singular solutions in casoratian form for two diï¬€erential

Chaos, Solitons and Fractals 23 (2005) 1333–1350

www.elsevier.com/locate/chaos

Singular solutions in Casoratian form for twodifferential-difference equations

Da-jun Zhang *

Department of Mathematics, Shanghai University, Shanghai 200436, PR China

Accepted 15 June 2004

Communicated by Prof. M. Wadati

Abstract

Negatons, positons, rational solutions and mixed solutions in Casoratian form for the Toda lattice and the differ-

ential-difference KdV equation are obtained. Some characteristics of the obtained singular solutions are investigated

through density graphics.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that soliton solutions can be represented in Wronskian form [1–3] and the Wronskian technique [4–

7] is an efficient and direct way to finding soliton solutions for nonlinear evolution equations. This technique admits

direct and simple verifications of the solutions, and there have been some generalizations surrounding it. For example,

it has been generalized to find rational solutions and mixed soliton-rational solutions in Wronskian (or Casoratian, the

discrete analogue of Wronskian) form for few soliton equations [8,9] by following the idea of long-wave limitation [10].

Recently, we also developed some new determinantal identities to obtain exact solutions for some soliton equations

with self-consistent sources [11–14]. Another meaningful generalization came from Sirianunpiboon, Howard and

Roy (SHR) [15] who altered the conditions satisfied by Wronskian entries and derived a more general Wronskian solu-

tion for the KdV equation. SHR�s generalization enables us to write more sorts of solutions into Wronskian form.

From their results one can not only obtain soliton solutions [4] and rational solutions [8], but also obtain the so-called

positons [16,17], a typical kind of singular solutions. In two very recent papers, the SHR�s procedure [15] was respec-

tively used to discuss various solutions of the KdV equation [18] and the KdV equation with loss and nonuniformity

terms [19].

One of purposes in this paper is to generalize the SHR�s procedure to discrete cases to obtain various solutions, such

as solitons, rational solutions and other singular solutions, and mixed solutions, in Casoratian form. Further than that,

we are very interested in the characteristics of these singular solutions, although some of their physical meanings are still

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.06.034

* Tel.: +86 21 66132182; fax: +86 21 66133292.

E-mail address: [email protected].

mailto:[email protected].

1334 D.-j. Zhang / Chaos, Solitons and Fractals 23 (2005) 1333–1350

not clear. In this paper, we study two differential-difference equations, the Toda lattice and the differential-difference

KdV equation. With regard to the Toda lattice, we first generalize the conditions satisfied by Casoratian entries to obtain

more solutions in Casoratian form. Some obtained solutions are novel. We also discuss classifications of the obtained

solutions and relations between them. Then we analyze the characteristics of some singular solutions with the help of

density graphics drawn through Mathematica. This kind of graphics enables us to describe clearly the singularity traces

and find their special properties. Similarly, we derive and discuss solutions of the differential-difference KdV equation.

The paper is organized as follows. In Section 2, we generalize the SHR�s procedure to the Toda lattice and obtain a

series of solutions in Casoratian form. In Section 3, we investigate the characteristics of some singular solutions with the

help of density graphics. In Section 4, we derive and discuss solutions of the differential-difference KdV equation.

2. Solutions in Casoratian form of the Toda lattice

In this section, we generalize the SHR�s procedure to the Toda lattice. Let us first list some concerned results about

this lattice. The Toda lattice is [20]

xn;tt ¼ exn�1�xn � exn�xnþ1 : ð2:1Þ

By employing new potentials un ¼ exn�1�xn and vn ¼ �xn;t, it reads

un;t ¼ unðvn � vn�1Þ;vn;t ¼ unþ1 � un:

ð2:2Þ

This equation is Lax integrable and has the following Lax pair

Unþ1 þ unUn�1 þ vnUn ¼ kUn; ð2:3aÞ

Un;t ¼ Un � unUn�1: ð2:3bÞ

Another form of the Toda lattice (2.1) is given as

½lnð1þ V nÞ�tt ¼ V nþ1 � 2V n þ V n�1; ð2:4Þ

where

V n ¼ e�yn � 1; yn ¼ xn � xn�1: ð2:5Þ

And further through the transformation

V n ¼ ðln fnÞtt; ð2:6Þ

Hirota gave the bilinear form of the Toda lattice [21]

fnfn;tt � f 2n;t � fn�1fnþ1 þ f 2

n ¼ 0: ð2:7Þ

Now we derive solutions to the bilinear equation (2.7). To generalize the SHR�s procedure to the Toda lattice, we first

give the following lemma which can be proved by using determinant definition.

Lemma 1. Suppose that jAj is an N · N determinant, P is some operator, PcðjÞ j A j means P only acts on every entry in

the jth column of jAj and PrðjÞ j A j means P only acts on every entry in the jth row of jAj. Then we have

XN XN
j¼1
PcðjÞ j A j¼j¼1

PrðjÞ j A j : ð2:8Þ

By virtue of this lemma, we can prove the following result about the Casoratian solutions of the Toda lattice.

Theorem 1. The following Casoratian fn solves the bilinear equation (2.7):

fn ¼ Casðw1ðn; tÞ;w2ðn; tÞ; . . . ;wN ðn; tÞÞ ¼j 0; 1; . . . ;N � 1 j¼j dN� 1 j; ð2:9Þ
and here each wj(n, t) enjoys
wjðnþ 1; tÞ þ wjðn� 1; tÞ ¼Xjs¼1

ajsðtÞwjðn; tÞ; ð2:10Þ

D.-j. Zhang / Chaos, Solitons and Fractals 23 (2005) 1333–1350 1335

�wj;tðn; tÞ ¼ awjðnþ 1; tÞ þ bwjðn; tÞ þ cwjðn� 1; tÞ; ð2:11Þ

where each ajs(t) is a real number or a function of t but independent of n, b is an arbitrary real number and the constant pair

(a, c) is equal to ð12;� 1

2Þ or (1,0) or (0,1).

The verification of the above theorem is similar to Ref. [7] and we need make use of the formulas
XNk¼1
akkðtÞ j dN� 1 j¼j dN� 2;N j þ j �1; gN� 1 j;

XNk¼1

akkðtÞ j dN� 2;N j¼j dN� 3;N � 1;N j þ j dN� 2;N þ 1 j þ j dN� 1 j þ j �1; gN� 2;N j;

and
XNk¼1
akkðtÞ j �1; gN� 1 j¼j dN� 1 j þ j �1; gN� 2;N j þ j �2; gN� 1 j þ j �1; 0;N � 1 j :

These formulas can be generated from Eq. (2.10) and Lemma 1. Here we let gN� j; gN� j and N � j indicate the sets ofconsecutive columns 0,1,2, . . .,N � j, consecutive columns 1,2, . . .,N � j and consecutive columns 2, . . . ,N � j,

respectively.

The function wj(n, t) which satisfies the conditions of (2.10) and (2.11) can be taken as

wjðn; tÞ ¼ cosh gj ð2:12Þ
or
wjðn; tÞ ¼ sinh gj; ð2:13Þ

where

gj ¼ kjnþ t sinh kj þ gð0Þj ; ð2:14Þ

and both kj and gð0Þj are real constants; or

wjðn; tÞ ¼ cos hj ð2:15Þ
or
wjðn; tÞ ¼ sin hj; ð2:16Þ

where

hj ¼ kjnþ t sin kj þ hð0Þj ; ð2:17Þ

and hð0Þj is also a real constant; or [7]

wjðn; tÞ ¼ aþj ekjnþtekj þ a�j e

�kjnþte�kj ð2:18Þ

or [9]

wjðn; tÞ ¼ aþj ekjnþtðekj�1Þ þ a�j e

�kjnþtðe�kj�1Þ; ð2:19Þ

where both aþj and a�j are real constants; or [22]

wjðn; tÞ ¼ et cos kj cos hj ð2:20Þ

or

wjðn; tÞ ¼ et cos kj sin hj: ð2:21Þ

We note that if we substitute �t for t in the above functions, the Casoratian (2.9) is still a solution to (2.7).

The classical soliton solutions can be generated from (2.12) and (2.13). In fact, similar to the procedure in the Ref.

[11,12], we can rewrite the Casoratian

fn ¼ Cas cosh g1; sinh g2; cosh g3; . . . ;1

2½egN � ð�1ÞNe�gN �

� �ð2:22Þ


into

fn ¼ 2�NYN

16 j<l

2 sinhkl � kj

2

!exp �

XNj¼1

gj þN � 1

2kj

� �( )Xl¼0;1

expXNj¼1

2ljnj þXN16 j<l

ljllAjl

( ); ð2:23aÞ

where 0<k1<k2< � � �<kN, the sum over l = 0,1 refers to each of the lj = 0,1, j = 0,1, . . . ,

eAjl ¼sinh

kl�kj2

sinhklþkj2

!2

; ð2:23bÞ

and

nj ¼ gj þN � 1

2kj �

1

4

XNl¼1;l 6¼j

Ajl: ð2:23cÞ

Eq. (2.23a) can be considered as the same as the so-called N-soliton solution in Hirota form [21]. A detailed proof for

(2.23) can be found in Appendix A.

Besides the soliton solutions, we can also derive singular solutions on the basis of Theorem 1. For simplicity, here-

after, we set gð0Þj ¼ 0 and hð0Þj ¼ 0 in (2.14) and (2.17) and aþj =a�j ¼ �1. Now, we reconsider g as a function of k, n and t,

i.e.,

g ¼ gðk; n; tÞ ¼ knþ t sinh k; ð2:24Þ

then we can have the expansion

wðk þ d; n; tÞ ¼ cosh gðk þ d; n; tÞ ¼Xþ1

s¼0

Esðk; n; tÞds; ð2:25aÞ

where

Esðk; n; tÞ ¼1

s!os

okscosh gðk; n; tÞ ¼ Es: ð2:25bÞ

Noticing that

wðk; nþ 1; tÞ þ wðk; n� 1; tÞ ¼ 2 cosh k � wðk; n; tÞ; ð2:26Þ

and

wtðk; n; tÞ ¼1

2wðk; nþ 1; tÞ � 1

2wðk; n� 1; tÞ; ð2:27Þ

it is easy to find that each Es satisfies

Esðk; nþ 1; tÞ þ Esðk; n� 1; tÞ ¼ 2Xsj¼0

asjEjðk; n; tÞ; ð2:28Þ

and

Es;tðk; n; tÞ ¼1

2Esðk; nþ 1; tÞ � 1

2Esðk; n� 1; tÞ; ð2:29Þ

where

asj ¼1

ðs� jÞ!os�j

oks�j cosh k ¼ 1

ðs� jÞ! coshðs�jÞk: ð2:30Þ

So, according to Theorem 1, the Casoratian

fn ¼ CasðE0ðk; n; tÞ;E1ðk; n; tÞ; . . . ;Esðk; n; tÞÞ ð2:31Þ

is a solution to Eq. (2.7). Similarly, it is not difficult to find that the following Casorati determinants also solve (2.7):

fn ¼ CasðF 0ðk; n; tÞ; F 1ðk; n; tÞ; . . . ; F sðk; n; tÞÞ; ð2:32Þ

fn ¼ CasðG0ðk; n; tÞ;G1ðk; n; tÞ; . . . ;Gsðk; n; tÞÞ; ð2:33Þ


and

fn ¼ CasðH 0ðk; n; tÞ;H 1ðk; n; tÞ; . . . ;Hsðk; n; tÞÞ; ð2:34Þ
where
F sðk; n; tÞ ¼1

s!os

okssinh gðk; n; tÞ ¼ F s; ð2:35Þ

Gsðk; n; tÞ ¼1

s!os

okssin hðk; n; tÞ ¼ Gs; hðk; n; tÞ ¼ knþ t sin k; ð2:36Þ

Hsðk; n; tÞ ¼1

s!os

okscos hðk; n; tÞ ¼ Hs: ð2:37Þ

Similar to Ref. [23], we denote the solutions generated from (2.31)–(2.34) by [Cs], [Ss], ½eSs� and ½eCs�, respectively.Rational solutions can also be derived from Theorem 1. Let g (k,n, t) be defined as (2.24) and we expand

cosh gðk; n; tÞ as

cosh gðk; n; tÞ ¼X1j¼0

Qjðn; tÞk2j; ð2:38aÞ

where

Qjðn; tÞ ¼X2js¼0

ns

s!p2j�sð~tÞ; ð2:38bÞ

and

psð~tÞ ¼Xkak¼s

~ta

a!; ð2:38cÞ

a ¼ ða1; a3; a5; . . .Þ; aj P 0; ðj ¼ 1; 3; 5; . . .Þ;kak ¼ a1 þ 3a3 þ 5a5 þ � � � ; a! ¼ a1!a3!a5! � � � ;~t ¼ ðt1; t3; t5; . . .Þ; tj ¼ t

j! ; ~ta ¼ ta11 ta33 t

a55 � � � :

Using Eqs. (2.26) and (2.27), we can get

Qjðnþ 1; tÞ þ Qjðn� 1; tÞ ¼Xjs¼0

2

½2ðj� sÞ�!Qsðn; tÞ; ð2:39Þ

Qj;tðn; tÞ ¼1

2Qjðnþ 1; tÞ � 1

2Qjðn� 1; tÞ: ð2:40Þ

That means, the following Casoratian

fn ¼ CasðQ0ðn; tÞ;Q1ðn; tÞ; . . . ;Qmðn; tÞÞ; ð2:41Þ

solves the bilinear equation (2.7). Solutions are generated from (2.41) and (2.6) are rational solutions of the Toda lattice

(2.4). Similarly, for sinh gðk; n; tÞ, we have

sinh gðk; n; tÞ ¼X1j¼0

Rjðn; tÞk2jþ1; ð2:42aÞ

where

Rjðn; tÞ ¼X2jþ1

s¼0

ns

s!p2jþ1�sð~tÞ; ð2:42bÞ

and the Casoratian

fn ¼ CasðR0ðn; tÞ;R1ðn; tÞ; . . . ;Rmðn; tÞÞ ð2:43Þ

can also lead to rational solutions.


By now, we have obtained a few functions that meet conditions of (2.10) and (2.11), including the functions defined

by (2.12)–(2.20), and function sets {E0,E1, . . . ,Es}, {F0,F1, . . . ,Fs}, {G0,G1, . . . ,Gs}, {H0,H1, . . . ,Hs}, {Q0,Q1, . . . ,Qm},

and {R0,R1, . . . ,Rm}. These functions can generate various mixed solutions in Casoratian form, for example,

fn ¼ Casðw1; . . . ;wm;E0ðks1Þ; . . . ;Es1ðks1Þ; . . . ;E0ðksiÞ; . . . ;EsiðksiÞ;F 0ðkm1

Þ; . . . ; F m1ðkm1

Þ; . . . ; F 0ðkmjÞ; . . . ; F mjðkmj Þ;G0ðkn1Þ; . . . ;Gn1ðkn1Þ;. . . ;G0ðknlÞ; . . . ;GnlðknlÞ;Q0;Q1; . . . ;Qh1 ;R0;R1; . . . ;Rh2Þ;

ð2:44Þ

where {wj = wj(n, t)} are some functions defined in (2.12)–(2.20).

In the rest part of this section, let us discuss the classifications of the obtained solutions and the relations between

them.

For the spectral problem (2.3a) where (un,vn) = (1,0) and k = x + x�1, i.e.,

Unþ1 þ Un�1 ¼ xþ 1

x

� �Un; ð2:45Þ

(2.12) and (2.13) are solutions to (2.45) when x is a real constant ekj , i.e., jkj>2, and in this case, the classical soliton

solutions are generated [22]. On the other hand, when x is taken as a complex constant eikj , i.e., jkj<2, Eq. (2.45) has

solutions (2.15) and (2.16) which generate solutions with singularities. Comparing with the Schrodinger spectral

problem

�/xx þ u/ ¼ k/; ð2:46Þ

where k<0 leads to classical soliton solutions to the KdV equation and k>0 generates singular solutions called positons

of the KdV equation [16,17], Matveev and coworker [22] called solutions ½eSs� and ½eCs� positons of the Toda lattice. In

addition, following the concept of negatons of the KdV equation [23], we name [Cs] and [Ss] to be negatons of the Toda

lattice. This kind of solutions correspond to the multi-poles solutions obtained from the inverse scattering transform

[24,25].

[Cs] and [Ss] are physically different and we can see their different characteristics in the next section, while ½eSs� and½eCs� only have the difference of a phase shift.

Similar to the derivations of [Cs] and [Ss], we can also derive singular solutions from (2.18)–(2.21), but these solu-

tions are not new and they are the same as [Cs] or [Ss] or ½eSs� or ½eCs�. To explain this fact, we give the following lemma.

Lemma 2. Suppose that

/ðk; n; tÞ ¼ gðk; tÞwðk; n; tÞ; ð2:47Þ

where g(k, t) is independent of n, and /(k,n, t), g(k, t) and w(k,n, t) are (s � 1)-order differentiable with respect to k. Let

Cas½/ðk; n; tÞ; s� ¼ Casð/ðk; n; tÞ;/ð1Þðk; n; tÞ; . . . ;/ðs�1Þðk; n; tÞÞ; ð2:48Þ

where

/ðjÞðk; n; tÞ ¼ oj

okj/ðk; n; tÞ:

Then we have

Cas½/ðk; n; tÞ; s� ¼ gsðk; tÞ � Cas½wðk; n; tÞ; s�: ð2:49Þ

Proof. By using Leibnitz�s rule for high order derivatives, it is not difficult to obtain

Cas½/ðk; n; tÞ; s� ¼j A j � Cas½wðk; n; tÞ; s�;

where A = (aij) is a lower triangular s · s matrix and its entries are defined as

aij ¼0; i < j;

Cj�1i�1g

ði�jÞðk; tÞ; iP j:

�
That means (2.49) is valid and we complete the proof. h
By virtue of this lemma and noticing that

eknþtek þ e�knþte�k ¼ 2et cosh k coshðknþ t sinh kÞ; ð2:50Þ


eknþtek � e�knþte�k ¼ 2et cosh k sinhðknþ t sinh kÞ; ð2:51Þ

eknþtðek�1Þ þ e�knþtðe�k�1Þ ¼ 2etðcosh k�1Þ coshðknþ t sinh kÞ; ð2:52Þ

eknþtðek�1Þ � e�knþtðe�k�1Þ ¼ 2etðcosh k�1Þ sinhðknþ t sinh kÞ; ð2:53Þ

we know that solutions generated from these four functions are the same as [Cs] or [Ss]. Of course, (2.20) and (2.21) lead

to the same solutions as ½eCs� and ½eSs� respectively. We also note here that because

cos hðk; n; tÞ ¼ cosh gðik; n; tÞ; ð2:54Þ

where h(k,n, t) and g(k,n, t) are respectively defined by (2.36) and (2.24), cos hðk; n; tÞ and cosh gðk; n; tÞ lead to same

rational solutions, and so do sin hðk; n; tÞ and sinh gðk; n; tÞ. Besides, functions (2.18)–(2.21) can not lead to new rational

solutions either. We take (2.52) as an example to explain this fact. Rewrite (2.52) as

/ðk; n; tÞ ¼ gðk; tÞwðk; n; tÞ;

where

/ðk; n; tÞ ¼ eknþtðek�1Þ þ e�knþtðe�k�1Þ; gðk; tÞ ¼ 2etðcosh k�1Þ;wðk; n; tÞ ¼ coshðknþ t sinh kÞ:

As /(k,n, t) is an even function with respect to k, the rational solutions can be denoted by

fn ¼ Casð/ðk; n; tÞ;/ð2Þðk; n; tÞ; . . . ;/ð2s�2Þðk; n; tÞÞk¼0: ð2:55Þ

Noticing that g(k, t) and w(k,n, t) are also even functions of k and similar to the proof of Lemma 2, we have

fn ¼ gsð0; tÞ � Casðwðk; n; tÞ;wð2Þðk; n; tÞ; . . . ;wð2s�2Þðk; n; tÞÞk¼0: ð2:56Þ

That means rational solutions derived from (2.52) can also formally be denoted by the Casoratian (2.41). h

To sum up, in this section we have obtained a few solutions in Casoratian form, including solitons, negatons, posi-

tons, rational solutions and various mixed solutions.

3. Singular solutions of the Toda lattice

In this section, we analyze the characteristics of some singular solutions with the help of density graphics drawn

through Mathematica. This kind of graphics enables us to describe clearly the behaviors and traces of the waves with

movable singularities. We note here that when we make these graphics we suppose n to be a continuous variable so that

we can observe the trend and motion of waves in a better way.

Let us first consider positons. The simplest positon is ½eS 0� i.e.,

V n ¼ � sin2k

sin2h; ð3:1Þ

which is generated from

fn ¼ sin h: ð3:2Þ

In Fig. 1 we give the wave shape and density graphics of (3.1). Fig. 1(b) is a density graphics where white background

denotes values of Vn being near to zero, and those parallel dark stripes denote the movable singularities where Vn tends

to negative infinity. It is clear to see that solution (3.1) has infinitely many singularities where Vn tends to negative infin-

ity, the distance between each two singularities are unchanged and all the singularities move from right to left with con-

stant speed � sin kk .

The positon ½eS 1� is given by

V n ¼�½2 sin k cosðk þ 2hÞ � sin 2k�2 þ 4sin2k½ð1þ 2nÞ sin k þ t sin 2k � sinðk þ 2hÞ� sinðk þ 2hÞ

½ð1þ 2nÞ sin k þ t sin 2k � sinðk þ 2hÞ�2; ð3:3Þ

which corresponds to the Casoratian

fn ¼ Casðsin h; sinð1ÞhÞ ¼ � 1

2½ð1þ 2nÞ sin k þ t sin 2k � sinðk þ 2hÞ�: ð3:4Þ

Fig. 1. Zero-positon as given by (3.1) for k = 1. (a) Shape of zero-positon at t = �1. (b) Density image of zero-positon for n2 [�15,15]

and t2 [�10,10].

Fig. 2. One-positon as given by (3.3) for k = 0.75. (a) Shape of one-positon at t = 2. (b) Density image of one-positon for n2 [�25,25],

t2 [�30,30] and plot range of [�2.2,0.8].


The shape and motion of (3.3) are described as Fig. 2. In Fig. 2(b), i.e., the density graphics, the grey–white stripes

denote the alternant changes of negative and positive amplitudes as it is described in Fig. 2(a), the dark black stripe just

like the diagonal of the graphics denotes the movable singularities where Vn !� 1, while the several brightest stripes

on both sides of the dark black stripe, represent waves with large positive amplitudes.

In Fig. 2(b), along with the singularity trace there are three special points which we call the singularity oscillation

points. These points appear periodically and some interesting facts are related to them. First, there will exist two pos-

itive-amplitude wave peaks at the same time and the wave will add two more zeros when n and t near to the singularity

oscillation points. Taking (0,0) as an example, we record this process in Fig. 3. The second interesting fact is that, cou-

pled with the change of zeros, the number of singularities is changed. From Fig. 2(b), one can find the singularity moves

continuously from right to left with constant speed except in the areas of the singularity oscillation points. Also taking

(0,0) as an example, as we see in Fig. 4, there are three singularities at the same time of t = 0 and they correspond to

n = �1,�0.5 and 0. In addition, roughly speaking, the singularity motion changes direction for a very short time when

the wave passes these singularity oscillation points. We can observe this change from Fig. 2(b) and from the first and the

sixth graphics in Fig. 3.

Obviously, these special changes of zeros and singularities are different from the 1-positon of the KdV equation of

which both the number of singularities and the number of zeros are unchanged [23].

In what follows we begin to discuss negatons. The simplest negaton [S0] is

V n ¼ � sinh2k

sinh2gð3:5Þ

-4 -2 2 4

-30

-20

-10

10

t =- 0.4

-10 -5 5 10

-4

-3

-2

-1

1

2

3t =- 0.1

-10 -5 5 10

-3

-2

-1

1

2

3t =- 0.09

-2 -1.5 -1 -0.5 0.5 1

-1000

-800

-600

-400

-200

200

400

t = 0

-4 -2 2 4

-3

-2

-1

1

2

3t = 0.1

-4 -2 2 4

-25

-20

-15

-10

-5

5

10

t =0.5

Fig. 3. Shape and motion of one-positon as given by (3.3) when k = 0.75 and t is near to zero.

-3 -2 -1 1 2 3

-0.1

-0.05

0.05

0.1

Fig. 4. Shape and zeros of fn as given by (3.4) for k = 0.75 and t = 0.


with the related Casoratian

fn ¼ sinh g: ð3:6Þ

In the corresponding density graphics Fig. 5(b), white background means the value of Vn near to zero while the black

strip denotes singularity trace. There is no zero and the single singularity moves from right to left with constant speed

� sinh kk .

The negaton [S1] is given as

V n ¼�4sinh2k½ðcosh k � coshðk þ 2gÞÞ2 þ ð1þ 2nþ 2t cosh kÞ sinh k sinhðk þ 2gÞ � sinh2ðk þ 2gÞ�

½ð1þ 2nÞ sinh k þ t sinh 2k � sinhðk þ 2gÞ�2; ð3:7Þ

and the related Casoratian is

fn ¼ � 1

2½ð1þ 2nÞ sinh k þ t sinh 2k � sinhðk þ 2gÞ�: ð3:8Þ

The shape and motion of [S1] are described in Fig. 6. In the density graphics Fig. 6(b), grey background denotes the

value of Vn near to zero, dark areas still denote negative singularity trace while two bright stripes denote the waves with

positive amplitudes. For [S1] with k = 0.7, the point (n, t) = (0,0) is a singularity oscillation point, i.e., two positive wave

peaks appear and the wave adds two more new zeros when it is near the point, the number of singularities becomes from

one to three when t is near to 0, and there is an oscillation of singularity when the wave passes the point. Fig. 7(a) shows

Fig. 6. One-negaton as given by (3.7) for k = 0.7. (a) Shape of one-negaton at t = 2. (b) Density image of one-negaton for n2 [�20,20],

t2 [�15,15] and plot range of [�3,1].

Fig. 5. Zero-negaton as given by (3.5) for k = 1. (a) Shape of zero-negaton at t = 2. (b) Density image of zero-negaton for n2 [�14,14]

and t2 [�10,10].


four zeros and three singularities of negaton [S1] at t = 0, and these characteristics are caused by three zeros of Casor-

atian (3.8) for k = 0.7 and t = 0, as it is described in Fig. 7(b).

Now we consider negatons [Cs]. Negaton [C0] is just 1-soliton solution while [C1] has singular properties. [C1] is given

as

V n ¼�4sinh2kfðcosh k þ coshðk þ 2gÞÞ2 � sinhðk þ 2gÞ½ð1þ 2nÞ sinh k þ t sinh 2k þ sinhðk þ 2gÞ�g

½ð1þ 2nÞ sinh k þ t sinh 2k þ sinhðk þ 2gÞ�2; ð3:9Þ


fn ¼1

2½ð1þ 2nÞ sinh k þ t sinh 2k þ sinhðk þ 2gÞ�: ð3:10Þ

In the corresponding density graphics Fig. 8(b), grey background still denotes the value of Vn being near to zero,

dark area denotes negative singularity trace while two bright stripes denote the waves with positive amplitudes. Orig-

inally, the singular point moves towards (n, t) = (0,0) with constant speed. And accompanied by the motion of singu-

larity, one positive-amplitude wave decays gradually while another positive-amplitude wave increases gradually. These

two positive amplitudes change their roles in the area near to (n, t) = (0,0) where the speed of the singularity becomes a

bit slow. Then, the singularity leaves the area and moves forward again with the original constant speed.

So far we have investigated some singular properties of positons and negatons. A typical characteristic for ½eS 1� and[S1] is that, in the area of singularity oscillation point, both the number of zeros and the number of singularities increase,

and the singularity trace has an oscillation.

-2 -1.5 -1 -0.5 0.5 1

-600

-400

-200

200

400

600

t = 0

-1.5 -1 -0.5 0.5

-0.1

-0.05

0.05

0.1

(a) (b)

Fig. 7. One-negaton as given by (3.7) for k = 0.7 and t = 0, and Casoratian fn as given by (3.8) for k = 0.7 and t = 0. (a) Shape of one-

negaton at t = 0. (b) Shape and zeros of fn for t = 0.

Fig. 8. One-negaton as given by (3.9) for k = 1. (a) Shape of one-negaton at t = 2. (b) Density image of one-negaton for n2 [�35,25],

t2 [�20,30] and plot range of [�4,1].


Similar to positons and negatons, the rational solutions that we have obtained also have singular properties. For the

sake of convenience, we denote the rational solutions generated from Casoratian (2.41) and (2.43) by ½Rmc � and ½Rm

s �respectively.

½R1c � is given by

V n ¼ � 1

12þ nþ t

� �2 ; ð3:11Þ


fn ¼1

2þ nþ t: ð3:12Þ

Its shape and motion are quite similar to the negaton [S0].

Figs. 9 and 10 correspond to the rational solutions ½R2c � and ½R1

s �. ½R2c � is given as

V n ¼�144 3

4þ 2ðnþ tÞ þ ðnþ tÞ2

h i2� 1

6ð1þ nþ tÞ½3þ 11nþ 9t þ 12ðnþ tÞ2 þ 4ðnþ tÞ3�

� �½3þ 11nþ 9t þ 12ðnþ tÞ2 þ 4ðnþ tÞ3�2

; ð3:13Þ

derived from the related Casoratian

fn ¼1

4þ 11n

12þ 3t

4þ ðnþ tÞ2 þ 1

3ðnþ tÞ3; ð3:14Þ

Fig. 9. Two-rational solution as given by (3.13). (a) Shape of two-rational solution at t = 2. (b) Density image of two-rational solution

for n2 [�35,35], t2 [�20,20] and plot range of [�2,0.5].

Fig. 10. One-rational solution as given by (3.15). (a) Shape of one-rational solution at t = 2. (b) Density image of one-rational solution

for n2 [�35,35], t2 [�20,20] and plot range of [�2,0.4].


and ½R1s � is given as

V n ¼�36½ðnþ tÞ þ ðnþ tÞ2�2 � 6½1þ 2ðnþ tÞ�½nþ 3ðnþ tÞ2 þ 2ðnþ tÞ3�

½nþ 3ðnþ tÞ2 þ 2ðnþ tÞ3�2; ð3:15Þ

with the related Casoratian

fn ¼n6þ ðnþ tÞ2

2þ ðnþ tÞ3

3: ð3:16Þ

From these two figures, we can find that both ½R2c � and ½R1

s � travel following the similar shape, motion and singular

properties to [S1], including the roles change of positive-amplitude waves and the singularity oscillation point. The points

ð� 12; 0Þ and (0,0) are respectively the singularity oscillation points of ½R2

c � and ½R1s �. In the area near such a point, there

exist two positive amplitude waves at the same time, the number of zeros changes from two to four, the number of sin-

gularities changes from one to three, and there is a sort of oscillation for the singularity trace.

To sum up, we have analyzed some characteristics of singularities for positons, negatons and rational solutions. The

singularities show some interesting behaviors when t is near to zero. For example, the positon ½eS 1� has periodically infi-

nitely many singularity oscillation points; besides, negaton [S1] and rational solutions ½R2c � and ½R1

s � also have their own

singularity oscillation points at t = 0.

4. Solutions in Casoratian form of the differential-difference KdV equation

The differential-difference KdV (DDKdV) equation was introduced by Wadati, which was derived from the Volterra

system and related to a sort of ladder circuit [26]. This equation reads [26,21,27,7]


� W n

1þ W n

� �t

¼ W n�12� W nþ1

2ð4:1Þ

and can be written in the bilinear form [27]

sinh1

4Dn

� �Dt � 2 sinh

1

2Dn

� �� fn � fn ¼ 0; ð4:2Þ

or

fn;tfn�12� fnfn�1

2;t � fn�1fnþ1

2þ fn�1

2fn ¼ 0; ð4:3Þ

with the transformation

W n ¼ðcosh 1

2DnÞfn � fnf 2n

� 1 ¼fnþ1

2fn�1

2

f 2n

� 1; ð4:4Þ

where Dn is defined as [21]

e�Dnan � bn ¼ anþ�bn�e:

It is well known that both the Toda lattice and the DDKdV equation have similar solution representations in Hirota

form and Wronskian form. Here, for the DDKdV equation, we have the following results similar to Theorem 1.

Theorem 2. The following Casoratian fn solves the bilinear equation (4.3):

fn ¼ Casðw1ðn; tÞ;w2ðn; tÞ; . . . ;wN ðn; tÞÞ; ð4:5Þ

and each wj(n, t) enjoys

wjðn; tÞ þ wjðnþ 1; tÞ ¼Xjs¼1

ajsðtÞwj nþ 1

2; t

� �; ð4:6Þ

wj;tðn; tÞ ¼ wjðnþ 1; tÞ ð4:7aÞ

or

wj;tðn; tÞ ¼ �wjðn� 1; tÞ ð4:7bÞ

or

wj;tðn; tÞ ¼1

2wjðnþ 1; tÞ � 1

2wjðn� 1; tÞ; ð4:7cÞ

where each ajs(t) is a real number or an even function of t.

Proof.When each wj(n, t) meets the conditions of (4.6) and (4.7a), on the basis of Lemma 1 and Ref. [7], it is not difficult

to verify the Casoratian (4.5) to be a solution of Eq. (4.3). For (4.7b) and (4.7c), we choose an alternative proof instead

of direct verifications. First, the bilinear equation is formally unchanged under the symmetric transformation with

respect to the origin (n, t) = (0,0), i.e.,

n ! �n; t ! �t: ð4:8Þ

In fact, under the above transformation, (4.3) turns out

�f�n;tf�nþ12þ f�nf�nþ1

2;t � f�nþ1f�n�1

2þ f�nþ1

2f�n ¼ 0; ð4:9Þ

where for convenience we still employ f to denote the new function transformed from the original one. Eq. (4.9) is just

the same as (4.3) after imposing n a shift of 2n� 12. Similarly, under (4.8) the condition (4.6) is also unchanged while

(4.7a) is just changed to (4.7b). That means the Casoratian (4.5) solves (4.3) when each wj(n, t) meets (4.6) and

(4.7b). Next, noticing that (4.3) is linear with respect to of/ot, the Casoratian (4.5) also solves (4.3) as well when each

wj(n, t) satisfies (4.6) and (4.7c). Thus we complete the proof. h

Here we note that, similar to Theorem 1, we can add an extra term bwj(n, t) to the conditions (4.7a)–(4.7c) where b is

an arbitrary real number; but unlike the Toda lattice, Theorem 2 is not valid if we substitute �t for t.


The entries of the Casoratian fn (4.5) can also be taken as the same as the functions (2.12)–(2.21), and by virtue of

Lemma 2, it is enough to consider solutions generated from (2.12)–(2.17). In addition, it is not difficult to prove that

each Es, Fs, Gs, Hs, Qj and Rj, given respectively by (2.25b), (2.35), (2.36), (2.37), (2.38b) and (2.42b), satisfy that

Fig. 11

t2 [�6

Esðk; nþ 1; tÞ þ Esðk; n; tÞ ¼ 2Xsj¼0

asjEj k; nþ 1

2; t

� �; asj ¼

1

2s�jðs� jÞ!os�j

oks�j coshk2;

F sðk; nþ 1; tÞ þ F sðk; n; tÞ ¼ 2Xsj¼0

asjF j k; nþ 1

2; t

� �;

Gsðk; nþ 1; tÞ þ Gsðk; n; tÞ ¼ 2Xsj¼0

csjGj k; nþ 1

2; t

� �; csj ¼

1

2s�jðs� jÞ!os�j

oks�j cosk2;

Hsðk; nþ 1; tÞ þ Hsðk; n; tÞ ¼ 2Xsj¼0

csjHj k; nþ 1

2; t

� �;

Qjðnþ 1; tÞ þ Qjðn; tÞ ¼ 2Xjk¼0

bjkQk nþ 1

2; t

� �; bjk ¼

1

4j�k ½2ðj� kÞ�!;

Rjðnþ 1; tÞ þ Rjðn; tÞ ¼ 2Xjk¼0

bjkRk nþ 1

2; t

� �;

and

HtðnÞ ¼1

2Hðnþ 1Þ � 1

2Hðn� 1Þ;

where H(n) can be considered as Es, Fs, Gs, Hs, Qj and Rj.

Here we list some solutions in Casoratian form of the DDKdV equation.

Soliton solution is denoted by (2.22) and it is the same as the Hirota�s result [27]. Negatons [Cs] and [Ss] are respec-

tively denoted by (2.31) and (2.32), and positons ½eSs� and ½eCs� are respectively denoted by (2.33) and (2.34). Rational

solutions ½Rmc � and ½Rm

s � are respectively denoted by (2.41) and (2.43). Of course, the DDKdV equation can also have

mixed solutions in the form of (2.44).

Taking the advantage of density graphics, we can discuss properties of singularities of the above obtained singular

solutions. Here, we only consider some solutions which shows different characteristics from those of the Toda lattice.

Let us first consider negaton [S1] which is given as

W n ¼An

f 2n

; ð4:10aÞ

. One-negaton as given by (4.10) for k = 1. (a) Shape of one-negaton at t = 1. (b) Density image of one-negaton for n2 [�10,10],

,6] and plot range of [�2.5,0.4].

Fig. 12. Images of one-negaton as given by (4.10a) and related functions as k = 1 and t = 0. (a) Shape of one-negaton at t = 0 and plot

range of [�120,60]. (b) Shape and zeros of An as given by (4.10b) t = 0. (c). Shape and zeros of fn as given by (3.8) t = 0.


where fn is given by (3.8) and

Fig. 1

n2 [�3

An ¼1

4ð2n sinh k þ t sinh 2k � sinh 2gÞ½2ð1þ nÞ sinh k þ t sinh 2k � sinh ð2gþ 2kÞ� � f 2

n : ð4:10bÞ

From the corresponding graphics, i.e., Figs. 11 and 12, we can find that the area near to the origin (n, t) = (0,0) is a

special area where the wave performs different behaviors from the Toda lattice. Comparing with Fig. 6, we call the point

(n, t) = (0,0) a weak singularity oscillation point. In fact, there are still an oscillation of the singularity trace and two pos-

itive amplitudes in the area near to this point; and as t approaches zero, the number of zeros begins to change and even

reaches four from two. However, as it is described in Fig. 12(a), when t = 0, there are only two zeros; comparing with a

typical singularity oscillation point which has three singularities, the number of singularities here is only two but not

three; and no other zeros and singularities are there between these two singularities. (For this reason we call

(n, t) = (0,0) a weak singularity oscillation point.) Mathematically, this kind of difference maybe due to the fact that

An has the same zeros as fn at t = 0, as it is described in Fig. 12(b) and (c).

Weak singularity oscillation points are also found for some other singular solutions of the DDKdV equation. Fig. 13

corresponds to positon ½eS 1� which is given as

W n ¼An

f 2n

; ð4:11aÞ


An ¼1

4ð2n sin k þ t sin 2k � sin 2hÞ½2ð1þ nÞ sin k þ t sin 2k � sinhð2hþ 2kÞ� � f 2

n : ð4:11bÞ

The wave has periodically infinitely many weak singularity oscillation points.

In addition, rational solutions ½R2c � and ½R1

s � also possess weak singularity oscillation points: (�1/2,0) and (0,0), respec-

tively; and their shape and motion are quite similar to negaton [S1]. ½R2c � is given as

3. One-positon as given by (4.11) for k = 0.7. (a) Shape of one-positon at t = �1. (b) Density image of one-positon for

0,30], t2 [�35,35] and plot range of [�0.5,0.1].


W n ¼An

f 2n

; ð4:12aÞ


An ¼ � 1

48ð1þ nþ tÞ½3þ 11nþ 15t þ 12ðnþ tÞ2 þ 4ðnþ tÞ3�; ð4:12bÞ

and ½R1s � is given as

W n ¼An

f 2n

; ð4:13aÞ


An ¼ � 1

48ð1þ 2nþ 2tÞ½nþ 3t þ 6ðnþ tÞ2 þ 2ðnþ tÞ3�: ð4:13bÞ

Thus, in this section, we have derived solitons, negatons, positons, rational solutions and mixed solutions for the

DDKdV equation. Some singular solutions have typical weak singularity oscillation points which are different from

the singularity oscillation points of the Toda lattice.

5. Conclusions

In this paper we generalized conditions which Casoratian entries satisfy and obtained various solutions in Casora-

tian form for the Toda lattice and the DDKdV equation. These solutions include solitons, negatons, positons, rational

solutions and mixed solutions which can described various interactions between different types of waves. We showed the

uniformity of soliton solutions in Hirota form and in Casoratian form, and this uniformity can acts as an alternative

proof for Hirota�s result in Refs. [21,27]. We also discussed the relations between solutions generated from some dif-

ferent Casoratian entries. In addition, with the help of density graphics, we analyzed characteristics for some obtained

singular solutions. This kind of graphics enables us to describe clearly singularity traces. We introduced the name of

singularity oscillation point to describe characteristics of some solutions. For the Toda lattice, positon ½eS 1�, negaton[S1] and rational solutions ½R2

c � and ½R1s � have singularity oscillation points. The typical characteristic of such a point

is that in the area near to this point, there exist two positive-amplitude waves at the same time, the number of zeros

changes from two to four; the number of singularities changes from one to three when t = 0; and there is a sort of oscil-

lation for the singularity trace. For the DDKdV equation, negaton [S1], positon ½eS 1�, rational solution ½R2c � and ½R1

s � haveweak singularity oscillation points. There are still an oscillation of singularity trace and two positive amplitudes in the

area near to such a point; and as t approaches zero, the number of zeros begins to change and even reaches four from

two. However, when t = 0, there are only two zeros and two singularities. To sum up, we have derived some solutions in

Casoratian form and discussed them by means of density graphics. Although the matching physical backgrounds are

not found for these singular properties, it is still mathematically interesting to describe these characteristics. The pro-

cedure used in this paper can apply to other differential-difference systems.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (10371070), the Youth Foundation

of Shanghai Education Committee and the Special Funds for Major Specialities of Shanghai Education Committee.

Appendix A

The proof of Eq. (2.23). By using the addition rule of determinants, the Casoratian (2.22) can be represented by the sum

of 2N Vandermonde determinants, i.e.,

fn ¼ 2�NX�¼�1

�2�4 � � � �2 N2½ �Dðe

�1k1 ; e�2k2 ; . . . ; e�N kN ÞeP

Nj¼1�jgj ; ðA:1Þ


where the Vandermonde determinant

Dðe�1k1 ; e�2k2 ; . . . ; e�N kN Þ ¼Y

16 j<l6N

ðe�lkl � e�jkj Þ: ðA:2Þ

Then we have

fn ¼ 2�NX�¼�1

YN16 j<l

�lðe�lkl � e�jkjÞ" #

eP

Nj¼1�jgj

¼ 2�NYN

16 j<l

2 sinhkl � kj

2

!Xl¼0;1

YN16 j<l

ellklþljkj�klþkj

2

! YN16 j<l

sinhkl�kj2

sinhklþkj2

!2ljll�lj�ll24 35ePNj¼1ð2lj�1Þgj ;

where lj = (�j + 1)/2 and we have made use of

�l sinh�lkl � �jkj

2¼ sinh

kl þ kj2

sinhkl�kj2

sinhklþkj2

!2ljll�lj�llþ1

:

Next, noticing that
YN16 j<l
ellklþljkj�klþkj

2 ¼ exp �XNj¼1

N � 1

2kj � ljðN � 1Þkj

� �( );

and
YN16 j<l
sinhkl�kj2

sinhklþkj2

!�lj�ll

¼ exp �XNj¼1

lj

XNl¼1;l6¼j

Ajl

2

!( );

we can further obtain Eq. (2.23) and finish the proof. h

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