The long wave limiting of the discrete nonlinear evolution equations

Chaos, Solitons and Fractals 42 (2009) 2965–2972

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Chaos, Solitons and Fractals

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

The long wave limiting of the discrete nonlinear evolution equations

Yi Zhang a,*, Hai-qiong Zhao b, Ji-bin Li a

a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR Chinab Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, PR China

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

Article history:Accepted 1 April 2009

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.04.047

* Corresponding author.E-mail address: [email protected] (Y. Zhang).

We show here that rational, positon, negaton, breather solutions of some discrete nonlin-ear evolution equations are presented via long wave limiting method. The discrete nonlin-ear evolution equations concerned are 1D Toda lattice, differential-difference KdV,differential-difference analogue KdV equations.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Recent studies of theory of soliton have shown that multi-soliton solutions of nonlinear evolution equations can be ob-tained by various methods. For example, the inverse scattering method [1], the Hirota’s bilinear method [2], the Darboux andB€acklund transformation [3]. More remarkable is that many physically important solutions can be presented explicitly,among which are rational, positon, negaton, breather [4] and complexiton [5] solutions. The rational solution of the KdVequation was first discovered by Airaut et al. [6]. Further results were obtained by Ablowitz and Satsuma [7], Narita [8], Nim-mo and Freeman [9], Hu [10] and so on. Later, Carstea obtained a class of nonsigular rational solutions from the 1D Volterrasystem [11]. In 1992, Matveev first introduced the concept of a positon and negaton as a new solution to the KdV equation[12]. On the basis of Matveev’s idea, the positon and negaton solutions were found in other integrable systems [14–16]. It isnoted that the positon solution is long-range analogues of solitons and is slowly decreasing, oscillating solutions of nonlinearintegrable equations. Under a proper choice of the scattering data, the positon solution has a following interesting property:the corresponding reflection coefficient is zero, but the transmission coefficient is unity [17]. For the negaton solution, thenumber of singularities and zeros are finite and they show very interesting time dependence. The general motion is in thepositive x direction, except for the certain negaton solution which exhibit one oscillation around the origin [18].

Jaworski recently pointed out that the positon and negaton solutions of some soliton equations can also be derived by aspecial limiting procedure of a classical soliton solution [19]. In fact this technique is the extension of Ablowitz and Sat-suma’s method for finding rational solution. In this paper, we unified call these as long wave limiting method. It’s a pity thatthe method applicable to the discrete nonlinear evolution equations is missing. Our main goal is to test whether the samelong wave limiting method is suitable for discrete nonlinear evolution equations to obtain rational, positon, negaton andbreather solutions. Here we give the examples of 1D Toda lattice, differential-difference KdV, and discrete analogue KdVequations. By starting with the Hirota’s bilinear method, we obtain the classical 2-soliton solution expression of these dis-crete nonlinear evolution equations. For the rational solution, we only choose suitable phase factors of the classical solitonsolutions and then take the wavenumbers to zero. But for the positon and negaton solutions we should set complex conju-gate or complex conjugate-like wavenumbers corresponding to appropriate phase factors. At last, collision of the negatonand soliton solutions also be displayed.

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2966 Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 2965–2972

2.1. The 1D Toda lattice equation

The 1D Toda lattice equation is given by [20]

o2

ot2 yn ¼ �eynþ1 � eyn�1 þ 2eyn ; ð2:1Þ

then substituting the dependent variable transformation

yn ¼ lnf 2n

fnþ1fn�1

� �: ð2:2Þ

The Eq. (2.1) can be written in bilinear form

D2t fn � fn ¼ 2ðcosh Dn � 1Þfn � fn; ð2:3Þ

where the bilinear operator Dx and Dt are defined by

Dmx Dn

t f � g ¼ ðox � ox0 Þmðot � ot0 Þnf ðx; tÞgðx0; t0Þjx0¼x;t0¼t : ð2:4Þ

By the perturbation method, we obtain 1-soliton and 2-soliton solutions

fn ¼ 1þ en1 ; ð2:5aÞfn ¼ 1þ en1 þ en2 þ en1þn2þA12 ; ð2:5bÞ

where n1; n2 is given by

n1 ¼ x1t þ k1nþ n01; n2 ¼ x2t þ k2nþ n0

2;x1 ¼ 2 sinh12

k1

� �;x2 ¼ 2 sinh

12

k2

� �ð2:6Þ

with arbitrary phase constants n01; n

02 and

eA12 ¼sinh2 k1�k2

4

� �sinh2 k1þk2

4

� � : ð2:7Þ

The fact that one can recover rational solution relies on our freedom of choosing the arbitrary phase constants. For exam-ple, if choosing en0

1 ¼ �1 and k1 ! 0, we have 1-rational solution

fn ¼ t þ n: ð2:8Þ

The same idea can also be applied to 2-soliton solution to obtain 2-rational solution. Taking en01 ¼ �en0

2 ¼ k1þk2k1�k2

andk1; k2 ! 0, we have 2-rational solution

fn ¼ t3 � 3t2nþ 3n2 � 34

� �t � n3 þ 1

4n: ð2:9Þ

Then we try to take 1-positon and 1-negaton solutions from 2-soliton solution. In what follows choosing k1 ¼ aþ bi,

k2 ¼ a� bi, n01 ¼

sinha2

sinb2

i, n02 ¼ �

sinha2

sinb2

i and substituting these into the Eq. (2.5b), then assembleing these we get breather-like

solution of the Eq. (2.1)

fn ¼ 1� e4 sinha2 cosb

2tþ2na �sinh a

2

sin b2

e2 sinha2 cosb

2tþna sin 2 cosha2

sinb2

t þ nb� �

: ð2:10Þ

This can be roughly described as an oscillating wave-packet modulated by a singular soliton-like solution envelope. Fig. 1illustrates this phenomenon.

At the same time, 1-positon solution of the Toda lattice equation also be obtained by taking the limit a! 0

fn ¼ 4 sinb2

cosb2

t þ n� �

þ 2 sin 2 sinb2

t þ nb� �

: ð2:11Þ

A multiplicative factor sinha2

sinb2

is discarded in fn, because it plays no effect on the Eq. (2.2) when we recover the solution (2.11)into the Eq. (2.2). If assuming b ¼ 2; t ¼ 0, Fig. 2 is illustrated.

On the other hand, if we choose k1 ¼ aþ bi; k2 ¼ �aþ bi, and take n01 ¼

sinb2

sinha2i, n0

2 ¼ �sinb

2sinha

2i, then let b! 0, 1-negaton solu-

tion of the Toda lattice equation is given by

fn ¼ 4 sinh a cosha2

t þ n� �

þ 2 sinh 2 sinha2

t þ na� �

: ð2:12Þ

At the same time, we substitute the Eq. (2.12) into the Eq. (2.2), and Fig. 3 is illustrated assuming a = 2, t = 0. From Figs. 2 and3, we know that all these positon and negaton solutions are singular.

1

2

3

4

5

2 4 6 8 10n

Fig. 2. A positon solution of the 1D Toda equation, yn for b = 2, t = 0.

0

1

2

3

4

5

2 4 6 8 10n

Fig. 3. A negaton solution of 1D Toda equation, yn for a = 2, t = 0.

0

1

2

3

4

2 4 6 8 10n

Fig. 1. A breather solution of the 1D Toda equation, yn for a = 1, b = 4, t = 0.

Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 2965–2972 2967

2.2. Collision solutions

The collision of all kinds of solutions is very interesting and can be studied in higher-order solutions. To describe the phe-nomenon more clearly, we choose the 1-soliton, 1-negaton collision solution to illuminate it. More complex collision solu-tions can be performed in the same method but more algebra computing. Here we only give the 3-soliton solution

fn ¼ 1þ en1 þ en2 þ en1þn2þA12 þ en3 þ en1þn3þA13 þ en2þn3þA23 þ en1þn2þn3þA123 : ð2:13Þ


In order to obtain negaton-soliton solution, choosing k1; k2; n01 and n0

2 the same as the negaton solution, the negaton-sol-iton solution is obtained by

fn ¼ 4 sinh a cosha2

t þ n� �

þ 2 sinh 2 sinha2

t þ na� �

þ en3sinh2 a�k3

4

sinh2 aþk34

e2 sinha2tþna �

sinh2 aþk34

sinh2 a�k34

e�2 sinha2t�na � 4 sinh

a2

cosha2

t þ n� � !

: ð2:14Þ

3.1. The differential-difference Korteweg-de Vries equation

The so-called differential-difference Korteweg-de Vires equation (DDKdV) is [21]

o

otun

1þ un

� �¼ un�1

2� unþ1

2: ð3:1Þ

Let us express un

un ¼fnþ1

2ðtÞfn�1

2ðtÞ

f 2n ðtÞ

� 1; ð3:2Þ

the Eq. (3.1) is transformed into [13]

sinhDn

4

� �Dt þ 2 sinh

Dn

2

� �� fn � fn ¼ 0; ð3:3Þ

where Dn;Dt is definition as the Eq. (2.5). The N-soliton soluton can be obtained from the Eq. (3.3), but here only need 1-sol-iton and 2-soliton solutions


n1 ¼ x1t � k1nþ n01; n2 ¼ x2t � k2nþ n0

2; ð3:5Þx1 ¼ sinh k1;x2 ¼ sinh k2; ð3:6Þ

eA12 ¼sinh2 k1�k2

2

sinh2 k1þk22

: ð3:7Þ

In order to obtain a 1-rational solution, we only take n01 ¼ �1 and k1 ! 0 to find

fn ¼ t � n: ð3:8Þ

Similarly, if selecting en01 ¼ �en0

2 ¼ k1þk2k1�k2

and k1; k2 ! 0, we obtain a 2-rational solution

fn ¼ t3 � 3t2n� �3n2 þ 34

� �t � n3 þ 1

4n; ð3:9Þ

With the choosing conjugate complex k1 ¼ aþ bi; k2 ¼ a� bi and n01 ¼ sinh a

sin b i; n02 ¼ � sinh a

sin b i, we obtain

fn ¼ 1� e2 sinh a cos bt�2na þ 2esinh a cos bt�na sinh a sinð� cosh a sin bt þ nbÞsin b

; ð3:10Þ

0

0.2

0.4

0.6

2 4 6 8 10n

Fig. 4. A breather solution of the DDKdV equation, un for a = 1, b = 2, t = 0.

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14 16 18 20n

Fig. 5. A positon solution of the DDKdV equation, un for b = 2, t = 0.


then put into the Eq. (3.2) to obtain a breather-like solution of the DDKdV equation (See Fig. 4).But if we choose the limiting of the a! 0, yield positon solution (See Fig. 5)

fn ¼ sin bðcos bt � nÞ � sinð� sin bt þ nbÞ: ð3:11Þ

In Fig. 5, un ¼fnþ1

2ðtÞf

n�12ðtÞ

f 2n ðtÞ

� 1 is illustrated.

On the other hand, if we choose k1 ¼ aþ bi; k2 ¼ �aþ bi and phase factors n01 ¼ sin b

sinh a i; n02 ¼ � sin b

sinh a i and the same limitingprocedure b! 0, yield 1-negaton solution (See Fig. 6).

fn ¼ sinh aðcosh at � nÞ þ sinhðsin at � naÞ: ð3:12Þ

3.2. Soliton-negaton interactions

The scattering of a kink soliton solution by the negaton solution can be studied by a N-soliton solution expression. Herefor the simple, we only study via a 3-soliton solution. The reasoning follows closely that of 1D Toda equation to obtain a 1-soliton and 1-negaton interaction solution (See Fig. 7).

fn ¼ sinh aðcosh at � nÞ þ sinhðsin at � naÞ þ en3sinh2 a�k3

2

sinh2 aþk32

esinh at�na �sinh2 aþk3

2

sinh2 a�k32

e� sinh atþna � 2 sinh aðcosh at � nÞ !

:

ð3:13Þ

4.1. The differential-difference analogue of the Korteweg-de Vries equation

The differential-difference analogue of the Korteweg-de Vries equation [DDAKdV] is [22]

d�1 un t þ d2

� �1þ un t þ d

2

� �� un t � d2

� �1þ un t � d

2

� � !

¼ un�12� unþ1

2; ð4:1Þ

0

0.05

2 4 6 8 10n

Fig. 6. A negaton solution of the DDKdV equation, un for a = 2, t = 0.

0

0.05

0.1

0.15

0.2

108642n

Fig. 7. Collision of soliton and negaton solution of DDKdV equation, un for a ¼ 1; k3 ¼ 2; t ¼ 0; n03 ¼ 0.


where d is an arbitrary parameter. The Eq. (4.1) can be reduced to the Eq. (3.1) by the small limit of d! 0. Substitutingtransformation

unðtÞ ¼fnþ1

2ðtÞfn�1

2ðtÞ

fnðt þ d2Þfnðt � d

2Þ� 1; ð4:2Þ

the bilinear form is in the following

sinhdDt þ Dn

4

� �d�1 sinh

dDt

2

� �þ sinh

Dn

2

� �� fn � fn ¼ 0: ð4:3Þ

The 1-soliton and 2-soliton solutions are


n1 ¼ x1t � k1nþ n01; n2 ¼ x2t � k2nþ n0

2; ð4:5Þsinhðdx1Þ ¼ d sinhðk1Þ; sinhðdx2Þ ¼ d sinhðk2Þ; ð4:6Þ

eA12 ¼sinh2 k1�k2�dðx1�x2Þ

2

� �sinh2 k1þk2�dðx1þx2Þ

2

� � : ð4:7Þ

By the same procedure as the 1D Toda and DDKdV equations, we set n01 ¼ �1 and k1 ! 0 to get a 1-rational solution

fn ¼ t � n: ð4:8Þ

For the 2-rational solution, choosing en01 ¼ �en0

2 ¼ k1þk2k1�k2

and k1; k2 ! 0, we obtain the 2-rational solution

fn ¼ t3 � 3nt2 � �3n2 þ 34þ 1

4d2

� �t � n3 þ 1

4þ 3

4d2

� �n: ð4:9Þ

By choosing of the wavenumbers k1 ¼ aþ bi; k2 ¼ a� bi and the phase factors n01 ¼

sinhðk1þk2�dðx1þx2 Þ2 Þ

sinhðk1�k2�dðx1�x2 Þ2 Þ

i, n02 ¼ �

sinhðk1þk2�dðx1þx2 Þ2 Þ

sinhðk1�k2�dðx1�x2 Þ2 Þ

i

and after some algebra calculations, a pair of counter propagating breather solution is given by

fn ¼ 1� e2At�2na þ 2eAt�na sinhð�aþ AtdÞ sinð�Bt þ nbÞsinð�bþ BtdÞ ; ð4:10aÞ

where

A ¼ �arcsinhðd sinhð�aþ biÞÞ þ arcsinhðd sinhðaþ biÞÞ2d

; ð4:10bÞ

B ¼ �iarcsinhðd sinhðaþ biÞÞ � iarcsinhðd sinhð�aþ biÞÞ2d

: ð4:10cÞ

At the same time, take the limit a! 0 to obtain a 1-positon solution

fn ¼sinðb� arcsinðd sinðbÞÞÞð2d cosðbÞt � 2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðd sinðbÞÞ2

qÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� ðd sinðbÞÞ2q

� d cosðbÞ� 2 sinðarcsinðd sinðbÞÞt � nbÞ: ð4:11Þ


On the other hand, selecting k1 ¼ aþ bi; k2 ¼ �aþ bi, and the limit b! 0, the final result is obtained

fn ¼sinhða� arcsinhðd sinhðaÞÞÞð2 coshðaÞt � 2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðd sinhðaÞÞ2

qÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ðd sinhðbÞÞ2q

� d coshðaÞþ 2 sinhðarcsinhðd sinhðaÞÞtd�1 � naÞ: ð4:12Þ

4.2. Interaction solutions

We are now presenting example of long wave limiting interaction solutions to the DDAKdV equation. Choosing 3-solitonsolution and selecting of the wavenumbers and phase factors the same as the positon solution, the interaction solution canbe obtained by

fn ¼sinhða� arcsinhðd sinhðaÞÞÞð�2 coshðaÞt þ 2n



1þ ðd sinhðaÞÞ2q

� d coshðaÞþ 2 sinhðarcsinhðd sinhðaÞÞtd�1 � naÞ

þ en3sinh2 a�k3�arcsinhðd sinhðaÞÞþdx3

2

� �sinh2 aþk3�arcsinhðd sinhðaÞÞ�dx3

2

� � earcsinhðd sinhðaÞÞtd�1�na �sinh2 aþk3�arcsinhðd sinhðaÞÞ�dx3

2

� �sinh2 a�k3�arcsinhðd sinhðaÞÞþdx3

2

� � e�arcsinhðd sinhðaÞÞtd�1þna

8<:

þsinh a� arcsinhðd sinhðaÞÞð Þð�2 coshðaÞt þ 2n



1þ ðd sinhðaÞÞ2q

� d coshðaÞ

9>=>;: ð4:13Þ

5. Conclusion and discussion

We know that long wave limiting method is powerful to obtain rational, positon, negaton, and breather solutions. At thesame time we conjecture the skill can also used in a higher-order soliton solution to obtain N-positon, M-negaton and col-lision solutions. Ma recently find a novel class of explicit exact solution called complexiton [23], and we believe that the longwave limiting method can also be applied to obtain this novel solution and will be discussed elsewhere. In a word, the pro-cedure is robust and the phenomenon is likely to be universal.

Acknowledgements

The authors thank the referees for their helpful suggestions. This work is supported by the National Natural Science Foun-dation of China (Nos. 10771196 and 10831003), the Natural Science Foundation of Zhejiang Province (No. Y7080198).

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The long wave limiting of the discrete nonlinear evolution equations