mixed rational-soliton solutions to the toda lattice equation

MIXED RATIONAL-SOLITON SOLUTIONS TO THE TODA LATTICE EQUATION WEN-XIU MA

We present a way to solve the Toda lattice equation using the Casoratian technique, and construct its mixed rational-soliton solutions. Examples of the resulting exact solutions are computed and plotted. Copyright © 2006 Wen-Xiu Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Differential and/or difference equations serve as mathematical models for various phenomena in the real world. The study of differential and/or difference equations enhances our understanding of the phenomena they describe. One of important fundamental questions in the subject is how to solve differential and/or difference equations. There are mathematical theories on existence and representations of solutions of linear differential and/or difference equations, especially constant-coefficient ones. Soliton theory opens the way to studies of nonlinear differential and/or difference equations. There are different solution methods for different situations in soliton theory, for example, the inverse scattering transforms for the Cauchy problems, Bäcklund transformations for geometrical equations, Darboux transformations for compatibility equations of spectral problems, Hirota direct method for bilinear equations, and truncated series expansion methods (including Painlevé series, and sech and tanh function expansion methods) for Riccati-type equations. Among the existing methods in soliton theory, Hirota bilinear forms are one of the most powerful tools for solving soliton equations, a kind of nonlinear differential and/or difference equations. In this paper, we would like to construct mixed rational-soliton solutions to the Toda lattice equation:

a˙ n = an bn−1 − bn ,

b˙ n = an − an+1 ,

(1.1)

where a˙ n = dan /dt and b˙ n = dbn /dt. The approach we will adopt to solve this equation is the Casoratian technique. Its key is to transform bilinear forms into linear systems Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 711–720

712

Mixed rational-soliton solutions to the Toda lattice

of solvable differential-difference equations. For the Toda lattice equation (1.1), we will present the general solutions to the corresponding linear systems and further generate mixed rational-soliton solutions. 2. Constructing solutions using the Casoratian technique Let us start from the Toda bilinear form. Under the transformation an = 1 +

d2 τ τ logτn = n+1 2n−1 , dt 2 τn

bn =

d τ τ˙ τ − τ τ˙ log n = n n+1 n n+1 , dt τn+1 τn τn+1

(2.1)

the Toda lattice equation (1.1) becomes

Dt2 − 4sinh2

Dn 2

τn · τn = 0,

(2.2)

− τn+1 τn−1 + τn2 = 0.

(2.3)

where Dt and Dn are Hirota’s operators. That is, 2

τ¨n τn − τ˙n

In the Casoratian formulation, we use the Casorati determinant φ1 (n) φ (n) 2 τn = Cas φ1 ,φ2 ,...,φN = .. . φN (n)

φ1 (n + N − 1) φ2 (n + N − 1) .. . φN (n + N − 1)

φ1 (n + 1) · · · φ2 (n + 1) · · · .. .. . . φN (n + 1) · · ·

(2.4)

to construct exact solutions, and we call such solutions Casoratian solutions. It is known [7] that such a τ-function τn solves the bilinear Toda lattice equation (2.3) if

φi (n + 1) + φi (n − 1) = 2εi coshαi φi (n),

φi (n) t = φi (n + 1),

(2.5)

where εi = ±1, αi are nonzero constants and (φi (n))t = ∂t φi (n) = ∂t φi (n,t). The resulting solutions are negatons, that is, a kind of solutions only involving exponential functions of the space variable n. There are other type solutions such as rational solutions [3], positons [5, 8], and complexitons [2]. Similar to [2, 3], we can prove that τn is a solution to the bilinear Toda lattice equation (2.3) if φi (n + 1) + φi (n − 1) =

N j =1

λi j φ j (n),

φi (n) t = ζφi (n + δ),

(2.6)

where ζ = ±1, δ = ±1 (i.e., |ζ | = |δ | = 1, ζ,δ ∈ R), and λi j are arbitrary constants. Under the transformation t → −t, the bilinear Toda lattice equation (2.3) is invariant, and (φi (n))t = φi (n + δ) becomes (φi (n))t = −φi (n + δ). Therefore, we only need to consider one of the cases ζ = ±1 while constructing solutions, since the replacement of t with −t generates solutions from one case to the other. We will only consider the case of ζ = 1 below.

Wen-Xiu Ma 713 Let us now begin to analyze the system of differential-difference equations (2.6) with ζ = 1. The corresponding system can be compactly written as

ΦN (n + 1,t) + ΦN (n − 1,t) = ΛΦN (n,t),

ΦN (n,t) t = ΦN (n + δ,t),

(2.7)

where ΦN = ΦN (n,t) := (φ1 (n,t),...,φN (n,t))T and Λ := (λi j )N ×N . Note that a constant similar transformation for the coefficient matrix Λ does not change the resulting Casoratian solution. Actually, if we have M = P −1 ΛP for an invertible constant matrix P, then

N = PΦN satisfies Φ

N (n + δ,t).

N (n,t) = Φ Φ t

N (n + 1,t) + Φ

N (n − 1,t) = M Φ

N (n,t), Φ

(2.8)

N have just a constantObviously, the Casorati determinants generated from ΦN and Φ factor difference, and thus the transformation (2.1) leads to the same Casoratian solutions

N . Therefore, as in the KdV case [4], we can focus on the following two from ΦN and Φ types of Jordan blocks of Λ: ⎡

λi ⎢ ⎢1 ⎢ ⎢. ⎢ .. ⎣ 0

0 λi .. . ···

..

. 1

λi

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

ki ×ki

Ai ⎢ ⎢ I2 ⎢ ⎢. ⎢ .. ⎣ 0

0 Ai .. .

..

···

I2

. Ai

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

αi Ai = βi

−βi

αi

,

(2.9)

l i ×l i

where λi , αi and βi > 0 are all real constants, I2 is the identity matrix of order 2, and ki and li are positive integers. A Jordan block of the first type has the real eigenvalue λi with algebraic multiplicity ki , √and a Jordan block of the second type has the pair of complex eigenvalues λi,± = αi ± βi −1 with algebraic multiplicity li . Case ki = 1 of type 1. The representative systems read as follows:

φi (n + 1) + φi (n − 1) = λi φi (n),

φi (n) t = φi (n + δ),

(2.10)

where δ = ±1 and λi = consts. Their eigenfunctions are classified as

φi = C1i εin eεi t + C2i n + εi δt εin eεi t ,

φi = C1i et cosαi cos αi n + δt sinαi

λi = 2εi ,

+ C2i et cosαi sin αi n + δt sinαi , δα

−δαi

φi = C1i εin eαi n+εi te i + C2i εin e−αi n+εi te

εi = ±1,

λi = 2cosαi , ,

αi = mπ, m ∈ Z,

λi = 2εi coshαi ,

(2.11)

αi = 0,

where C1i and C2i are arbitrary constants. The above three sets of eigenfunctions generate rational solutions, positon solutions, and negaton solutions, respectively. Generally, the following two results provide ways to solve the linear system (2.6). The detailed proof will be published elsewhere.

714


Theorem 2.1 (λi = ±2). Let ε = ±1 and δ = ±1 (i.e., |ε| = |δ | = 1, ε,δ ∈ R). If ( f (n,t))t = f (n + δ,t), then the nonhomogeneous system

φ(n + 1,t) + φ(n − 1,t) = 2εφ(n,t) + f (n,t),

φ(n,t) t = φ(n + δ,t),

(2.12)

has the general solution

φ(n,t) = α(n)t + β(n) +

ts 0

0

f (n + δ,r)e−εr dr ds eεt ,

(2.13)

where α(n) and β(n) are determined by α(n + δ) − εα(n) = f (n + δ,0),

β(n + δ) − εβ(n) = α(n).

(2.14)

Theorem 2.2 (λi = ±2). Let λ = ±2 and δ = ±1. (a) The homogeneous system

φ(n + 1,t) + φ(n − 1,t) = λφ(n,t),

φ(n,t) t = φ(n + δ,t),

(2.15)

has its general solution δ

−δ

φ(λ;c,d)(n) = cωn etω + dω−n etω ,

(2.16)

where c and d are arbitrary constants, and λ = ω + ω −1 ,

(2.17)

ω2 − λω + 1 = 0.

(2.18)

that is,

(b) Define fk = φ(λi ;ck ,dk ), 1 ≤ k ≤ ki , where ck and dk are arbitrary constants. The nonhomogeneous system φk (n + 1,t) + φk (n − 1,t) = λi φk (n,t) + φk−1 (n,t),

φk (n,t) t = φk (n + δ,t),

(2.19)

where 1 ≤ k ≤ ki and φ0 = 0, has its general solution

k −1

k −1 1 ∂ p fk− p 1 ∂ p φ λi ;ck− p ,dk− p φk = , p = p p! ∂λi p! ∂λi p =0 p =0

1 ≤ k ≤ ki .

(2.20)

= 0) corresponds to ωi = eαi in (a). The Remark 2.3. The soliton case of λi = 2coshαi (αi nonhomogeneous system in (b) is associated with one Jordan block of type 1. Begin with

⎡

2ε

⎢ ⎢∗ ⎢ Λ=⎢ ⎢ ... ⎣ ∗

0 2ε .. .

..

···

∗

. 2ε

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ki ×ki

,

∗-arbitrary consts.,

(2.21)

Wen-Xiu Ma 715 where ε = ±1. By the general solution formula in Theorem 2.1, we can have φi (n,t) = εn eεt ψi (n,t),

1 ≤ i ≤ N,

(2.22)

where ψi (n,t) are polynomials in n and t. Thus,

τn = Cas ψ1 ,...,ψN

(2.23)

presents polynomial solutions to the bilinear Toda lattice equation (2.3) and thus rational solutions to the Toda lattice equation (1.1) through (2.1). Theorem 2.4. The Jordan block of type 1 with λi = ±2 leads to rational solutions to the Toda lattice equation (1.1). This adds one case of λ = −2 to the result in [3]. A few examples of rational solutions associated with one Jordan block case with λ = 2 were presented in [3]. 3. Mixed rational-soliton solutions Let us now show a way to construct mixed rational-soliton solutions. We use the following procedure. Step 1. Solve the triangular systems whose coefficient matrices possess Jordan blocks of = 0) to form a set of eigenfunctions (φ1 ,...,φN ). type 1 with λi = ±2 or λi = 2coshαi (αi Step 2. Evaluate the τ-function τn = Cas(φ1 ,...,φN ). Step 3. Evaluate an and bn by the transformation (2.1), to obtain mixed rational-soliton solutions to the Toda lattice equation (1.1). The τ-functions generated above are quite general. In what follows, we would like to present two sets of special eigenfunctions required in forming such τ-functions. Special eigenfunctions yielding rational solutions. We take two specific Taylor expansions as in [9]: k

−k

φ+ (n,t) = ekn+te + e−kn+te =

∞

i=0 kn+tek

φ− (n,t) = e

−kn+te−k

−e

=

∞

i =0

ai+1 (n,t)k2i , (3.1) ai+1 (n,t)k

2i+1

,

the coefficients of which satisfy ai (n + 1,t) + ai (n − 1,t) =

i

2 ai− j+1 (n,t), (2 j)! j =0

ai (n,t) t = ai (n + 1,t),

(3.2)

where i ≥ 1. These two sets of functions are given by ai+1 (n,t) = et

2i 2n2i− j β j (t), (2i − j)! j =0

ai+1 (n,t) = et

2i+1

2n2i+1− j β j (t), (2i + 1 − j)! j =0

(3.3)

716


respectively. The functions β j (t) above are defined by ∞

j =0

j

β j (t)k =

∞ tp

p =0

p!

∞ 1

q =1

q!

p

k

q

.

(3.4)

This is a rational solution case, since λii = 2, 1 ≤ i ≤ N. Special eigenfunctions yielding solitons. We start from the same eigenfunctions k

−k

k

−k

φ+ (n,t) = ekn+te + e−kn+te = 2et coshk cosh(kn + t sinhk),

(3.5)

φ− (n,t) = ekn+te − e−kn+te = 2et sinh k cosh(kn + t sinhk), which solve

φ(n,t) t = φ(n + δ,t).

φ(n + 1,t) + φ(n − 1,t) = 2(coshk)φ(n,t),

(3.6)

Computing derivatives of the above system with the parameter k leads to a set of eigenfunctions as follows: bi (n,t) = ∂ik−1 φ(n,t),

1 ≤ i ≤ N,

i

(3.7)

which satisfies bi (n + 1,t) + bi (n − 1,t) =

j =1

where 1 ≤ i ≤ N and λi j = 2( ij−−11 )( ∂ki− j 2coshk, 1 ≤ i ≤ N.

λi j b j (n,t),

coshk ).

bi (n,t) t = bi (n + δ,t),

(3.8)

This is a soliton case if k = 0, since λii =

Examples of mixed rational-soliton solutions. Mixed rational-soliton solutions can now be computed, for example, by

τn = Cas e±t a1 ,...,e±t a p ,b1 ,...,bN − p .

(3.9)

In particular (in the case of φ+ ), we have

τn = Cas e−t a1 ,b1 = 4et coshk cosh k(n + 1) + t sinhk − cosh(kn + t sinhk) ,

τn = Cas e−t a1 ,e−t a2 ,b1

= 4et coshk 2(n + t + 1)cosh k(n + 2) + t sinhk − 4(2n + 2t + 1)cosh k(n + 1) + t sinhk

+ (2n + 2t + 3)cosh(kn + t sinhk) .

(3.10)

Wen-Xiu Ma 717 ×103

3 2.5

20 15 −a 10 5 0 0 0.5

t

2 1.5 1

1

1.5 2 t

0 2.5

5 3

−5

−10

0.5 0

n

−10

−5

0

5

10

5

10

n

10

(a)

(b) Figure 3.1. 3D and density plots of an .

3

×102

2.5

2 1

b

t

0

2 1.5 1

−1

0 0.5 1 t

−5

1.5 2

0 2.5

3

10

5

−10

0.5 0 −10

n

−5

0 n

(a)

(b) Figure 3.2. 3D and density plots of bn .

3

3

2.5

2.5

2 t

t

1.5

2 1.5

1

1

0.5

0.5

0 −4

−3

−2

−1

−5

−4

−3

n

(a)

−2

−1

n

(b)

Figure 3.3. Contour plots of (a) an and (b) bn .

The solution from τn = Cas(e−t a1 ,b1 ) with k = 1 is depicted in Figures 3.1, 3.2, and 3.3, and the solution from τn = Cas(e−t a1 ,e−t a2 ,b1 ) with k = −1 in Figures 3.4, 3.5, and 3.6.

718

Mixed rational-soliton solutions to the Toda lattice ×102

20 15 −a 10 5 0 0 0.5

3 2.5 2 t

1.5 1

1 t

1.5

2

0

2.5

2

3

−4

−2

−6

0.5 0 −6

n

−4

−2

0

2

n

(a)

(b) Figure 3.4. 3D and density plots of an .

3 2.5

×102

2

0 b

t

−0.5 −1 −1.5

0

0.5

1 t

1.5

2

2.5

2

3

0

−2

−4

1.5 1 0.5

−6

0 −6

n

−4

−2

0

2

n

(a)

(b) Figure 3.5. 3D and density plots of bn .

3

3

2 t

2

1

t

0

1 0

−1 −2 −1

0

1

2

3

−2

−1

n

(a)

0

1

2

n

(b)

Figure 3.6. Contour plots of (a) an and (b) bn .

4. Discussions A careful analysis based on Theorems 2.1 and 2.2 can prove that the Jordan blocks of type 1 with λi = ±2, |λi | > 2, and |λi | < 2 generate rational solutions, negatons, and positons,

Wen-Xiu Ma 719 respectively; and that the Jordan blocks of type 2, which possess complex eigenvalues, generate complexitons. Moreover, we can have another case of conditions on eigenfunctions: φi (n + 1) + φi (n − 1) =

N j =1

λi j φ j (n), (4.1)

1 φi (n,t) t = ζ φi (n + 1,t) − φi (n − 1,t) , 2

where ζ = ±1 and λi j are arbitrary constants. An analysis is left for future publication, on this case of conditions and its representative system φ(n + 1,t) + φ(n − 1,t) = λφ(n,t) + f (n,t),

1 1 φ(n,t) t = φ(n + 1,t) − φ(n − 1,t), 2 2

(4.2)

where ( f (n,t))t = 1/2 f (n + 1,t) − (1/2) f (n − 1,t), which will lead to different mixed rational-soliton solutions to the Toda lattice equation (1.1). The above construction of mixed rational-soliton solutions is direct and much easier than the existing approach by computing long-wave limits of soliton solutions [1, 6]. The basic idea can also be applied to other integrable lattice equations, for example, the Volterra lattice equation:

u˙ n = un un+1 − un−1 .

(4.3)

The transformation of un = τn+2 τn−1 /τn+1 τn puts the Volterra lattice equation into the following bilinear form: τ˙n+1 τn − τn+1 τ˙n − τn+2 τn−1 + τn+1 τn = 0.

(4.4)

The Casorati determinant τn = Cas(φ1 ,...,φN ) solves this equation if φi (n + 1,t) + φi (n − 1,t) =

N j =1

λi j φ j (n,t),

φi (n,t) t = φi (n + 2,t),

(4.5)

where 1 ≤ i ≤ N and λi j are arbitrary constants. Therefore, this allows us to construct Casoratian solutions to the Volterra lattice equation in a simple and direct way. The details of constructing Casoratian solutions will be published elsewhere. Acknowledgment The author would like to thank Professors K. Maruno and Y. You for stimulating discussions.

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References [1] A. S. Cârstea and D. Grecu, On a class of rational and mixed soliton-rational solutions of the Toda lattice, Progress of Theoretical Physics 96 (1996), no. 1, 29–36. [2] W. X. Ma and K.-I. Maruno, Complexiton solutions of the Toda lattice equation, Physica A 343 (2004), no. 1–4, 219–237. [3] W. X. Ma and Y. You, Rational solutions of the Toda lattice equation in Casoratian form, Chaos, Solitons & Fractals 22 (2004), no. 2, 395–406. , Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans[4] actions of the American Mathematical Society 357 (2005), no. 5, 1753–1778. [5] K. Maruno, W. X. Ma, and M. Oikawa, Generalized Casorati determinant and positon-negatontype solutions of the Toda lattice equation, Journal of the Physical Society of Japan 73 (2004), 831–837. [6] K. Narita, Solutions for the Mikhailov-Shabat-Yamilov difference-differential equations and generalized solutions for the Volterra and the Toda lattice equations, Progress of Theoretical Physics 99 (1998), no. 3, 337–348. [7] J. J. C. Nimmo, Soliton solution of three differential-difference equations in Wronskian form, Physics Letters. A 99 (1983), no. 6-7, 281–286. [8] A. A. Stahlhofen and V. B. Matveev, Positons for the Toda lattice and related spectral problems, Journal of Physics. A. Mathematical and General 28 (1995), no. 7, 1957–1965. [9] H. Wu and D.-J. Zhang, Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form, Journal of Physics. A. Mathematical and General 36 (2003), no. 17, 4867–4873. Wen-Xiu Ma: Department of Mathematics, University of South Florida, Tampa, FL 33620, USA E-mail address: [email protected]