Theoretical and Mathematical Physics, 133(2): 1539–1548 (2002)

INVERSE SPECTRAL TRANSFORM FOR THE q-DEFORMED VOLTERRA EQUATION S. Lombardo∗ We use the inverse spectral transform to study a q-deformation of the Volterra equation. The q-deformed time dependence of the spectral data is computed, and the one-soliton solution is explicitly constructed.

Keywords: q-equations, q-deformations

1. Introduction The name Volterra equation or Volterra model usually refers to the differential–difference equation dcn = cn (cn+1 − cn−1 ), dt

n ∈ Z,

cn (t) > 0.

(1)

Vito Volterra used this model to study ecological problems [1], but it proved rather universal and has many applications [2]. Recently, there has been a growing interest in different versions of this model. In particular, it was found that the model is related to the Liouville equation, and its proper Hamiltonian interpretation provides the lattice deformation of the Virasoro algebra [3]. Its quantum version was also investigated [4], and a q-deformation of the model arises naturally as a similarity reduction of the q-deformed Painlev´e-I equation [5]. This q-deformed version of the Volterra equation is the starting point of our investigation. To our knowledge, the q-deformation of the Volterra model q 2 Dt

Rn = qq22 Tt (Rn Rn+1 ) − Rn Rn−1 ,

n ∈ Z,

(2)

first appeared in a paper by Nijhoff devoted to establishing a relation between the q-deformation of the discrete Painlev´e-I equation and q-orthogonal polynomials [6]. Here, qa Dt is the q-differentiation operator q a Dt

and

q a Tt

f (t) ≡

f (tq a ) − f (t) = (q a − 1)t

qa Tt f (t) − f (t) , (q a − 1)t

a ∈ Z,

(3)

is the q-dilatation qa Tt f (t)

≡ f (tq a ),

a ∈ Z.

(4)

We actually study the system of equations q 2 Dt

Rn = Qn − Qn+1 ,

(q2 Tt Rn )Qn−1 = Qn Rn−2 ,

(5) (6)

which can be reduced to (2) by setting Rn = Rn + (1 − q 2 )tQn+1 . ∗ Department

of Applied Mathematics, University of Leeds, Leeds, UK, e-mail: [email protected].

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 133, No. 2, pp. 259–269, November, 2002. c 2002 Plenum Publishing Corporation 0040-5779/02/1332-1539$27.00


1539

Equations (5) and (6) can be represented as the compatibility condition for the system of linear equations ψˆn+1 + Rn ψˆn−1 = λψˆn ,

(7)

ψˆn = Qn ψˆn−2 ,

(8)

q 2 Dt

where Eq. (7) is closely related to the standard auxiliary linear problem for the Volterra model (Sec. 3) solved many years ago (see, e.g., [2]) and Eq. (8) is a q-differential equation defining the time dependence of the spectral data. Our aim in this paper is to find the time dependence of the spectral data and to construct the onesoliton solution of system (7), (8) explicitly. We show that the solution has some soliton-like features although it travels with a nonconstant velocity. As q → 1, the solution tends to the standard one-soliton solution of Eq. (1). Before proceeding, we discuss a very important feature of equations of this type that characterizes qequations in general. Despite the similarity of the q-derivative and the partial derivative in time (actually, in any variable), they have quite different properties. Indeed, while the equation ∂t f (t, · ) = 0 immediately implies that f (t, · ) is independent of the time t, there are infinitely many time-dependent functions that belong to the Kernel of q Dt , i.e., Ker(q Dt ) ≡ {f (t, · ) : q Dt f (t, · ) = 0}. For example, any function f (log t/ log q), where f (y) = f (y + 1), belongs to Ker(q Dt ). It is therefore necessary to generalize the Cauchy problem to this case, thus providing the existence and uniqueness of the solution. To solve the problem, we confine ourselves to the class of functions that are meromorphic in the vicinity of t = 0. The following lemma allows formulating the Cauchy problem for ordinary and partial q-equations. Lemma. Let g(t) be a meromorphic function of t in an open neighborhood of t = 0. Also let g(t) ∈ Ker(q Dt ), i.e., q Dt g(t) = 0. Then g(t) is a constant, i.e., is independent of t. Proof. If g(t) is meromorphic near t = 0, it can be represented by a convergent Laurent series +∞


g(t) =

g k tk .

(9)

k=−n

It then follows from q Dt g(t) = 0 that

q Dt

g(t) =

+∞
k=−n

gk tk−1

qk − 1 = 0, q−1

(10)

which implies qk − 1 = 0, q−1

(11)

g(t) = g0 = const .

(12)

gk tk−1 and hence gk = 0 for any k = 0. Therefore,

In what follows, we assume that all functions are meromorphic in the vicinity of t = 0. 1540

2. Lax representation We give a detailed derivation of q-Volterra equation (2), (and (5), (6)). We first take the q-time derivative of (7), q 2 Dt

Assuming that

q 2 Dt

ψˆn+1 +q2 Dt (Rn ψˆn−1 ) = λ q2 Dt ψˆn .

(13)

λ = 0 and recalling the q-modified Leibnitz rule

qα Dt (f

· g) = (qα Dt f )g(t) + f (q α t)(qα Dt g) = (qα Dt f )g(q α t) + f (t)(qα Dt g),

(14)

Qn+1 ψˆn−1 + q2 Dt Rn ψˆn−1 + Rn (q 2 t)Qn−1 ψˆn−3 = λQn ψˆn−2 .

(15)

we obtain

Using Eq. (7) solved for ψˆn−2 (λψˆn−2 = ψˆn−1 + Rn−2 ψˆn−3 ), we can then rewrite the last expression as (Qn+1 + q2 Dt Rn − Qn )ψˆn−1 + (Rn (q 2 t)Qn−1 − Qn Rn−2 )ψˆn−3 = 0.

(16)

Finally, by setting the coefficients at the respective terms ψˆn−1 and ψˆn−3 to zero, we obtain Eqs. (5) and (6). It follows from (5) that q2 Tt Rn

= Rn + (q 2 − 1)t(Qn − Qn+1 ).

(17)

Equation (6) can now be rewritten as Rn Qn−1 + (q 2 − 1)t(Qn − Qn+1 )Qn−1 = Qn Rn−2 ,

(18)

and rearranging the terms, we obtain Qn Rn = , Qn−1 Rn−2

(19)

Rn = Rn + (1 − q 2 )tQn+1 .

(20)

where

It follows from (19) that Qn can be represented as Qn = −cRn Rn−1 ,

(21)

where the factor c is independent of n but can depend on time. From (5), we then obtain the equation q 2 Dt

Rn = q 2 q2 Tt (c(t)Rn Rn+1 ) − c(t)Rn Rn−1 .

(22)

The choice c = const (c = 1 in [5]) leads to Eq. (2), but in this case, Qn and Rn have time-dependent asymptotic forms as |n| → ∞. We want to study the problem in which Rn and Qn have constant and time-independent asymptotic forms. We therefore choose Qn → −1, Rn → 1 as |n| → ∞; this immediately implies that c(t) = [1 − (1 − q 2 )t]−2 .

(23)

1541

3. The direct and inverse scattering transform We assume that all Rn are positive, Rn → 1 as |n| → ∞, and the infinite product +∞ 

∆∞ =

Rn < ∞

(24)

n=−∞

converges. It turns out that if the functions Rn satisfy system (5), (6), then ∆∞ is independent of time; therefore, if it was bounded initially, it remains bounded for any time t. Indeed, the following proposition holds.  Proposition. If ∆∞ = +∞ −∞ Rk is an analytic function in a neighborhood of t = 0, then ∆∞ is a constant. Proof. Recalling the lemma, all we must prove is that

q2 Tt ∆∞

= q 2 Tt

  ∞

q 2 Dt

 Rk

=

k=−∞

Recalling (6), we have relation.

q 2 Tt

∆∞ = 0, i.e.,

∞ 

q 2 Tt

q 2 Tt

∆∞ = ∆∞ . We have

Rk .

(25)

k=−∞

Rk = Qk Rk−2 /Qk−1 , which gives the result when substituted in the previous

A simple gauge transformation

ψn =

 ∆n ψˆn ,

∆n =

∞ 

Rk

(26)

k=n+1

maps problem (7) into the corresponding problem   Rn+1 ψn+1 + Rn ψn−1 = λψn for the self-adjoint operator Lnm =

  Rn δn,m+1 + Rm δn+1,m .

(27)

(28)

The direct and inverse scattering transform associated with this operator have been studied in detail (see, e.g., [2]). Here, we briefly recall basic facts and definitions required for our study. Eigenvalue problem (27) has a twice-degenerate continuous spectrum filling the interval I = {λ; −2 ≤ λ ≤ 2}; the corresponding eigenfunctions are bounded for all n. In general, problem (27) can also have real, discrete eigenvalues. We assume that they are finite in number, i.e., J = {λ1 , . . . , λN ; |λk | > 2}. This set is usually called the discrete spectrum, and the corresponding eigenfunctions are square summable. If λ ∈ I, the asymptotic behavior of ψn as |n| → ∞ is given by ψn = c1 z −n + c2 z n + o(1),

|z| = 1,

λ = z + z −1 .

(29)

We can introduce Jost functions, i.e., special solutions of Eq. (27) defined by the asymptotic forms ψn = z n + o(1) as n → +∞, 1542

φn = z −n + o(1) as n → −∞.

(30)

The functions ψn and φn are analytic inside the unit circle in the complex z plane. A second pair of special solutions ψ˜n and φ˜n is defined by the asymptotic forms ψ˜n = z −n + o(1) as n → +∞,

φ˜n = z n + o(1) as n → −∞.

(31)

They are analytic outside the unit circle. We note that these functions, being solutions of a real-valued equation with asymptotic forms given z ), φ¯n (z) = φn (¯ z ), etc. Evidently, on the by real-valued functions of z, satisfy the relations ψ¯n (z) = ψn (¯ unit circle, we have ψ˜n (z) = ψ¯n (z),

φ˜n (z) = φ¯n (z),

|z| = 1.

(32)

Continuing these relations to the complex z plane, we obtain ψ˜n (z) = ψ¯n (¯ z −1 ) = ψn (z −1 ),

φn (z −1 ) = φ˜n (z) = φ¯n (¯ z −1 ).

(33)

In the z plane, the continuous spectrum is on the unit circle (λ = z + z −1 ). For |z| = 1, the solutions ψn and ψ¯n are linearly independent, and because Eq. (27) has at most two linearly independent solutions, any solution of (27) must be a linear combination of ψn and ψ¯n . In particular, φn = α(z)ψ¯n + β(z)ψn , |α(z)|2 − |β(z)|2 = 1,

|z| = 1, |z| = 1.

(34) (35)

The points zk , |z| < 1, k = 1, . . . , N , where α(z) = 0, are in a one-to-one correspondence with the eigenvalues of the discrete spectrum of the operator L: λk = zk + zk−1 . At these points, φn (zk ) = bk ψn (zk ),

(36)

i.e., φn (zk ) are the corresponding eigenfunctions. We call the set   β(z) ; k = 1, 2, . . . , N S(n; z) = zk , bk , r(z) ≡ α(z)

(37)

the spectral transform and its elements the scattering data or spectral data for the operator L. We note that for every n, ψn (−z) = (−1)n ψn (z),

φn (−z) = (−1)n φn (z),

(38)

and therefore α(−z) = α(z),

β(−z) = β(z),

|z| = 1.

(39)

In terms of the spectral data, the operator L is therefore chosen by the condition r(−z) = r(z), |z| = 1. Furthermore, the points zk form a set symmetric with respect to z = 0, and bk are the same at symmetric points of this set. There is a one-to-one map between the spectral data and the sequence of Rn ; the problem of finding the scattering data for a given sequence {Rn } is called the direct spectral transform. The solution of the inverse spectral problem, i.e., the reconstruction of Rn for a given collection S(n, t; z) is described in detail in [2]. 1543

The inverse spectral problem can be explicitly solved for reflectionless potentials, where r(z) is identically equal to zero. We introduce the notation φn (z) = z −n θn (z)

 ∆n ,

ψ˜n (z) = z −n χ ˜n (z)

 ∆n ,

ψn (z) = z n χn (z)

 ∆n .

(40)

By virtue of (33), we have χ ˜n (z) = χn (z −1 ).

(41)

˜n (z) are analytic inside and outside the unit circle; they satisfy The respective functions θn (z) and χ zχ ˜n−1 + z −1 Rn+1 χ ˜n+1 = (z + z −1 )χ ˜n

(42)

(the same equation holds for θn (z)). It follows from the analytic properties and from Eq. (41) that Rn =

χ ˜n−1 (0) . χ ˜n (0)

(43)

Recalling Eq. (6), we then find Qn = −γ(t)

χ ˜n−2 (0) , Tt χ ˜n (0)

(44)

q2

where the multiplier γ(t) can be determined by the asymptotic forms of χ ˜n and Qn , and in our case, it turns out to be equal to 1. The solution is then given by Rn =

χ ˜n−1 (0) , χ ˜n (0)

Qn = −

χ ˜n−2 (0) , Tt χ ˜n (0)

(45)

q2

where χ ˜n is obtained by solving the inverse problem. For reflectionless potentials, the solution is determined by the characteristics of the discrete spectrum only; in general, α(z) has N zeros and is given by   N N z − zk . zk α(z) = sgn zzk − 1 k=1

(46)

k=1

In this case, the function χ ˜n (z) is rational and can be represented as

χ ˜n (z) = 1 +

N 2n −1
zm bm χ ˜n (zm ) ,  α (z )(z − z ) m m m=1

(47)

where the coefficients χ ˜n (zk−1 ), k = 1, . . . , N , are solutions of the closed system of linear algebraic equations χ ˜n (zk−1 ) = 1 +

N


2n −1 zm bm χ ˜n (zm ) . −1  α (z )(z − z m m) k m=1

(48)

The details of this derivation can be found in [2]. In the next section, we find the time dependence of the spectral data. In particular, we show that if Rn and Qn satisfy the q-Volterra equation, then {zk } and α(z) are independent of time, while r(z) and bk are explicit functions of time. 1544

In the simplest case (one-soliton solution), α(z) has two zeros: z1 = z0 and z2 = −z0 , |z0 | < 1, and the system of algebraic equations (48) consists of one equation only. Its solution is given by

χ ˜n (z1−1 )

=

χ ˜n (z2−1 )

 −1 2f (t)z02n = An = 1 + −2 , (z0 − z02 )

f (t) = −

z0  α (z

0)

b0 (t).

(49)

Hence, f (t)z02n−1 An (t) f (t)z02n−1 An (t) + , (z − z0 ) (z + z0 )

χ ˜n (z; t) = 1 −

χ ˜n (z; t)|z=0 = 1 + 2f (t)z02n−2 An (t).

(50) (51)

It then follows from (43) that

Rn =

1 + 2f (t)z02n−4 An−1 (t) χ ˜n−1 (0) = . χ ˜n (0) 1 + 2f (t)z02n−2 An (t)

(52)

Similarly, it follows from (44) that

Qn = −

1 + 2f (t)z02n−6 An−2 (t) χ ˜n−2 (0) =− . ˜n (0) 1 + 2f (q 2 t)z02n−2 An (q 2 t) q 2 Tt χ

(53)

In particular, for the one-soliton solution, we have

∆∞ =

χ ˜−∞ (0) = z0−4 . χ ˜+∞ (0)

(54)

4. Time evolution of the spectral data To find the time evolution of the spectral data, we must take Eq. (8) into account, which is a qdifferential time evolution. We note that the spectral data have been defined for problem (27), which differs from (7) by gauge transformation (26). Using this transformation, we can identify the spectral problems for the operators corresponding to (7) and (27). We have ¯ φˆn = α(z)ψˆn + β(z)ψˆn ,

(55)

¯ˆ and ψˆn satisfy (7) and have the asymptotic forms where the functions φˆn , ψ, φˆn → z −n (∆∞ )−1/2 ψˆn → z n ,

¯ ψˆn → z −n

as n → −∞,

(56)

as n → ∞.

(57)

Similarly to the standard approach [2], the compatibility of Eqs. (7) and (8), the asymptotic forms of the Jost functions, and the proposition imply that q 2 Dt

φˆn − Qn φˆn−2 = z 2 q2 Tt φˆn .

(58) 1545

Substituting (55) in (58), we find the equations for α(z) and β(z), q 2 Dt

α = −z 2 α + z 2 q2 Tt α,

(59)

q 2 Dt

β = −z −2 β + z 2 q2 Tt β.

(60)

It follows from (59) that q2 Dt α = 0, and applying the lemma, we therefore see that α(z) is constant in time. We now consider Eq. (60). Using the definition of q-derivative (3), we obtain the equation

q 2 Dt

β=

z −2

1 − z −4 β + (1 − q 2 )t

(61)

for β(z, t). Similarly, we find that bk (z, t) satisfies the same equation

q 2 Dt b k

=

z −2

1 − z −4 bk . + (1 − q 2 )t

(62)

In particular, it is easy to verify that 2 q2 Dt (|α|

− |β|2 ) = 0,

|z| = 1.

(63)

To solve Eqs. (61) and (62), we rewrite them (the one for β, for instance) as

q 2 Tt

1 − (q 2 − 1)z −2 t β, 1 − (q 2 − 1)z 2 t

(64)

1 − (q 2 − 1)z −2 t β(t). 1 − (q 2 − 1)z 2 t

(65)

k  1 − (q 2 − 1)z −2 q 2j t β(t). 1 − (q 2 − 1)z 2 q 2j t j=0

(66)

β=

i.e., β(q 2 t) = Iterating (65) k times, we obtain

β(q 2k t) = Hence, assuming |q| < 1, we have

β(0) =

∞  1 − (q 2 − 1)z −2 q 2j t β(t) 1 − (q 2 − 1)z 2 q 2j t j=0

(67)

((q 2 − 1)z 2 t; q 2 )∞ β(0), ((q 2 − 1)z −2 t; q 2 )∞

(68)

in the limit as k → ∞, which gives β(t) = where we use the notation (a; q)∞ ≡

∞  k=0

1546

(1 − aq k ),

(a; q)0 = 1,

|q| < 1,

a ∈ C.

(69)

Fig. 1.

Density plot of the function Rn : the soliton solution of the Volterra model is on the left, and

the soliton solution of the q-deformed Volterra model is on the right (the axes represent t versus n).

Eq. (68) can be rewritten in terms of q-exponentials [6] defined as ∞


eq (ζ) ≡

k=0

Eq (ζ) ≡

ζk 1 = , (q; q)k (ζ; q)∞

∞
q k(k−1)/2 ζ k k=0

(q; q)k

|ζ| < 1,

= (−ζ; q)∞ ,

(70)

ζ ∈ C.

(71)

The series eq (ζ) and Eq (ζ), which are the most elementary q-hypergeometric series, are the q-analogues of the standard exponential in accordance with the limit formulas lim Eq ((1 − q)ζ) = eζ = lim eq ((1 − q)ζ).

q→1

(72)

q→1

Here, the limit must be considered termwise and is sometimes referred to as limq↑1 . With this notation, we can finally express the solutions of Eqs. (61) and (62) in terms of elementary q-functions, β(t) = eq2 ((q 2 − 1)z −2 t)Eq2 ((1 − q 2 )z 2 t)β(0),

(73)

bk (t) = eq2 ((q 2 − 1)zk−2 t)Eq2 ((1 − q 2 )zk2 t)bk (0).

(74)

In particular (recalling (49)), we obtain f (t) = eq2 ((q 2 − 1)z0−2 t)Eq2 ((1 − q 2 )z02 t)f (0),

(75)

which is obviously analytic in the vicinity of t = 0. Finally, substituting (75) in (49), (52), and (44), we obtain the one-soliton solution of the q-Volterra equation explicitly. We note that using (72), we have lim β(t) = e(z 2

q ↑1

2

−z −2 )t

β(0),

−2

lim bk (t) = e(zk −zk 2 2

q ↑1

)t

bk (0),

(76)

i.e., we obtain the standard time dependence (see [2]). Hence, we recover the one-soliton solution of Eq. (1) in the limit q → 1. Comparing the q-deformed solution with the solution of Eq. (1) (see, e.g., [2]), we conclude that for fixed t, the shape of both solutions (as functions of n) is the same. But the time dependence is different (see Fig. 1): the q-deformed solution travels with a nonconstant velocity because of the singularities (poles) of eq2 (eq (ζ) is a meromorphic function with simple poles at ζ = q −k , k ∈ Z). 1547

Acknowledgments. The author thanks A. V. Mikhailov for drawing attention to this problem, for the useful advice, and for encouraging the publication of this paper. The author also thanks V. Kuznetsov and F. Nijhoff for the stimulating discussions. The author is grateful to the University of Leeds and to the Isaac Newton Institute for Mathematical Sciences, Cambridge, where a part of this work was completed during the program “Integrable Systems,” for their hospitality. REFERENCES 1. V. Volterra, Lec¸on sur la th´eorie math´ematique de la lutte pour la vie, Gautier-Villars, Paris (1931). 2. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Plenum, New York (1984). 3. L. D. Faddeev and L. A. Takhtajan, “Liouville model on the lattice,” in: Field Theory, Quantum Gravity, and Strings (Lect. Notes Phys., Vol. 246, H. J. de Vega and N. S´ anchez, eds.), Springer, Berlin (1986), pp. 166–179. 4. A. Yu. Volkov, Phys. Lett. A, 167, 345–355 (1992); A. Antonov, Theor. Math. Phys., 113, 1520–1529 (1997); hep-th/9607031 (1996). 5. F. W. Nijhoff, Lett. Math. Phys., 30, 327–336 (1994). 6. T. H. Koornwinder, “q-Special function, a tutorial,” math.CA/9403216 (1994); “Special function and q-commuting variables,” q-alg/9608008 (1996).

1548