An extended two-dimensional Toda lattice hierarchy and two ...

JOURNAL OF MATHEMATICAL PHYSICS 49, 093506 共2008兲

An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources Xiaojun Liu,1,a兲 Yunbo Zeng,2,b兲 and Runliang Lin2,c兲 1

Department of Applied Mathematics, China Agricultural University, Beijing 100083, People’s Republic of China 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China 共Received 8 September 2007; accepted 8 August 2008; published online 10 September 2008兲

We extend the two-dimensional Toda lattice hierarchy 共2DTLH兲 by its squared eigenfunction symmetries. This extended 2DTLH 共ex2DTLH兲 includes the twodimensional Toda lattice equation with self-consistent sources 共2DTLSCS兲 as its first nontrivial equation. Lax representation of ex2DTLH is also presented. With the help of the Lax representation, we construct a nonauto-Bäcklund Darboux transformation 共DT兲 for 2DTLSCS by applying the variation of constants to 2DTLSCS auto-Bäcklund DT. This DT enables us to find many solutions to 2DTLSCS, including solitons, rational solutions, positons, negatons, and complexitons. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2976685兴

I. INTRODUCTION

In this paper, we will discuss an extension to the well-known two-dimensional Toda lattice hierarchy1 共2DTLH兲 by using its squared eigenfunction symmetry. This extended hierarchy will contain the two-dimensional Toda lattice equation with self-consistent sources.2 The 2DTLH is a very important integrable hierarchy studied by Ueno and Takasaki,1 the construction of which was inspired by the studies of the KP hierarchy. It includes the twodimensional Toda lattice equation3 and involves infinite collection of Lax-type equations in two sets of independent variables. The role of pseudodifferential operators in the machinery and construction of the KP hierarchy is played by shift operators in the 2DTLH. The importance of 2DTLH is not only represented by its physical applications 共e.g., in interface growth4兲, geometrical applications 关in connection with Laplace–Darboux transformation 共DT兲 for general second order partial differential equations5 and classification of certain surfaces in space related as focal or caustic surfaces6兴, and theoretical values 共connection with infinite dimensional Lie algebras1,7兲 but also by the links between it and other integrable hierarchies. For instance, 2DTLH can be embedded into the two-component KP hierarchy and can be reduced to the one-dimensional Toda lattice hierarchy, to sinh-Gordon equation, etc.1 Squared eigenfunction symmetry is a widely used object in integrable systems. “Squared” usually means a product of eigenfunction and its adjoint associated with the auxiliary linear problem of the nonlinear integrable equations. In integrable equations of one spatial 共continuous or discrete兲 and one temporal 共i.e., 1 + 1兲 dimensions, squared eigenfunction arises from the variational derivative of the eigenvalue that is associated with the spectral problem of the Lax pair. It is related to symmetries of the integrable equation. Such an idea is crucial in nonlinearization procedure and 共high order兲 restricted flows.8,9 By its identification with an usual symmetry of the a兲

Electronic mail: [email protected]. Electronic mail: [email protected]. Electronic mail: [email protected].

b兲 c兲

0022-2488/2008/49共9兲/093506/19/$23.00

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J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

integrable equation, it leads to finite dimensional integrable systems8–10 共or integrable symplectic maps11兲. Many of such systems reported can be derived from a general r-matrix structure.12 The same idea was generalized to the reduction in 共2 + 1兲-dimensional integrable systems. In the KP and 2DTLH cases, the x-derivative of squared eigenfunction ␺␺ⴱ was shown to be symmetry generation functions.13 The symmetry reduction in the KP hierarchy utilizes the squared eigenfunction to produce the Ablowitz-Kaup-Newell-Segur 共AKNS兲 hierarchy, Yajima–Oikawa hierarchy, etc. In the Sato theory, ␺␺ⴱ is naturally connected with pseudodifferential operators ␺⳵−1␺ⴱ. So the above reductions can be described as k-constrained KP hierarchy14,15 by using pseudodifferential operators. The restricted flows and constrained KP hierarchy also play important roles in obtaining soliton equations with self-consistent sources 共SESCSs兲. SESCSs have wide physical applications in fields such as plasma, hydrodynamics, and solid-state physics.16–18 In the 共2 + 1兲-dimensional case such as the KP hierarchy, the KP hierarchy with self-consistent sources was generated by identifying its stationary equation with constrained KP hierarchies.19 However, this approach is incomplete in the sense that although it is effective in obtaining the KP equation with selfconsistent sources 共called the first-type KPSCS兲, it cannot obtain the other 共called the second兲 type of KP with self-consistent sources, which also appeared in Ref. 16 and rediscoverd with the source generating approach by Ref. 20. Our extended KP hierarchy21 solved this problem. The idea of extended KP was to replace one particular flow Ltk = 关Bk , L兴 of the KP hierarchy by L␶k = 关Bk + 兺qi⳵−1ri , L兴, where the squared eigenfunction symmetries were used. This idea is quite close to the work of Oevel,22 Oevel and Carillo,23 and Aratyn et al.,24 where they considered an extra flow Lz = 关兺qi⳵−1ri , L兴 for KP. However, this is the first time that the above extension is used to implement the KP with sources of both the first and second types. In this paper, we focus on the same extension of 2DTLH based on Ref. 21. It will help us get the two-dimensional Toda lattice equation with self-consistent sources 共2DTLSCS兲 and its Lax presentation simultaneously. Due to the close relationship between 2DTLH and interface growth,4 it is reasonable to speculate on the potential application of 2DTLSCS in the same field. However, it is beyond the scope of the current paper. To show the extended 2DTLH 共ex2DTLH兲, as the first step, the squared eigenfunction symmetry for 2DTLH is studied in the framework of difference operator via Sato’s approach. Although the squared eigenfunction symmetry and symmetry generating function of KP are studied extensively for a long time, this is not the case for 2DTLH. So it is still an interesting topic to study. First let us briefly recall Ueno and Takasaki’s1 construction of 2DTLH: Let x = 共x1 , x2 , . . .兲 and y = 共y 1 , y 2 , . . .兲 be two series of independent continuous variables and n 苸 Z be a discrete variable. Then 2DTLH is defined by four infinite collections of Lax-type equations: Lxm = 关Bm,L兴,

共1a兲

Lym = 关Cm,L兴,

共1b兲

M xm = 关Bm,M兴,

共1c兲

M ym = 关Cm,M兴,

m = 1,2, . . . .

共1d兲

The difference operators L and M are defined by L = ⌳ + u0 + u1⌳−1 + u2⌳−2 + ¯ ,

M = v−1⌳−1 + v0 + v1⌳ + v2⌳2 + ¯ ,

where ⌳ stands for a shift operator, which satisfies ⌳f共n兲 = f共n + 1兲⌳.1 Potentials ui and vi are 1

Here the notation is different from Ref. 1: instead of an infinite dimensional matrix we use a shift operator.

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J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

functions of x, y, and n, Bm = L+m denotes the positive part 共ⱖ0兲 of Lm with respect to the powers of ⌳ and Cm = M −m denotes the negative part 共⬍0兲 of M m, and the bracket stands for the usual commutator of operators. The key point for the definition of 共1兲 is that the four sets of flows can be proven to mutually commute, the compatibilities of which give a zero curvature form for the 2DTLH: Bk,xm − Bm,xk + 关Bk,Bm兴 = 0,

共2a兲

Ck,ym − Cm,yk + 关Ck,Cm兴 = 0,

共2b兲

Bk,ym − Cm,xk + 关Bk,Cm兴 = 0.

共2c兲

When m = k = 1, 共2兲 lead to the equations uy = v − v共1兲 ,

共3a兲

vx = v共u − u共−1兲兲.

共3b兲

␺x = B共␺兲 = 共⌳ + u兲共␺兲,

共4a兲

␺y = C共␺兲 = 共v⌳−1兲共␺兲,

共4b兲

The Lax pair for 共3兲 is

where x ª x1, y ª y 1, B ª B1, C ª C1, u ª u0, and v ª v−1. Eliminating u from 共3兲 and replacing v by v = exp共q − q共−1兲兲,

共5兲

we will obtain the so-called two-dimensional Toda lattice equation: qxy = exp共q − q共−1兲兲 − exp共q共1兲 − q兲.

共6兲

The 2DTLSCS was first presented in Ref. 25 by the source generating method as follows: qxy = eq−q

共−1兲

wi,y = eq−q

− eq

共−1兲

+ 兺 共wiwⴱi 兲y ,

共7a兲

共i = 1, . . . ,N兲,

共7b兲

共1兲−q

w共−1兲 i

wⴱi,y = − eq

共1兲−q

共1兲

wⴱi .

共7c兲

Our ex2DTLH can be presented briefly as follows: First we introduce a new vector field ⳵¯yk, which is a linear combination of all vector fields ⳵ym. The corresponding flow is proved to be connected with the squared eigenfunction of 2DTLH. Then we derive a new Lax-type equation accompanied with the condition of time evolutions of eigenfunctions. Commutativities of ⳵¯yk, ⳵ym, and ⳵xk flows give rise to our ex2DTLH. This hierarchy helps us derive the 2DTLSCS in a different way from Refs. 2 and 25 and to obtain their Lax representations. In the second part of our paper, we will solve 2DTLSCS by DT. Since the Lax representation for 2DTLSCS is obtained, it is not difficult to construct a DT for 2DTLSCS, which can transform solutions of 2DTLSCS to solutions with the same number of source terms. Unfortunately this auto-Bäcklund transformation cannot give a nontrivial solution from a trivial one. So we virtually need a nonauto-Bäcklund transformation for constructing solutions. The idea is to consider 2DTLSCS as 2DTL with nonhomogeneous terms 共i.e., self-consistent source terms兲. By applying the method of variation of constant 共VC兲 to the DT, we find a new nonauto-Bäcklund DT which

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Liu, Zeng, and Lin

transforms a solution of the two-dimensional Toda lattice with N self-consistent sources to a solution with 共N + 1兲 self-consistent sources. Next, we obtain the m-time DT formula by induction. With the help of this formula, we can find a number of solutions to 2DTLSCS, such as solitons, rational solutions, positons, negatons, and complexitons. This paper will be organized as follows: In Sec. II we present the new ex2DTLH, which includes 2DTLSCS. A crucial step in resolvent identities is proved. In Sec. III we first construct the usual DT for 2DTLSCS; then by applying VC to this DT, we find a DT which can increase the sources’ number by 1. The m-time DT formula follows by induction. Its final form is expressed in terms of Casorati determinants. In Sec. IV, we show some smooth and nonsmooth solutions of 2DTLSCS by DT. Some properties of these solutions are also illustrated. In Sec. V, we give conclusion and comments. II. NEW EXTENDED TWO-DIMENSIONAL TODA LATTICE HIERARCHY A. Sato approach and resolvent identities

First we show some useful notations and definitions. Definition 1 (Residue of shift operator): Let P = 兺i苸Z Pi⌳i. Then the residue of P is Res⌳ P = P0 . Definition 2 (Adjoint operator ⴱ): Pⴱ = 兺i苸Z⌳−i Pi. Definition 3 [Shift operator’s action P共␭n兲]: P共␭n兲 =

兺 Pi⌳i共␭n兲 = 冉 i苸Z 兺 P i␭ i冊 · ␭ n .

i苸Z

Definition 4 (Formal inverses of difference operator ⌬): For difference operator ⌬ = ⌳ − 1, the formal inverses are given by ⌬+−1 = − 兺 ⌳i

or

⌬−−1 =

iⱖ0

兺 ⌳i .

iⱕ−1

Define wave operators ˆ 共⬁兲 = b + b ⌳−1 + b ⌳−2 + ¯ , W 0 1 2

ˆ 共0兲 = c + c ⌳ + c ⌳2 + ¯ , W 0 1 2

where b0 = 1. Ueno and Takasaki1 proved that if L and M are solutions to 共1兲 then there are wave operators by which L and M can be written as ˆ 共⬁兲−1, ˆ 共⬁兲⌳W L=W

ˆ 共0兲⌳−1W ˆ 共0兲−1 M=W

共8a兲

and ˆ 共⬁兲 = − Lm W ˆ 共⬁兲, ⳵ xmW ⬍0 ˆ 共⬁兲 = M m W ˆ 共⬁兲, ⳵ y mW ⬍0

ˆ 共0兲 = Lm W ˆ 共0兲 , ⳵ xmW ⱖ0

共8b兲

ˆ 共0兲 = − M m W ˆ 共0兲 . ⳵ y mW ⱖ0

共8c兲

Define wave function ˆ 共⬁兲共␭n兲e␰共x,␭兲 = w共⬁兲 = W

冉兺 冊 iⱖ0

ˆ 共0兲共␭n兲e␰共y,␭ w共0兲 = W

−1兲

=

bi␭−i ␭ne␰共x,␭兲 ª wˆ共⬁兲␭ne␰共x,␭兲 ,

冉兺 冊

ci␭i ␭ne␰共y,␭

iⱖ0

−1兲

ª wˆ共0兲␭ne␰共y,␭

−1兲

and adjoint wave function

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J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

ˆ 共⬁兲ⴱ−1共␭−n兲e−␰共x,␭兲 ª wˆ共⬁兲ⴱ␭−ne−␰共x,␭兲 , w共⬁兲ⴱ = W ˆ 共0兲ⴱ−1共␭−n兲e−␰共y,␭−1兲 ª wˆ共0兲ⴱ␭−ne−␰共y,␭−1兲 , w共0兲ⴱ = W where ␰共x , ␭兲 = 兺iⱖ1xi␭i, ␰共y , ␭−1兲 = 兺iⱖ1y i␭−i. Then 共1兲 can also be written as the compatibilities of following eigenvalue problems: Lw共⬁兲 = ␭w共⬁兲,

Mw共0兲 = ␭−1w共0兲 ,

共9a兲

⳵xmw共⬁兲 = Bmw共⬁兲,

⳵xmw共0兲 = Bmw共0兲 ,

共9b兲

⳵ymw共⬁兲 = Cmw共⬁兲,

⳵ymw共0兲 = Cmw共0兲 .

共9c兲

Lemma 1 (Ueno and Takasaki1): Suppose P = 兺Pi⌳i, Q = 兺Q j⌳ j. Then Res⌳ P · Qⴱ = Res␭ ␭−1 P共␭n兲Q · 共␭−n兲. Proof: It is sufficient to prove a special case, say, P = Pi⌳i, Q = Q j⌳ j: Res⌳ PQⴱ = Res⌳ Pi⌳i−jQ j = ␦i,j PiQ j, while Res␭ ␭−1 P共␭n兲Q共␭−n兲 = Res␭ ␭−1 Pi␭n+iQ j␭−n−j = ␦i,j PiQ j. 䊐 Similar to the KP theory, in which the principal part of the resolvent can be expressed in terms of a quadratic form of the wave function and adjoint wave function,26 we have the following resolvent identities for 2DTLH. Proposition 1 (Resolvent identities): k k ␭k = − w共0兲⌬+−1w共0兲 , Lⱖ0 ␭−k = − w共⬁兲⌬+−1w共⬁兲 , 兺 M ⱖ0 兺 kⱖ0 k苸Z ⴱ

ⴱ

k k L⬍0 ␭−k = w共⬁兲⌬−−1w共⬁兲 , 兺 M ⬍0 ␭k = w共0兲⌬−−1w共0兲 . 兺 k苸Z k⬎0 ⴱ

ⴱ

ˆ 共⬁兲⌳W ˆ 共⬁兲−1, Proof: We prove one of them. The others are similar. Since L = W k ˆ 共⬁兲⌳kW ˆ 共⬁兲−1兲 L⬍0 = 共W ⬍0

= =

ˆ 共⬁兲⌳kW ˆ 共⬁兲 Res⌳共W 兺 mⱖ1

−1

⌳m兲⌳−m

兺 Res␭ ␭−1共Wˆ共⬁兲⌳k共␭n兲e␰共x,␭兲兲 · 共⌳−mWˆ共⬁兲

ⴱ−1

共␭−n兲e−␰共x,␭兲兲⌳−m

mⱖ1

=

Res␭ ␭k−1w共⬁兲⌳−mw共⬁兲 兺 mⱖ1

ⴱ

= Res␭ ␭k−1w共⬁兲⌬−−1w共⬁兲ⴱ . k ␭−k = w共⬁兲⌬−−1w共⬁兲ⴱ. So 兺k苸ZL⬍0

䊐

B. New extended two-dimensional Toda lattice hierarchy

For fixed k ⱖ 1, N ⬎ 0, we define a new time variable ¯y k such that the corresponding vector field is N

␭ij⳵y , 兺 i=1 j⬎1

⳵¯yk = ⳵yk + 兺

j

共10兲

where ␭i are distinct nonzero parameters. Then the ¯y k flow is given by

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J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

⳵ ¯ ,L兴, L = 关C k ⳵¯yk

⳵ ¯ ,M兴, M = 关C k ⳵¯yk

共11兲

with N

␭ijC j , 兺 i=1 jⱖ1

¯ =C +兺 C k k

which, according to Proposition 1, can be rewritten as N

¯ = C + 兺 w共0兲⌬−1w共0兲ⴱ . C k k − i i i=1

ⴱ

共0兲 For simplicity, we denote w共0兲 by wi and wⴱi . Then the compatibility of 共1兲 and 共11兲 gives i and wi the following: Proposition 2: New ex2DTLH. For m ⫽ k:

Bm,xk − Bk,xm + 关Bm,Bk兴 = 0,

共12a兲

¯ ¯ Cm,y¯k − C k,y m + 关Cm,Ck兴 = 0,

共12b兲

¯ ¯ Bm,y¯k − C k,xm + 关Bm,Ck兴 = 0,

共12c兲

Bk,ym − Cm,xk + 关Bk,Cm兴 = 0,

共12d兲

wi,xm = Bm共wi兲,

wi,ym = Cm共wi兲

ⴱ ⴱ 共wⴱi 兲, = − Bm wi,x m

共i = 1, . . . ,N兲,

ⴱ ⴱ wi,y 共wⴱi 兲 = − Cm m

共12e兲 共12f兲

and for m = k: ¯ + 关B ,C ¯ Bk,y¯k − C k,xk k k兴 = 0,

共13a兲

⳵xkwⴱi = − Bⴱk 共wⴱi 兲

共13b兲

⳵xkwi = Bk共wi兲,

共i = 1, . . . ,N兲,

¯ = C + 兺N w ⌬−1wⴱ. where C k k i=1 i − i Note that hereafter wi and wⴱi do not need to be precisely the wave function and adjoint wave function. So we call them eigenfunction and adjoint eigenfunction. In fact, Eqs. 共12e兲 and 共12f兲 关or 共13b兲兴 ensure the closeness of Eqs. 共12a兲–共12d兲 关or 共13a兲兴. Furthermore, under conditions 共12e兲 and 共12f兲 关or 共13b兲兴, one can easily obtain the Lax representations of 共12a兲–共12d兲 as

␺xm = Bm共␺兲,

␺xk = Bk共␺兲,

共14a兲

␺ym = Cm共␺兲,

¯ 共␺兲 ␺¯yk = C k

共14b兲

¯ 共␺兲. ␺¯yk = C k

共15兲

or get the Lax representation of 共13a兲 as

␺xk = Bk共␺兲,

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093506-7

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

Example 1 (Two-dimensional Toda lattice equation with self-consistent sources): When m = k = 1, let u = u0, v = v−1, x = x1, y = ¯y 1, B1 = ⌳ + u, Then (13) becomes

冉

N

uy = − ⌬ v + 兺 wiwⴱi

C1 = v⌳−1 .

共−1兲

i=1

With substitutions u = qx, v = exp共q − q

vx = v共u − u共−1兲兲,

,

ⴱ wi,x = − Bⴱ1共wⴱi 兲,

wi,x = B1共wi兲, 共−1兲

冊

i = 1, . . . ,N.

共16a兲

共16b兲

兲, (16) yields

qxy = eq−q

共−1兲

− eq

N

+ 兺 共wiwⴱi 兲x ,

共17a兲

共i = 1, . . . ,N兲,

共17b兲

− qxwⴱi .

共17c兲

共1兲−q

i=1

wi,x = w共1兲 i + q xw i ⴱ = − wⴱi wi,x

共−1兲

This is the two-dimensional Toda lattice equation with N self-consistent sources. Analogously, let us introduce ¯xk such that N

␭−j 兺 i ⳵x , i=1 jⱖ1

⳵¯xk = ⳵xk + 兺

j

then we will get another new ex2DTLH as follows. For m ⫽ k, ¯ 兴 = 0, Bm,x¯k − ¯Bk,xm + 关Bm,B k

共18a兲

¯ 兴 = 0, Cm,x¯k − ¯Bk,ym + 关Cm,B k

共18b兲

Cm,yk − Ck,ym + 关Cm,Ck兴 = 0,

共18c兲

Bm,yk − Ck,xm + 关Bm,Ck兴 = 0,

共18d兲

wi,ym = Cm共wi兲,

wi,xm = Bm共wi兲,

ⴱ ⴱ wi,y = − Cm 共wⴱi 兲, m

i = 1, . . . ,N,

ⴱ ⴱ wi,x = − Bm 共wⴱi 兲, m

共18e兲 共18f兲

and for m = k, ¯ 兴 = 0, Ck,x¯k − ¯Bk,yk + 关Ck,B k

⳵ykwi = Ck共wi兲,

⳵ykwⴱi = − Cⴱk 共wⴱi 兲,

i = 1, . . . ,N,

共19a兲 共19b兲

N where ¯Bk = Bk − 兺i=1 wi⌬+−1wⴱi .

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093506-8

J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

Example 2 [2DTLSCS (Refs. 2 and 25)]: When m = k = 1, (19) leads to (7). It is interesting to see that (7) is equivalent to (17) under the transformations x → − y,

y → − x,

wi → − eqwⴱi ,

q → q,

wⴱi → e−qwi .

So hereafter we may concentrate on 2DTLSCS (17). This transformation was discovered by Hu. III. DARBOUX TRANSFORMATION FOR 2DTLSCS

In the second part of our paper, we discuss 2DTLSCS 共16兲. First recall the Lax pair of 2DTL 共4兲. For convenience, we denote B = B1, C = C1, u = u0, v = v−1, x = x1, y = y 1. A. Applying the method of variation of constant to Darboux transformation for 2DTLSCS

Let us first present the definition of the Casorati determinant: for m discrete functions h1 , . . . , hm, the Casorati determinant

Cas共h1, . . . ,hm兲 ª

冨

h1 h共1兲 1 ]

¯

hm

¯

共1兲 hm

]

]

共m−1兲 ¯ hm h共m−1兲 1

冨

.

DT for 2DTL 共3兲 is given by Ref. 27. Let us give a slightly different proof in the following lemma. Lemma 2: Let h be a particular eigenfunction of (4), D = ⌳ + ␴, ␴ ª −h共1兲 / h . Then DT ˜u ª u共1兲 + ␴ − ␴共1兲 ,

共20a兲

˜v ª v␴/␴共−1兲 ,

共20b兲

˜␺ ª D共␺兲 = Cas共h, ␺兲 h

共20c兲

˜ , ˜v兲 is a new solution to (3). gives a new solution to (4). Hence 共u −1 ˜ ˜ Proof: Since B ª ⌳ + ˜u, C ª ˜v⌳ , ˜␺ = D共␺兲, a sufficient condition to ensure 共4兲 is Dx + DB − ˜BD = 0,

共21a兲

˜ D = 0. Dy + DC − C

共21b兲

D共h兲 = 0

共22兲

Being aware of

and taking partial derivatives ⳵x, ⳵y to 共22兲, one gets Dx共h兲 + D共hx兲 = 共Dx + DB兲共h兲 = ˜BD共h兲 = 0, ˜ D共h兲 = 0. Dy共h兲 + D共hy兲 = 共Dy + DC兲共h兲 = C This means

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093506-9

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

共Dx + DB − ˜BD兲共h兲 = 0,

˜ D兲共h兲 = 0. 共Dy + DC − C

共23兲

From 共20a兲 and 共20b兲 we realize that the operator’s actions in 共23兲 are simply some function multiplications. So 共21兲 holds. 䊐 Lax representation for 2DTL with N self-consistent sources is

␺x = B共␺兲,

冉

共24a兲

冊

N

␺y = C + 兺 wi⌬−−1wⴱi 共␺兲. i=1

共24b兲

Notice that Lax representation 共24兲 holds under following conditions: wi,x = B共wi兲,

i = 1, . . . ,N,

ⴱ wi,x = − Bⴱ共wi兲.

共24c兲 共24d兲

Proposition 3 [DT for 2DTLSCS (16)]: Let h be a particular eigenfunction to (24), D = ⌳ + ␴, ␴ ª −h共1兲 / h. Based on DT (20), we define ˜ i ª D共wi兲 = w

Cas共h,wi兲 , h

˜ ⴱi ª Dⴱ 共wⴱi 兲 = − w −1

S共hwⴱi 兲 , h共1兲

共25a兲

共25b兲

where S ª ⌳⌬−−1. Then (20) and (25) give a new solution to (24). So we get a new solution of (16). ˜ i defined by 共25a兲 satisfies 共24c兲 共B replaced by Proof: From Lemma 2 it is easy to see that w ˜B兲. It is necessary to prove that w ˜ ⴱi satisfies 共24d兲. From the proof of Lemma 2 we know 共⳵x − ˜B兲D = D共⳵x − B兲. Taking the formal adjoint

ⴱ

to this equality and rewriting it as −1

−1

共− ⳵x − ˜Bⴱ兲Dⴱ = Dⴱ 共− ⳵x − Bⴱ兲, −1

then we find a sufficient condition for Dⴱ to be the DT for 共24d兲. As a result we have proved 共25b兲. Lastly we need to prove that the DT given by 共20兲 and 共25兲 makes 共24b兲 covariant. That is to say N

N

i=1

i=1

˜D − 兺 w ˜ i⌬−−1w ˜ ⴱi D = 0. Dy + DC + 兺 Dwi⌬−−1wⴱi − C

共26兲

Based on 共21a兲, we have to prove that the extra terms in 共26兲 with respect to wi, wⴱi are equal. We find

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093506-10

−

J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

ⴱ 共1兲 w共1兲 h共1兲 h共1兲 i S共hwi 兲 −1 ⴱ −1 ⴱ −1 ⴱ −1 ⴱ h ˜ ˜ ˜ ˜ + 2 wi⌬−−1共hwⴱi 兲 + w共1兲 w ⌳⌬ w − ⌬ w − w ⌬ w ⌳ + w ⌬ w i − i − i − − i i i i i h h h h

=−

ⴱ ⴱ ˜i w h共1兲 h共1兲 −1 ⴱ −1 ⴱ −1 S共hwi 兲 −1 S共hwi 兲 ˜ ˜ S共hwⴱi 兲 − wiwⴱi + w共1兲 w ⌳⌬ w − ⌬ w + w ⌬ ⌳ − w ⌬ i − i − i − − i i i h h h h共1兲 h

=−

˜i S共hwⴱi 兲 S共hwⴱi 兲 h共1兲 w −1 ⴱ ˜ i⌬−−1⌳ ˜ i⌬−−1 ˜ i⌳⌬−−1wⴱi −w S共hwⴱi 兲 + w共1兲 wi⌳⌬−−1wⴱi + w −w i ⌳⌬− wi − h h h h

=0 for each i. This ends the proof. 䊐 Theorem 1 [DT and method of VC for 2DTLSCS (16)]: Let f and g be two linear independent eigenfunctions of (24). Suppose a共y兲 is an arbitrary function of y. Let h ª f + a共y兲g, 共27a兲

˜ N+1 = c共y兲D共f兲, w ⴱ ˜ N+1 = w

d共y兲 , h共1兲

共27b兲

then (20), (25), and (27) give a new solution for (16) and (24) with 共N + 1兲 self-consistent sources, where c共y兲, d共y兲 satisfy the equality c共y兲d共y兲 = ⳵y log a共y兲. ⴱ ˜ N+1 satisfies 共24c兲. To prove that w ˜ N+1 satisfies 共24d兲, we have Proof: It is easy to see that w ⴱ ˜ N+1,x =− w

共−1兲

ⴱ ⴱ ˜ N+1 ˜ N+1 − ˜uw =− −w

d共y兲 共1兲 d共y兲 共2兲 + u共1兲h共1兲兲, 共1兲2 hx = − 共1兲2 共h h h

冉

冊

d共y兲 d共y兲 h共1兲 h共2兲 d共y兲 d共y兲h共2兲 ⴱ ˜ N+1,x − u共1兲 共1兲 = w . − u共1兲 − + 共1兲 共1兲 = − h h h h h共1兲2 h

Based on Proposition 3, we need to show that extra terms coming from

冉

N

Dy + D C + 兺

wi⌬−−1wⴱi

i=1

冊冉

˜+ − C

N+1

˜ ⴱi wi⌬−−1w 兺 i=1

冊

D

can be canceled out. This is true: −

冉

Cas共h, f兲 −1 共1兲 1 ay ayg共1兲 h共1兲ayg ⴱ ˜ N+1⌬−−1w ˜ N+1 + ⌬− h −1⌳ − −w 共⌳ − h共1兲/h兲 = 2 Cas共g,h兲 − c共y兲d共y兲 h h2 h h h =

冊

Cas共g, f兲 ay Cas共g, f兲 − c共y兲d共y兲a共y兲 = 0. h2 h2 䊐

B. m-time nonauto-Bäcklund Darboux transformations

Theorem 2: Let f j and g j 共j = 1 , 2 , . . . , m兲 be m pairs of independent eigenfunctions to (24). Suppose a j共y兲 are arbitrary functions of time. Let h j ª f j + a j共y兲g j . Then after m -time DTs (Theorem 1), we find a solution for (24) with 共N + m兲 self-consistent sources, which is

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093506-11

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

u关m兴 = u共m兲 +

v关m兴 = v


共1兲共h , . . . ,h 兲 Cas
共h , . . . ,h 兲 Cas 1 m 1 m − , Cas共1兲共h1, . . . ,hm兲 Cas共h1, . . . ,hm兲

共28a兲

Cas共1兲共h1, . . . ,hm兲Cas共−1兲共h1, . . . ,hm兲 , Cas2共h1, . . . ,hm兲

wi关m兴 =

Cas共h1, . . . ,hm,wi兲 , Cas共h1, . . . ,hm兲

wⴱi 关m兴 = 共− 1兲m

wN+j关m兴 = c j共y兲f j关m兴 = c j共y兲

共28b兲

共28c兲

i = 1, . . . ,N,

Cas共h1, . . . ,hm,wⴱi 兲 , Cas共1兲共h1, . . . ,hm兲 Cas共h1, . . . ,hm, f j兲 , Cas共h1, . . . ,hm兲

ⴱ 关m兴 = 共− 1兲m−jd j共y兲 wN+j

共28d兲

共28e兲

j = 1, . . . ,m

Cas共1兲共h1, . . . ,hˆ j, . . . ,hm兲 , Cas共1兲共h1, . . . ,hm兲

共28f兲

where


共h , . . . ,h 兲 = Cas 1 m

冨

h1

¯

hm

]

]

]

¯

共m兲 hm

共m−2兲 ¯ hm h共m−2兲 1

h共m兲 1

冨

,

Cas共h1, . . . ,hm, f兲 =

冨

S共h1 f兲 ¯ S共hm f兲 h共1兲 1

¯

]

]

h共m−1兲 1

¯

共1兲 hm

]

共m−1兲 hm

冨

and c j共y兲d j共y兲 = ⳵y log a j共y兲. Proof: Since each time DT has the form D = ⌳ + ␴, after m-time DTs, the corresponding DT operator has the form D共m兲 = ⌳m + ␴m−1⌳m−1 + ¯ + ␴0 . There are m undetermined coefficients ␴i, i = 0 , . . . , m − 1. Being aware of D共m兲共hi兲 = 0,

i = 1,2, . . . ,m,

we find the linear equation for undetermined coefficients,

冨

¯ h共m−1兲 h1 h共1兲 1 1

¯ h共m−1兲 h2 h共1兲 2 2 ]

]

]

]

共1兲 共m−1兲 ¯ hm hm hm

冨冤 冥 冤 冥 ␴0 ␴1 ] ␴m−1

h共m兲 1

=−

h共m兲 2 ]

.

共m兲 hm

By Cramer’s rule

␴0 = 共− 1兲m

Cas共1兲共h1, . . . ,hm兲 , Cas共h1, . . . ,hm兲

␴m−1 = −


共h , . . . ,h 兲 Cas 1 m . Cas共h1, . . . ,hm兲

In order to obtain 共28a兲 and 共28b兲, noticing that the transformations 共20a兲 and 共20b兲 are unchanged from 2DTL to 2DTLSCS, we may consider the m-time DTs for 2DTL temporary. So assuming that ˜ ˜␺, we have D共m兲 transforms ␺ to ˜␺ = ␺关m兴 and satisfies ˜␺x = ˜B˜␺ and ˜␺y = C

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093506-12

J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

D共m兲x + D共m兲B − ˜BD共m兲 = 0,

共29a兲

˜ D共m兲 = 0. D共m兲y + D共m兲C − C

共29b兲

Comparing the coefficient of ⌳m in 共29a兲 and the coefficient of ⌳−1 in 共29b兲, respectively, we find 共28a兲 and 共28b兲. ˜ = D共m兲共w兲 can be expressed in form 共28c兲 accordFor an arbitrary eigenfunction w, its DT w ing to the Laplace expansion formula. For 共28d兲 we need induction. Suppose that for any adjoint eigenfunction wⴱ the m-time DT formula is 共28d兲, then by 共25b兲, the 共m + 1兲th DT is wⴱ关m + 1兴 = −

S共hm+1关m兴wⴱ关m兴兲 hm+1关m兴共1兲

冋冉

S ⌬ = =

冊

hm+1关m − 1兴 S共hm关m − 1兴wⴱ关m − 1兴兲 − hm+1关m − 1兴共1兲wⴱ关m − 1兴共1兲 hm关m − 1兴 hm+1关m兴共1兲

册

hm+1关m − 1兴共1兲S共hm关m − 1兴wⴱ关m − 1兴兲 S共hm+1关m − 1兴wⴱ关m − 1兴兲 − . hm+1关m兴共1兲hm关m − 1兴共1兲 hm+1关m兴共1兲

By induction hypothesis ⴱ S共hm关m − 1兴wⴱ关m − 1兴兲 m+1 Cas共h1, . . . ,hm,w 兲 , = 共− 1兲 hm关m − 1兴共1兲 Cas共1兲共h1, . . . ,hm兲

ⴱ S共hm+1关m − 1兴wⴱ关m − 1兴兲 m+1 Cas共h1, . . . ,hm−1,hm+1,w 兲 , = 共− 1兲 hm+1关m − 1兴共1兲 Cas共1兲共h1, . . . ,hm−1,hm+1兲

we find wⴱ关m + 1兴 =

共− 1兲m+1 关Cas共h1, . . . ,hm,wⴱ兲Cas共1兲共h1, . . . ,hm−1,hm+1兲 Cas 共h1, . . . ,hm−1兲Cas共1兲共h1, . . . ,hm+1兲 共1兲

− Cas共h1, . . . ,hm−1,hm+1,wⴱ兲Cas共1兲共h1, . . . ,hm兲兴 =

共− 1兲m+1 Cas共1兲共h1, . . . ,hm−1兲Cas共1兲共h1, . . . ,hm+1兲

⫻

冨

S共h1wⴱ兲 ¯ S共hm−1wⴱ兲 S共hmwⴱ兲 S共hm+1wⴱ兲 h共1兲 1

¯

]

]

h共m−1兲 1 h共1兲 1

¯

]

]

h共m兲 1

= 共− 1兲m+1

¯ ¯

共1兲 hm−1

]

共1兲 hm

]

共m−1兲 hm−1 共1兲 hm−1

共m−1兲 hm 共1兲 hm

]

]

共m兲 hm−1

Cas共h1, . . . ,hm+1,wⴱ兲 . Cas共1兲共h1, . . . ,hm+1兲

共m兲 hm

0

¯

0

共1兲 hm+1

0

¯

0

]

]

共m−1兲 hm+1 共1兲 hm+1

h共1兲 1

]

]

共m兲 hm+1

0

h共m兲 1

] ¯

0

¯

共1兲 hm−1

¯

共m兲 hm−1

]

冨

Next we want to prove 共28e兲 and 共28f兲. First, it is easy to see that wN+j关m兴 = c j共y兲D共f j兲 can be ⴱ 关m兴 we use induction. Suppose that when j ⱕ m the formula for derived from 共28c兲. For wN+j ⴱ wN+j关m兴 is given by 共28f兲. Then

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093506-13

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

ⴱ wN+j 关m + 1兴 = −

ⴱ 关m兴兲 S共hm+1关m兴wN+j . 共1兲 hm+1关m兴

Because hm+1关m兴 is obtained by m-time DTs by using eigenfunctions h1 , . . . , hm sequentially, it is equivalent to m-time DTs by successively using h1 , . . . , h j−1 , h j+1 , . . . , hm and h j: hm+1关m兴 = hm+1关m − 1兴共1兲 −

h j关m − 1兴共1兲 hm+1关m − 1兴. h j关m − 1兴

ⴱ 关m兴 = d j共y兲 / h j关m − 1兴共1兲, Since wN+j

ⴱ wN+j 关m + 1兴 = −

冉

冊

共1兲 ˆ hm+1关m − 1兴 d j共y兲 m+1−j d j共y兲Cas 共h1, . . . ,h j, . . . ,hm+1兲 . S⌬ = 共− 1兲 Cas共1兲共h1, . . . ,hm+1兲 hm+1关m兴共1兲 h j关m − 1兴

When j = m + 1, ⴱ 关m + 1兴 = wN+m+1

dm+1共y兲Cas共1兲共h1, . . . ,hm兲 dm+1共y兲 . = hm+1关m兴共1兲 Cas共1兲共h1, . . . ,hm+1兲 䊐

Thus the proof ends. IV. SOLUTIONS FOR 2DTLSCS

Let us start from trivial solution q = 1, v = 1, u = 0, N = 0 of 共24兲. The Lax pair reads

␺x = ␺共1兲,

␺y = ␺共−1兲 .

共30兲

A. Solitons

Equations 共30兲 have linear independent eigenfunctions, f共n,x,y兲 = exp共n␻ + zx + z−1y兲,

g共n,x,y兲 = exp共− n␻ + z−1x + zy兲,

共z = e␻兲.

Let a共y兲 = e␣共y兲; then h = f + a共y兲g = 2 exp ⍀ cosh Z, where ⍀ = cosh ␻ · x + cosh ␻ · y + ␣/2,

Z = n␻ + sinh ␻ · x − sinh ␻ · y − ␣/2.

From 共28兲, taking m = 1 we have the following 1-soliton solution for 共16兲: u关1兴 =

cosh共Z + 2␻兲 cosh共Z + ␻兲 − , cosh共Z + ␻兲 cosh Z

v关1兴 =

cosh共Z + ␻兲cosh共Z − ␻兲 , cosh2 Z

w关1兴 = c共y兲

wⴱ关1兴 =

sinh ␻ · e⍀ , cosh Z

d共y兲e−⍀ , 2 cosh共Z + ␻兲

共c共y兲d共y兲 = ␣˙ 兲.

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093506-14

J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

0.4

0.4

0.2

0.2

u[1]

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -15

0

v[1]

-0.2

y

-0.4 -10

-0.6 -5

1.09 1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1 -15

0 -0.2

-5

-0.8

0

5

10

n

15

y

-0.4 -0.6

-10

-1

0

-0.8 5

10

n

15

-1

0.4

0.4

0.2

w[1]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -15

0.2

0

20 15 10 5 0 -5 -10 -15 -15

w*[1]

-0.2

y

-0.4 -10

-0.6 -5

0 -0.2 -0.4

-5

-0.8

0

5

10

n

15

y

-0.6

-10

0

-1

n

-0.8 5

10

15

-1

FIG. 1. Snapshot of 1-soliton at x = 0 with ␻ = 0.3, ␣共y兲 = sin共共14␲ / 3兲y兲, c共y兲 = 1, d共y兲 = ␣˙ .

The 1-soliton solution for 2DTLSCS is different from usual 1-soliton for 2DTL since the soliton travel speed is determined by ␣共y兲. In particular, it provides a soliton with a nonconstant speed during its propagation. Furthermore the amplitude of w关1兴wⴱ关1兴 is controlled by ␣˙ ; therefore the change in solitary wave speed leads to the synchronous change in the amplitude and vice versa. Similar phenomena were observed in other systems with self-consistent sources, such as KP 共Ref. 28兲 and Toda lattice.29 Figure 1 shows a solitary wave with nonconstant speed. If we take two pairs of independent eigenfunctions, with respect to z j = e␻ j 共j = 1 , 2兲 respectively, i.e., f j = exp共n␻ j + z jx + z−1 j y兲,

g j = exp共− n␻ j + z−1 j x + z j y兲,

j = 1,2,

let a j共y兲 = 共−1兲 je␣ j共y兲, then h1 = f 1 + a1g1 = 2 exp ⍀1 · sinh Z1,

h2 = f 2 + a2g2 = 2 exp ⍀2 · cosh Z2 ,

where ⍀ j = cosh ␻ j · x + cosh ␻ j · y + ␣ j/2,

Z j = n␻ j + sinh ␻ j · x − sinh ␻ j · y − ␣ j/2.

To simplify expressions, we define

Hk =

冏

sinh Z1

cosh Z2

sinh共Z1 + k␻1兲 cosh共Z2 + k␻2兲

= sinh

冉

冏

冊

k共␻1 − ␻2兲 k共␻1 + ␻2兲 k cosh Z1 + Z2 + 共␻1 + ␻2兲 + sinh 2 2 2

冉

k ⫻cosh Z1 − Z2 + 共␻1 − ␻2兲 2

冊

共k = 1,2兲.

Then the 2-soliton solution for 共16兲 is

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093506-15

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

1

1

0.5

u[2]

0.5

0

-10

-8

y

v[2] -10

-0.5

-6

-4

-2

0

2

4

-8

-6

y

-0.5 -4

-2

0

-1

6

n

0

2

4

n

6

-1

1

1

0.5

w1[2]

0.5

0

-10

-8

-6

y

w1*[2] -10

-0.5 -4

-2

0

2

4

n

6

0

-8

-6

y

-0.5 -4

-2

0

-1

2

4

n

6

-1

1

1

0.5

0.5

-w2[2]

0

-10

-8

-6

w2*[2] -10

-0.5 -4

-2

0

2

4

n

6

0

y -8

-6

y

-0.5 -4

-2

0

-1

2

n

4

6

-1

FIG. 2. Snapshot of 2-soliton at x = 0 with ␻1 = 1, ␻2 = 0.6, ␣1 = 5 sin共共11␲ / 3兲y兲, ␣2 = 6y, c j = 1, d j = ␣˙ j 共j = 1 , 2兲.

u关2兴 =

H共1兲 2 H共1兲 1

−

H2 , H1

v关2兴 =

共−1兲 H共1兲 1 H1

H21

,

w1关2兴 = c1共y兲a1共y兲

2 sinh ␻1共cosh ␻1 − cosh ␻2兲exp ⍀1 cosh共Z2 + ␻2兲 , H1

w2关2兴 = c2共y兲a2共y兲

2 sinh ␻2共cosh ␻1 − cosh ␻2兲exp ⍀2 sinh共Z1 + ␻1兲 , H1

wⴱ1关2兴 = −

wⴱ2关2兴 =

d1共y兲exp共− ⍀1兲cosh共Z2 + ␻2兲 2H共1兲 1 d2共y兲exp共− ⍀2兲sinh共Z1 + ␻1兲 2H共1兲 1

,

,

c1d1 = ␣˙ 1 ,

c2d2 = ␣˙ 2 .

The speed of two solitary waves are determined by ␣1 and ␣2, respectively. So it produces rich dynamic behaviors. For example, we can make the first solitary wave travel at a periodically changing speed. The 2-soliton interactions are shown by Fig. 2. The interaction

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J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

y

093506-16

n

FIG. 3. Contour graph of u关2兴.

makes phase shifts, which can be shown by the contour graph of u关2兴 in Fig. 3. B. Rational solution

In Eq. 共30兲, noticing that ⳵zk␺ is another eigenfunction, we find that g共n , x , y兲 = zn exp共zx + z y兲 and f k共n , x , y兲 ª ⳵zkg 共k ⱖ 1兲 are all independent eigenfunctions for 共30兲. So let ␰ ª zx + z−1y, −1

f 1共n,x,y兲 = ⳵zg = zn−1e␰共n + z␰z兲, f 2共n,x,y兲 = ⳵z2g = zn−2e␰共n2 + 2nz␰z − n + z2␰z2 + z2␰zz兲, f 3共n,x,y兲 = ¯ , let hk = f k + a共y兲g, and take k = 1, m = 1. We have a rational solution for 共16兲: u关1兴 = −

z , 共␩ + za + 1/2兲2 − 1/4

v关1兴 = 1 −

1 , 共␩ + za兲2

w关1兴 = c共y兲

wⴱ关1兴 = d共y兲

zn+1e␰a , ␩ + za

z−ne−␰ , ␩ + za + 1

where ␩ = n + z␰z, c共y兲d共y兲 = 共d / dy兲log a. This is a nonsmooth weak solution which has singularities. If take k = 2, m = 1, we find another rational solution:

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093506-17

J. Math. Phys. 49, 093506 共2008兲

An extended two-dimensional Toda hierarchy

u关1兴 ª z

v关1兴 ª

冉

冊

␩共2兲 + z2a ␩共1兲 + z2a , − ␩共1兲 + z2a ␩ + z 2a

共␩共1兲 + z2a兲共␩共−1兲 + z2a兲 , 共␩ + z2a兲2

w关1兴 ª 2c共y兲azn+1e␰

wⴱ关1兴 ª d共y兲

n + z␰z , ␩ + z 2a

z−n+1e−␰ , ␩共1兲 + z2a

where ␩ = n2 + 2nz␰z − n + z2␰z2 + z2␰zz, c共y兲d共y兲 = 共d / dy兲log a. C. Other weak solutions

We can obtain other types of weak solutions, such as negatons, positons, complexitons. Let f = znezx+z

−1 y

ª zneF共x,y,z兲,

g = z−nez

−1x+zy

ª z−neG共x,y,z兲

be pair of eigenfunctions to 共30兲. Then f z and gz are another pair of eigenfunctions. Let h = f + a共y兲g = 2 exp ⍀ · cosh Z, where ⍀, a共y兲, and Z are defined in Sec. IV A. Then hz = f z + a共y兲gz = 2⍀ze⍀ cosh Z + 2e⍀Zz sinh Z. Taking m = 2, we obtain the following solution. For simplicity, we define

冏

h h

hz

共k兲

hz共k兲

冏

= 4e2⍀Ck,

冉

where

Ck = Zz +

Cas共h,hz, f兲 = − 8a共x兲e3⍀

冊

k k sinh共k␻兲 + sinh共2Z + k␻兲, 2z 2z

sinh2 ␻ cosh共Z + ␻兲, z

Cas共h,hz, f z兲 = 4a共x兲e3⍀共D共1兲 1 cosh Z + D1 cosh共Z + 2␻兲 − D2 cosh共Z + ␻兲兲, where Dk = −

冉

n k + + Fz z 2z

冊冉

冊

n k k2 k⍀z + − Gz sinh共k␻兲 + 2 sinh共k␻兲 + cosh共k␻兲, z 2z 4z z

k = 1,2.

Then we find the following solution for 共16兲: u关2兴 =

C共1兲 2 C共1兲 1

−

C2 , C1

w1关2兴 = − 2c1共y兲a共y兲

w2关2兴 = c2共y兲a共y兲

v关2兴 =

共−1兲 C共1兲 1 C1

C21

,

sinh2 ␻e⍀ cosh共Z + ␻兲, zC1

e⍀共D共1兲 1 cosh Z + D1 cosh共Z + 2␻兲 − D2 cosh共Z + ␻兲兲 , C1

共31a兲

共31b兲

共31c兲

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093506-18

J. Math. Phys. 49, 093506 共2008兲

Liu, Zeng, and Lin

wⴱ1关2兴 = − d1共y兲

⍀z cosh共Z + ␻兲 + 共Zz + 1/z兲sinh共Z + ␻兲 2e⍀C共1兲 1

wⴱ2关2兴 = d2共y兲

cosh共Z + ⍀兲 2e⍀C共1兲 1

,

,

共31d兲

共31e兲

where c j共y兲d j共y兲 = 共d / dy兲log a共y兲. Equations 共31兲 represent two types of solution for 2DTLSCS depending on the choice of ␻ and ␣共y兲: when ␻ , ␣共y兲 苸 R, it denotes a negaton solution, while when ␻ , ␣共y兲 苸 iR, it denotes a positon solution. In general, ␻ can be any complex number, but the outcome may not be a real valued solution. Therefore, in the general case, we adopt the following method to get the real solution: Suppose ␻ = ␰ + i␩. Then the real 共or imaginary兲 part of f, g and derivatives of the real 共or imaginary兲 part of f and g are all eigenfunctions of 共30兲. So letting a共y兲 苸 R, we can get the real solution of 2DTLSCS by using 共for instance兲 Re共f兲, Re共f兲␰ and Re共h兲 = Re共f兲 + a共y兲Re共g兲,

Re共h兲␰ = Re共f兲␰ + a共y兲Re共g兲␰

in 共28兲. For example, the real part of h can be written as Re共h兲 = 2e⍀1关cosh Z1 cos Z2 cos ⍀2 − sinh Z1 sin Z2 sin ⍀2兴, where ⍀1 = 共x + y兲cosh ␰ cos ␩ + ␣/2, ⍀2 = 共x + y兲sinh ␰ sin ␩ , Z1 = n␰ + 共x − y兲sinh ␰ cos ␩ − ␣/2, Z2 = n␩ + 共x − y兲cosh ␰ sin ␩ . According to this expression, the real solution obtained will involve exponential and trigonometric functions and correspond to the complex eigenvalue of the associated Lax presentation. Thus it is a so-called one-complexiton solution.30 Furthermore, by using the series of eigenfunctions h1 , . . . , hk and their derivatives ⳵zj hi or i j ⳵␰ Re共hi兲, etc., we can get higher order positons, negatons, or complexitons or interaction solui tions.

V. CONCLUSION

We present an ex2DTLH by using squared eigenfunction symmetries. It helps us construct the two-dimensional Toda lattice equation with self-consistent sources in a different way from Refs. 2 and 25 as well as their Lax representations. Since the two-dimensional Toda lattice equation with self-consistent sources can be considered as a two-dimensional Toda lattice equation with nonhomogeneous terms, the method of VC can be applied to the ordinary DTs for 2DTLSCS to construct a nonauto-Bäcklund DT. Then it offers a different way to solve 2DTLSCS in contrast with Refs. 2 and 25. The 2DTLH offers various types of reductions, for example, the periodic reductions in the sinh-Gordan equation and dimension reductions in the Toda lattice equation. Then the following question arises: does ex2DTLH offers similar reductions in the above equation with self-consistent sources? We may discuss such problems elsewhere.

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093506-19

An extended two-dimensional Toda hierarchy

J. Math. Phys. 49, 093506 共2008兲

ACKNOWLEDGMENTS

The author thanks Professor X. B. Hu for valuable comments. This work is supported by the National Basic Research Program of China 共973 Program兲 共Grant No. 2007CB814800兲, National Natural Science Foundation of China 共Grant No. 10601028兲, and in part by the Key Laboratory of Mathematics Mechanization. 1

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