Applied Mathematics and Computation 218 (2011) 2277–2280

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Derivation of Korteweg–de Vries flow equations from the regularizedlong-wave (RLW) equation

Mehmet Naci Özer a, Murat Koparan b,⇑a Department of Mathematics, Art and Science Faculty, Eskis�ehir Osmangazi University, 26480 Eskis�ehir, Turkeyb Department of Elementary Education, Education Faculty, Anadolu University, 26470 Eskis�ehir, Turkey

a r t i c l e i n f o a b s t r a c t

Keywords:Regularized long-wave (RLW) equationKdV equationMultiple-time scales

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.07.045

⇑ Corresponding author.E-mail addresses: mnozer@ogu.edu.tr (M. Naci Ö

We perform a multiple-time scales analysis and compatibility condition to the regularizedlong-wave (RLW) equation. We derive Korteweg–de Vries (KdV) flow equation in the bi-Hamiltonian form as an amplitude equation.

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1. Introduction

The well known regularized long-wave (RLW) equation is an important nonlinear wave equation. Solitary waves are wavepackets or pulses propagating in nonlinear dispersive media. On account of dynamical balance between the nonlinear anddispersive effects, these waves retain a stable waveform. A soliton is a very specific type of solitary wave, which also keepsits waveform after collision with another soliton.The regularized long-wave (RLW) equation,

ut þ ux � uxxt � 6uux ¼ 0 ð1Þ

also known as the Peregrine [1] or Benjamin–Bona–Mahony [2] equation throughout scientific quarters, was originally intro-duced as an alternative for the Korteweg–de Vries (KdV) equation

ut þ ux � uxxx � 6uux ¼ 0: ð2Þ

Although above equations have quite different dispersion properties, these two equations possess an intimate relationship.For example, the linear dispersion relation of the RLW equation is

wðkÞ ¼ k

1þ k2 ; ð3Þ

which, by the way, is the same as the dispersion relation of the shallow water wave equation [3]. For long wavelengths, k issmall, and wðkÞ can be expanded according to

wðkÞ ¼ k� k3 þ Oðk5Þ: ð4Þ

The first two terms of this expansion coincide exactly with the complete linear dispersion of the KdV equation (2). For thisreason, for sufficiently long wavelengths, the traveling-wave solutions of Eqs. (1) and (2) are expected to be quite similar.Inspite of this, there is deep difference between these two cases, since a polynomial is definitely not equivalent to infiniteseries. This difference, appearing when higher-order terms of the dispersion relation expansion are considered, might alsoshow up at higher-order approximation in a perturbation theory [4]. As it is well known, the KdV equation (2) governs

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zer), mkoparan@anadolu.edu.tr (M. Koparan).

2278 M. Naci Özer, M. Koparan / Applied Mathematics and Computation 218 (2011) 2277–2280

the first relevant order of an asymptotic perturbation expansion, which describes weakly nonlinear dispersive waves. How-ever, to make sense of it as really governing such waves, the large time behavior of the perturbative series must be analyzed[5]. It has been shown to have solitary wave solutions and to govern a large number of important physical phenomena suchas shallow water waves and plasma waves [1,6].

In Section 2, we present some materials, recursion operators and multiple-time formalism for KdV equation (2). In Section3, we first give a multiple scales method and then we apply the method to the RLW equation with derivation of the KdV flowequations.

Throughout the paper, we make extensive use of Reduce to calculate and simplify our results.

2. Recursion operators and KdV flow equations

A function v ¼ P½u� is called a generalised symmetry of a given evolution equation if it is solves the linearised equation,

v t ¼ K 0½u�v ; ð5Þ

here P½u� is a function of u and its spatial derivatives, and the differential operator K ½u� is the Frechet derivative of K½u� whichis defined by

K 0½u�v ¼ dde

K½uþ ev �je¼0: ð6Þ

The evolution of a differential function of u;K½u�, whit respect to any independent variable s is given by

ðK½u�Þs ¼ K 0½u�us: ð7Þ

Therefore asking that v ¼ P½u� be a solution of (5) is just the requirement

ðP½u�Þt ¼ ðK½u�Þt; ð8Þ

that the flows (5) and

us ¼ P½u�; ð9Þ

commute. Note that u is now being thought of as a function of x and both flow times t and s [7,8].The KdV equation (2) has the hereditary recursion operator [9,10]

R½u� ¼ @2 þ 4uþ 2ux@�1: ð10Þ

This maps ux onto the KdV equation (2) itself. The infinite hierarchy of mutually commuting flows satisfies the relation:

ut2nþ1 ¼ Rn½u�ux ¼ K2nþ1½u�; n ¼ 0;1; . . . : ð11Þ

KdV equation (2) for n ¼ 1,

ut ¼ uxxxxx � 10uuxxx � 20uxuxx þ 30u2ux: ð12Þ

fifth order KdV equation for n ¼ 2 ,

ut ¼ �uxxxxxxx þ 14uuxxxxx þ 42uxuxxxx þ 70uxxuxxx � 280uuxuxx � 70u2uxxx � 70u3x þ 14u3ux: ð13Þ

seventh order KdV equation for n ¼ 3 respectively.

2.1. Multıple-time scales formalısm

In this section we are going to apply a multiple-time scales version [11] of the reductive perturbation method to studylong waves as governed by the RLW equation (1). To study the long-wavelength limit of the RLW equation (1), we put

k ¼ eK; ð14Þ

with e a small parameter. In this limit, the dispersion relation (3) can be expanded as

wðkÞ ¼ eK � e3K3 þ e5K5 � e7K7 þ � � � : ð15Þ

Accordingly, the solution of the corresponding linear RLW equation (1) can be written in the form

u ¼ a expfi½kx�wðkÞt�g � a expfi½Keðx� tÞ þ e3K3t � e5K5t þ e7K7t þ � � ��g; ð16Þ

where a is a constant. As given by this solution, we now define a slow space

n ¼ eðx� tÞ; ð17Þ

as well as an infinity of properly normalized slow time variables:

M. Naci Özer, M. Koparan / Applied Mathematics and Computation 218 (2011) 2277–2280 2279

s3 ¼ e3t; s5 ¼ e5t; s7 ¼ e7t; etc: ð18Þ

Consequently, we have

@

@x¼ e

@

@n; ð19Þ

and

@

@t¼ �e

@

@nþ e3 @

@t3 � e5 @

@t5 þ e7 @

@t7 � � � � : ð20Þ

It is important to note that the introduction of slow time variables normalized according to the dispersion relation expansionare such that they allow for an automatic elimination of the solitary-wave-related secular-producing terms appearing in theevolution equations for the higher-order terms of the wave field [4].

3. The multiple-time scales method

Following Zakharov and Kuznetsov [12,11] we use a multiple-time scales method to derive the KdV flow equation (2)from the RLW equation (1).

We now consider RLW equation (1) and we assume the following series expansions for solution:

uðx; tÞ ¼X1

n¼0

enþ1unðn0; n1; . . . ; nn; s0; s1; . . . ; snÞ; n ¼ 0;2;4; . . . ð21Þ

In order to study the long-wave limit of RLW equation (1), we will introduce slow space and multiple-time variables basedon the long-wave limit of the linear dispersion relation

wðkÞ ¼ k

1þ k2 ¼ kX1

n¼0

ð�1Þnk2n; ð22Þ

where k is the wave number. Based on this solution (21), we also define now the slow space and multiple-time variables withrespect to the scaling parameter e > 0 respectively as follows:

n ¼ eðx� tÞ; s3 ¼ a1e3t; s5 ¼ �a2e5t; s7 ¼ a3e7t; . . . ð23Þ

Introducing the expansions (19) and (20) into the field Eq. (1) and setting the coefficients of like powers of e equal to zero, weobtain the following sets of differential equations:

(i) For the coefficients of e2 , we find

a1u0s3 þ u0nnn � 6u0u0n ¼ 0: ð24Þ

(ii) For the coefficient of e4 , we find

u2nnn � 6u0u2n þ a1u2s3 � a1u0nns3 � 6u2u0n � a2u0s5 ¼ 0: ð25Þ

(iii) For the coefficients of e6 , we find

u4nnn � 6u0u4n þ a1u4s3 � a1u2nns3 � 6u2u2n � a2u2s5 � 2a2u0nns5 � 6u4u0n � a3u0s7 ¼ 0: ð26Þ

(iv) For the coefficients of e8 , we find

u6nnn�6u0u6nþa1u6s3 �a1u4nns3 �6u2u4n�a2u4s5 þa2u2nns5 �6u4u2n�a3u2s7 þa3u0nns7 �6u6u0n�a4u0s9 ¼0 ð27Þ

and so on.

3.1. The derivation of KdV flow equations

We now use (24) and take a1 ¼ 1, we obtain

u0s3 þ u0nnn � 6u0u0n ¼ 0; ð28Þ

the well known KdV equation (2) with u ¼ u0. It is known that taking in the Eq. (28) for travelling wave solution in the formu0ðx; tÞ ¼ /ðxþ ctÞ:

u0ðx; tÞ ¼ 2a2 sec h2aðxþ 4a2t þ hÞ; ð29Þ

where h is the phase [13]. We take u0 to be the solitary-wave solution of the KdV equation (28),

u0 ¼ �2k2 sec h2ðkn� 4k3s3Þ ð30Þ

2280 M. Naci Özer, M. Koparan / Applied Mathematics and Computation 218 (2011) 2277–2280

where choosing the phase h3 ¼ kðn� 4k2s3Þ.We now take in Eq. (25) a2 ¼ 1, we obtain the equation

u0s5 ¼ u2nnn � 6u0u2n þ u2s3 � u0nns3 � 6u2u0n; ð31Þ

and in this equation choosing u2 ¼ 4k2u0,

h5 ¼ kðn� 4k2s3 þ 16k4s5Þ;

the solitary wave solution

u0 ¼ �2k2 sec h2ðkn� 4k3s3 þ 16k5s5Þ

and use to imposing the natural (in the multiple-time formalism) compatibility condition

ðu0s3 Þs5¼ ðu0s5 Þs3

;

we obtain the evolution of u0 in the time s5 derive the fifth-order KdV equation with u ¼ u0

u0s5 ¼ u0nnnnn � 10u0u0nnn � 20u0nu0nn þ 30u20u0n: ð32Þ

We take in Eq. (26) a3 ¼ 1, we obtain the equation

u0s7 ¼ u4nnn � 6u0u4n þ u4s3 � u2nns3 � 6u2u2n � u2s5 � u0nns5 � 6u4u0n ð33Þ

and in this equation choosing u4 ¼ ð4k2u0Þ2,

h7 ¼ kðn� 4k2s3 þ 16k4s5 � 64k6s7Þ;

the solitary wave solution

u0 ¼ �2k2 sec h2ðkn� 4k3s3 þ 16k5s5 � 64k6s7Þ

and using to compatibility condition

ðu0s3 Þs7¼ ðu0s7 Þs3

;

we obtain the evolution of u0 in the time s7 derive the seventh-order KdV equation

u0s7 ¼ �u0nnnnnnn þ 14u0u0nnnnn þ 42u0nu0nnnn þ 70u0nnu0nnn � 280u0u0nu0nn � 70u20u0nnn � 70u3

0n þ 14u30u0n:

Extending this procedure, we obtain all equations of the KdV hierarchy.

4. Conclusion

We have used a multiple-time scales method to provide a new derivation of the KdV flow equations from the RLW equa-tion and then represent their solitary wave solution. The equations for the coefficients at each order in epsilon, contain nosecular terms in our derivation of KdV flow equations (2). Therefore no freedom is left in choosing coefficients at each orderin epsilon and the expansion is uniquely determined. Thus there exists a relation between the RLW equation (1) with theKdV flow equations (1). Details are given in [14].

References

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