝐾2𝑝, 𝑎0 = ∓𝑖𝐾

√2 𝑝

𝑞, 𝑎1 = 0, 𝑏1 = 𝑎0, (70)

Case 11,12:

Ω = −2𝐾2𝑝, 𝑎0 = ∓𝑖𝐾

√2 𝑝

𝑞, 𝑎1 = 0, 𝑏1 = −𝑎0. (71)

Therefore, we have another family of complex exact traveling wave solutions for NLS equation as

𝑢111,2 = 𝑖𝐾

√𝑝

2 𝑞(−1± cscℎ (𝑖𝐾(𝜉 +𝐾 𝑝 𝜏))− coth (𝑖𝐾(𝜉 +𝐾 𝑝𝜏))) , (72)

𝑢113,4 = 𝑖𝐾

√𝑝

2 𝑞(1∓ cscℎ (𝑖𝐾(𝜉 +𝐾 𝑝 𝜏))− coth (𝑖𝐾(𝜉 +𝐾 𝑝 𝜏))) , (73)

𝑢115,6 = 𝑖𝐾

√𝑝

2 𝑞(−1± cscℎ (𝑖𝐾(𝜉 −𝐾 𝑝 𝜏)) + coth (𝑖𝐾(𝜉 −𝐾 𝑝𝜏))) , (74)

𝑢117,8 = 𝑖𝐾

√𝑝

2 𝑞(1∓ cscℎ (𝑖𝐾(𝜉 −𝐾 𝑝 𝜏)) + coth (𝑖𝐾(𝜉 −𝐾 𝑝 𝜏))) , (75)

𝑢119,10 = ∓𝑖𝐾

√2 𝑝

𝑞(1 + coth (𝑖𝐾(𝜉 − 2𝐾 𝑝 𝜏))) , (76)

𝑢1111,12 = 𝑖𝐾

√2 𝑝

𝑞(∓1± coth (𝑖𝐾(𝜉 + 2𝐾 𝑝𝜏))) . (77)

To show the properties of the exact solutions for the NLS equation and the shallow water wave equations, we takesome of these solutions as illustrative examples and draw their graphs as Figs. 7, 8, 9 and 10.



(a) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (b) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 7: The graphs for the solutions of NLS equation (u117 ) at different values of Ursell parameter.

(c) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (d) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

(e) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (f) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 8: The graphs for the elevation of the free surface (𝜂7) at different values of Ursell parameter.From the above figures we notice that the real and imaginary parts of all solutions are in the form of solitary traveling

waves and their amplitudes increase as the Ursell parameter increases, see Figs. 7(a, b), 8(c, d), 8(e, f), 9(g, h) and 9(i, j).We see also that the conservation of power 𝑊 is satisfied all over the space 𝑥 and time 𝑡 except at very few spots where

it deviates from zero by the order of 10−14 and 10−18 depending on the Ursell parameter , see Figs. 6(k, l) and 10(k, l).This means that the effect of the Ursell parameter plays an important role on the wave profiles.

5 ConclusionsThe nonlinear water wave problem for an incompressible and inviscid fluid of constant depth is studied by including theeffects of the constant gravity acceleration and surface tension. The governing equations of this problem are converted



(g) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (h) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

(i) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (j) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 9: The graphs for the elevation of the free surface (u7) at different values of Ursell parameter.

(k) 𝜀 = 0.015, 𝛿 = 0.03,U𝑟 = 0.5 (l) 𝜀 = 0.03, 𝛿 = 0.035,U𝑟 = 0.86

Figure 10: The graphs for the solutions of NLS equation (W7) at different values of Ursell parameter.into shallow water equations by using some asymptotic expansions. We use the reductive perturbation method, which isone of the modern approximation methods, to derive the NLS equation from the nonlinear shallow water wave equations.We also derived the dispersion properties for these equations which are normal since the phase velocity is greater than thegroup velocity.

Based on the computerizing symbolic computation, in this paper we have presented a complex ansatz method forexact traveling wave solutions to NLS equation. Thus the physical variables of the problem are investigated. The resultsindicate that the wave profile is significantly influenced by the Ursell parameter. So that depending on this parameter, wedraw several figures for the exact solutions, which are in the form of bright and dark solitons. It is found that the wavepacket is stable in all the region of 𝑘-𝛿 plane except at the very small values.

In this study, the value of Bond number is very small (7.34× 10−6). This value indicates that the system is relativelyunaffected by surface tension effects, and the resulting waves are gravity water waves.

Finally, we observe that the conservation of power is satisfied which means that the suggested methods of solutions



are successfully applied to the tackled problem.

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The Derivation and Study of the Nonlinear Schrodinger ...internonlinearscience.org/upload/papers/20110308101224985.pdf · moving with a ﬁxed velocity. Of particular interest are