Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator

Physics Letters A 323 (2004) 241–250 www.elsevier.com/locate/pla

Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator M. Nieszporski a,b,∗ , P.M. Santini c , A. Doliwa d a Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Ho˙za 74, 00-682 Warszawa, Poland b Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, ul. Lipowa 41, 15-424 Białystok, Poland c Dipartimento di Fisica, Università di Roma “La Sapienza”, and Istituto Nazionale di Fisica Nucleare, Sezione di Roma,

Piazzale Aldo Moro 2, I-00185 Roma, Italy d Uniwersytet Warminsko-Mazurski w Olsztynie, Wydzial Matematyki i Informatyki, ul. Zolnierska 14 A, 10-561 Olsztyn, Poland

Received 24 July 2003; received in revised form 6 November 2003; accepted 2 February 2004 Communicated by A.P. Fordy

Abstract With this Letter we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrödinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.  2004 Elsevier B.V. All rights reserved. PACS: 04.60.Nc Keywords: Darboux transformations; Difference equations; Lattice integrable nonlinear σ -models

1. Introduction Linear differential operators admitting Darboux Transformations (DTs) play a crucial role in the theory of integrable systems [1]. (i) One can associate with such operators integrable nonlinear partial differential equations (PDEs). (ii) One can make use of the spectral theory of such linear operators to solve classical initial-boundary value problem for the nonlinear PDEs [2,3]. (iii) One can construct solutions of the nonlinear equations from * Corresponding author.

E-mail addresses: [email protected] (M. Nieszporski), [email protected] (P.M. Santini), [email protected] (A. Doliwa). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.003

242

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

simpler solutions, using the Darboux–Bäcklund transformations. (iv) One can often associate with the nonlinear PDE a geometric meaning, inherited by the geometric properties of the linear operator. It is natural to search for a discretization of this beautiful picture. In general, searching for distinguished discretizations of linear differential operators that admit general DTs (integrable discretizations) is not a trivial task. Several methods of “integrable discretization” have been used so far (see, e.g., [4]), but no one is fully satisfactory. To the best of our knowledge, none of the difference schemes that can be regarded as discretizations of elliptic equations was proved to be covariant under DTs. Our main goal is to change this situation and in this Letter we construct DTs for the following three basic discretizations L5 := am,n Tm + am−1,n Tm−1 + bm,n Tn + bm,n−1 Tn−1 − fm,n , Γm,n Γm−1,n −1 Γm,n Γm,n−1 −1 LSchInt := Tm + Tm + Tn + T − qm,n , Γm+1,n Γm,n Γm,n+1 Γm,n n

(1)

L7 := am,n Tm + am−1,n Tm−1 + bm,n Tn + bm,n−1 Tn−1 + sm+1,n Tm Tn−1 + sm,n+1 Tm−1 Tn − fm,n

(3)

(2)

of elliptic operators, where Tm and Tn are the translation operators with respect to the discrete variables (m, n) ∈ Z2 : Tm fm,n = fm+1,n ,

Tn fm,n = fm,n+1 ,

and the standard notation for complex-valued functions fm,n = f (m, n) is used throughout the Letter. In Section 2 we make general considerations concerning “integrable discretizations” of both hyperbolic and elliptic operators and we discuss some basic properties of the three operators above. In Sections 3, 4, and 5 we construct the DTs for the operators L5 , LSchInt , and L7 , respectively. In the last Section 6, devoted to the continuous limit of the results of the previous sections, we consider the following continuous limits: A∂x2 + A,x ∂x + B∂y2 + B,y ∂y − F,

(4)

∂x2 + ∂y2 − Q,

(5)

A∂x2

+ B∂y2

+ 2S∂x ∂y + (A,x +S,y )∂x + (B,y +S,x )∂y − F,

(6)

respectively, of the operators L5 , LSchInt , and L7 and we construct the corresponding DTs (the DTs for the operators (4) and (6) are also a novel result). To the best of our knowledge, the operators L5 and LSchInt make their entry, in this Letter, in the theory of integrable systems (for L7 a class of Laplace transformations is already known [5,6]). In particular, we would like to draw the attention of the reader on the discretization (2) of the 2D Schrödinger operator (5). Since the operator LSchInt admits DTs and reduces to the 2D Schrödinger operator (5) under the natural continuous limit, it can be considered as a distinguished integrable discretization of it. It is therefore the proper starting point in the search for the integrable discrete analogues of the nonlinear PDEs of elliptic type associated with (5) (like the Veselov–Novikov hierarchy [7], the nonlinear σ model [8] and the Ernst equation [9]) and in the search for a generalization of the Toda law [10] to a 2D lattice.

2. Difference hyperbolic and elliptic schemes In recent years the study of linear difference equations that admit DTs was undertaken. Most of the results are based on the 4-point difference scheme, i.e., a scheme that relates four neighbouring points ψm+1,n+1 = αm,n ψm+1,n + βm,n ψm,n+1 + γm,n ψm,n

(7)

(where α, β, γ are real functions of the discrete variables (m, n) ∈ Z2 ), which is proper for discretizing secondorder hyperbolic equations in the canonical form. In particular, the proper discrete analogue of the Laplace equation

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

243

for conjugate nets Ψ,uv + CΨ,u + DΨ,v = 0,

(8)

whose DTs were obtained in [11,12], turns out to be the 4-point scheme [13] ψm+1,n+1 = αm,n ψm+1,n + βm,n ψm,n+1 + (1 − αm,n − βm,n )ψm,n ,

(9)

describing a lattice with planar quadrilaterals [14,15], whose general DTs were extensively studied in recent years (see, e.g., [16–18] for the DTs and [6,14,19] for the Laplace transformations). The Moutard equation [20] Ψ,uv = F Ψ,

(10)

well-known self-adjoint reduction of Eq. (8) admitting DTs, is relevant in the description of asymptotic nets. It splits naturally in the discrete case into two mutually adjoint, with respect to bilinear form
f, g := fm,n gm,n , m,n

equations [21,22] ψm+1,n+1 + ψm,n = fm,n (ψm+1,n + ψm,n+1 ), ψm+1,n+1 + ψm,n = fm+1,n ψm+1,n + fm,n+1 ψm,n+1

(11)

admitting DTs. We observe that, in analogy with the continuous case, in which the complexification of the independent variables (u = x + iy and v = x − iy) in Eq. (10) is used to get surfaces of constant mean curvature, a 4-point scheme with a complexification of the discrete variables (m, n) has been recently used in [23] to obtain discrete lattices with constant discrete mean curvature. Our approach to the discretization of elliptic operators is different. In our opinion, a proper discretization of an elliptic operator, besides the obvious requirement of reproducing such operator in the natural continuous limit, should satisfy two basic properties. • It should be applicable to solve generic Dirichlet boundary value problems on a 2D lattice. • It should possess a class of DTs which must be (at least) as rich as that of its differential counterpart. It is easy to convince one-self that the first criterion cannot be satisfied by the 4-point scheme (7), which can be used instead in the discretization of hyperbolic operators (see, for instance, [15], for the identification of the proper initial-boundary value problem for (9) and for its geometric interpretation). It is also easy to convince one-self that at least 5-point difference schemes should be used to satisfy the first criterion (we refer to the book [24] for details). Therefore, we are naturally led to the search for discrete symmetries of Darboux type for 5-point (and higher point) schemes and to this goal is devoted most of the Letter. The only restriction which will be imposed on the associated linear operators is the property of self-adjointness. The 5-point operator L5 in (1), the first example we consider in the Letter, is just the most general formally self-adjoint operator on the star of a square lattice (see Fig. 1). In the particular case a = b = 1, this operator reduces to the 5-point operator LSch = Tm + Tm−1 + Tn + Tn−1 − fm,n ,

(12)

which is the simplest and most used, in numerical applications [24], discretization of the Schrödinger operator (5).

244

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

Fig. 1. The 5-point stencil.

The operator L5 exhibits the following covariance property (gauge invariance): L5 → L˜5 = gm,n L5 gm,n , am,n → a˜ m,n = am,n gm,n gm+1,n ,

bm,n → b˜m,n = bm,n gm,n gm,n+1 ,

2 fm,n → f˜m,n = fm,n gm,n , (13)

which can be used to draw it into the 2D integrable Schrödinger operator LSchInt (2). Indeed, if am,n and bm,n are g a = gm,n+1 and, under the gauge transformation (13), we given functions, one can find a function gm,n such that bm,n m,n m+1,n 1 1 2 get a˜ m,n = b˜m,n =: Γm,n . Finally, the operator L˜ 5 is of the wanted form (2). Therefore: Γm,n

LSchInt =

Γm,n

gm,n gm,n L5 , Γm,n Γm,n

(14)

2 /Γ 2 ), where g and Γ are defined by: with qm,n = fm,n (gm,n m,n

am,n gm+1,n = bm,n gm+1,n ,  √ Γm,n = am,n gm,n gm+1,n = bm,n gm,n gm,n+1 .

(15a) (15b)

We will use the gauge transformation (14) in Section 4 to derive DTs for LSchInt operator. Besides the fact that the operator LSchInt possesses DTs, it is also symmetric with respect to the exchange m ↔ n (if the potentials Γ and q possess this symmetry) and reduces to the two-dimensional Schrödinger operator (5) under the natural continuous limit (see Section 6). For these reasons, as it was mentioned in the introduction, it can be considered as a distinguished “integrable” discretization of the two-dimensional Schrödinger operator. The 7-point operator L7 in (3), whose Laplace transformations have been already mentioned in the introduction, can be interpreted as the most general self-adjoint scheme on the star of a regular triangular lattice [5] and plays a relevant role in a recently developed discrete complex function theory [25]. We finally remark that several discretizations of the 1D Schrödinger operator have appeared in the literature throughout the years (see, for instance, [26–29]).

3. 5-point self adjoint operator and its Darboux transformation In this section we consider the 5-point operator L5 together with the associated difference equation: am,n ψm+1,n + am−1,n ψm−1,n + bm,n ψm,n+1 + bm,n−1 ψm,n−1 = fm,n ψm,n . The operator L5 admits the following DTs.

(16)

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

245

Let θ be another solution of (16), i.e.: am,n θm+1,n + am−1,n θm−1,n + bm,n θm,n+1 + bm,n−1 θm,n−1 = fm,n θm,n ;

(17)

then: fm,n =

1 θm,n

(am,n θm+1,n + am−1,n θm−1,n + bm,n θm,n+1 + bm,n−1 θm,n−1 ).

(18)

Eliminating fm,n from (16) and (17) we get $m (am−1,n ψm,n θm−1,n − am−1,n θm,n ψm−1,n ) + $n (bm,n−1 ψm,n θm,n−1 − bm,n−1 θm,n ψm,n−1 ) = 0.

(19)

It means that there exists a function α such that $n α = am−1,n θm,n θm−1,n $−m

ψm,n , θm,n

$m α = −bm,n−1 θm,n θm,n−1 $−n

ψm,n , θm,n

(20)

where: $m fm,n = fm+1,n − fm,n ,

$n fm,n = fm,n+1 − fm,n ,

$−m fm,n = fm−1,n − fm,n ,

$−n fm,n = fm,n−1 − fm,n .

Setting  = ψm,n

αm,n , θm,n

 we find that ψm,n satisfies the following equation:           am,n ψm+1,n + am−1,n ψm−1,n + bm,n ψm,n+1 + bm,n−1 ψm,n−1 = fm,n ψm,n ,

(21)

where θm,n θm,n  , bm,n−1 = , bm−1,n−1 θm−1,n−1 am−1,n−1 θm−1,n−1   1 1 1 1      . fm,n = θm,n am,n + am−1,n + bm,n + bm,n−1 θm+1,n θm−1,n θm,n+1 θm,n−1  = am−1,n

(22) (23)

Comparing Eqs. (18) and (23), we also infer that θ  = 1/θ is a solution of (21).

4. A two-dimensional Schrödinger operator and its Darboux transformation Combining the gauge transformation (14) with the DT of the previous section, one obtains the following DT for the discrete analogue (2) of the Schrödinger operator (5). Let Nm,n be a solution of the integrable discrete analogue of the 2D Schrödinger equation: Γm,n Γm−1,n Γm,n Γm,n−1 Nm+1,n + Nm−1,n + Nm,n+1 + Nm,n−1 = fm,n Nm,n , Γm+1,n Γm,n Γm,n+1 Γm,n and let ϑ be another solution of it. It means that   1 Γm−1,n Γm,n Γm,n−1 Γm,n ϑm+1,n + ϑm−1,n + ϑm,n+1 + ϑm,n−1 . fm,n = ϑm,n Γm+1,n Γm,n Γm,n+1 Γm,n

(24)

(25)

246

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

 , defined by Eliminating fm,n from (25) and (24) we get that there exists a function Nm,n     ϑm,n  Γm,n−1 Nm,n Nm,n−1  Γ N ϑm,n ϑm,n−1 − = , $m Γm,n m,n m,n Γm,n ϑm,n ϑm,n−1     ϑm,n  Γm−1,n Nm,n Nm−1,n $n Γ N ϑm,n ϑm−1,n − =− , Γm,n m,n m,n Γm,n ϑm,n ϑm−1,n

(26)

which satisfies  Γm,n

 Γm+1,n

 Nm+1,n

+

 Γm−1,n  Γm,n

 Nm−1,n

+

 Γm,n

 Γm,n+1

 Nm,n+1

+

 Γm,n−1  Γm,n

   Nm,n−1 = fm,n Nm,n ,

(27)

where Γm,n Γm−1,n−1 ϑm−1,n−1 , ϑm,n         ϑ Γm−1,n Γm,n Γm,n Γ Γ m,n  + fm,n =   Γm,n Γm+1,n Γ  ϑ m+1,n Γm,n Γ  ϑ m−1,n        Γm,n−1 Γm,n Γ Γ +  +  Γm,n+1 Γ  ϑ m,n+1 Γm,n Γ  ϑ m,n−1        Γ 1 ϑ 1 Γ = (Γ ϑ)m−1,n−1 + . + + (Γ ϑ)m−1,n (Γ ϑ)m,n−1 Γ ϑ m−1,n ϑ m,n−1 2 Γm,n =

(28)

 Now the function ϑm,n = (Γ /Γ  ϑ)m,n is a solution of (27). While one does not know DTs for the standard discretization LSch in (12) of the Schrödinger operator, we have identified in this Letter the operator LSchInt , a suitably dressed version of LSch , which possesses DTs and, in the natural continuous limit, reduces to the celebrated Schrödinger operator in 2D.

5. 7-point self-adjoint operator and its Darboux transformations The construction of DTs presented in the previous two sections applies to the self-adjoint 7-point scheme associated with L7 : am,n ψm+1,n + am−1,n ψm−1,n + bm,n ψm,n+1 + bm,n−1 ψm,n−1 + sm+1,n ψm+1,n−1 + sm,n+1 ψm−1,n+1 = fm,n ψm,n ,

(29)

where am,n , bm,n , sm,n and fm,n are given functions, which is a discretization of the most general second order, self-adjoint, linear, differential equation in two independent variables (see Section 6). Let θm,n be another solution of Eq. (29): am,n θm+1,n + am−1,n θm−1,n + bm,n θm,n+1 + bm,n−1 θm,n−1 + sm+1,n θm+1,n−1 + sm,n+1 θm−1,n+1 = fm,n θm,n . Eliminating fm,n from (29) and (30) we get      ψm,n ψm−1,n ψm,n−1 ψm−1,n + sm,n θm−1,n θm,n−1 − − $m am−1,n θm,n θm−1,n θm,n θm−1,n θm,n−1 θm−1,n      ψm,n ψm,n−1 ψm−1,n ψm,n−1 + sm,n θm−1,n θm,n−1 = 0. − − + $n bm,n−1 θm,n θm,n−1 θm,n θm,n−1 θm−1,n θm,n−1

(30)

(31)

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

247

It means that there exists a function ψ  such that ψm,n ψm,n − sm,n θm−1,n θm,n−1 $−n , θm,n θm,n ψm,n ψm,n  $m (ψm,n θm,n ) = −(bm,n−1 θm,n θm,n−1 + sm,n θm−1,n θm,n−1 )$−n + sm,n θm−1,n θm,n−1 $−m , θm,n θm,n

 $n (ψm,n θm,n ) = (am−1,n θm,n θm−1,n + sm,n θm−1,n θm,n−1 )$−m

(32) and function

 ψm,n

satisfies the following equation:

        ψm+1,n + am−1,n ψm−1,n + bm,n ψm,n+1 + bm,n−1 ψm,n−1 am,n       + sm+1,n ψm+1,n−1 + sm,n+1 ψm−1,n+1 = fm,n ψm,n ,

(33)

where the new fields are given by θm,n θm+1,n am−1,n θm,n θm,n+1 bm,n−1  , bm,n = , θm,n−1 pm,n θm−1,n pm,n sm−1,n−1 θm−1,n θm,n−1  = , sm,n θm−1,n−1 pm−1,n−1  1 1 1 1      fm,n = θm,n am,n + am−1,n + bm,n + bm,n−1 θm+1,n θm−1,n θm,n+1 θm,n−1  1 1   + sm+1,n + sm,n+1 θm+1,n−1 θm−1,n+1  am,n =

(34)

 and where pm,n = θm,n am−1,n bm,n−1 + θm−1,n sm,n am−1,n + sm,n θm,n−1 bm,n−1 . Again θm,n = 1/θm,n is a solution of (33).

6. Continuous limit and DTs for second-order partial differential operators The Darboux transformations for the operator L7 constructed in the previous section reduce, in the natural continuous limit, to the DTs for the most general self-adjoint, second order, partial differential operator in two independent variables. This result is, to the best of our knowledge, new also in the continuous case. Moreover, the continuous limit of the gauge-equivalent operators L5 and LSchInt needs some attention, yielding two partial differential operators which are not gauge-equivalent. For these two reasons we have decided to dedicate the last section of the paper to the continuous limit of the results constructed in the previous sections. 6.1. DTs for elliptic second-order PDEs In the natural continuous limit ((m, (n) → (x, y), in which the lattice spacing ( goes to zero and     ψm,n Ψ (x, y)

am,n = A(x, y) + O ( 3 , bm,n B(x, y)  

1 fm,n = 2(A + B) − ((A,x + B,y ) + ( 2 F (x, y) + (A,xx + B,yy ) + O ( 3 , 2

(35)

(36a) (36b)

248

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

the operator L5 and the associated 5-point scheme (16) reduce respectively to the following structures: L := A∂x2 + A,x ∂x + B∂y2 + B,y ∂y − F,

(37a)

LΨ := (AΨ,x ),x + (BΨ,y ),y −F Ψ = 0,

(37b)

whose gauge covariance L → L˜ = GLG, B˜ = G2 B, A˜ = G2 A,

 F˜ = G2 F − G (AG,x ),x + (BG,y ),y

(38)

can also be obtained from (13) in the natural continuous limit gm,n = G(x, y) + O((). We remark that the Ψ,xy term, the only second order term present in Eqs. (8) and (10), is missing in (37b). Eq. (37b) is elliptic if AB > 0. In the continuous limit (35), with θm,n = Θ(x, y) + O(( 3 ) and with fm,n expanded according to (18) to get (36b), we finally obtain the DT: 1 (ΘΨ  ),y = −AΘ(Ψ/Θ),x , Θ from the solution space of equation

1 ΘΨ  ,x = BΘ(Ψ/Θ),y Θ

(AΨ,x ),x + (BΨ,y ),y = F Ψ

(39)

(40)

to the solution space of equation (A Ψ  ,x ),x + (B  Ψ  ,y ),y = F  Ψ  ,

(41)

where Θ is another solution of (40), so that 1 (AΘ,x ),x + (BΘ,y ),y , Θ and the new “potentials” A , B  and F  are related to the old ones as follows: F=

1 , B 1 B = , A 

F  = Θ A (1/Θ),x ,x + B  (1/Θ),y ,y . A =

(42)

(43a) (43b) (43c)

Comparing Eqs. (41) and (43c), we also infer that Θ  = 1/Θ is a solution of (41). 6.2. Moutard transformation for the Schrödinger equation In the natural continuous limit (35), with     Ψ (x, y) Nm,n

ϑm,n = Θ(x, y) + O ( 3 , J (x, y) Γm,n  

J,x +J,y (J,x )2 + (J,y )2 + (2 qm,n = 4 − 2( + Q(x, y) + O (3 , 2 J J

(44a) (44b)

the discrete Schrödinger equation (24) reduces to the celebrated Schrödinger equation Ψ,xx +Ψ,yy = QΨ

(45)

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

249

and the DTs of the discrete Schrödinger equation reduce to the classical Moutard transformations 1 1 (ΘΨ  ),x = Θ(Ψ/Θ),y , (ΘΨ  ),y = −Θ(Ψ/Θ),x Θ Θ from a solution of Eq. (45) to a solution of equation

(46)

Ψ  ,xx + Ψ  ,yy = Q Ψ  ,

(47)

where Θ is another solution of (45), so that  1 Q = Θ (1/Θ),xx + (1/Θ),yy . Q = (Θ,xx + Θ,yy ), Θ Furthermore, Θ  = 1/Θ is a solution of (47). We remark that, while the discrete Schrödinger operator LSchInt in (2) is gauge-equivalent, via (14), to the general 5-point operator L5 in (1), the Schrödinger operator (5) (smooth continuous limit of LSchInt ) is not gaugeequivalent, via (38), to the operator L in (37a) (smooth continuous limit of L5 ). This contradiction is only apparent, and it is due to the fact that the gauge function gm,n , defined in (15a), does not have a smooth continuous limit if a, b have the limits (36), with A = B, and therefore it cannot reduce to the smooth function G(x, y) appearing in (38). 6.3. DTs for the general second-order self-adjoint operator The 7-point scheme (29) reduces, in the continuous limit,     ψm,n Ψ (x, y) 3

 am,n + sm,n   A(x, y)   = +O ( , B(x, y) bm,n + sm,n sm,n −S(x, y) fm,n = 2(A + B + S) − ((A,x + B,y + 2S,x + 2S,y ) + (

(48a)  2



1 F (x, y) + (A,xx + B,yy ) + O ( 3 , 2

(48b)

to the most general second order self-adjoint partial differential equation in two independent variables: (AΨ,x ),x + (SΨ,y ),x + (BΨ,y ),y + (SΨ,x ),y = F Ψ.

(49)

= Θ(x, y) + O(( 3 )

and fm,n is expanded according to (30) to obtain (48b), the DTs of the If, in addition, θm,n 7-point scheme constructed in Section 5 reduce to the following DTs: 1 1 (ΘΨ  ),y = −AΘ(Ψ/Θ),x − SΘ(Ψ/Θ),y , (ΘΨ  ),x = BΘ(Ψ/Θ),y + SΘ(Ψ/Θ),x Θ Θ from the solution space of Eq. (49) to the solution space of equation (A Ψ  ,x ),x + (B  Ψ  ,y ),y + (S  Ψ  ,y ),x + (S  Ψ  ,x ),y = F  Ψ  , where Θ is a fixed solution of (49), so 1 (AΘ,x ),x + (BΘ,y ),y + (SΘ,y ),x + (SΘ,x ),y , F= Θ and the new “potentials” A , B  , S  and F  are related to the old ones as follows: A B S , B = , S = , 2 2 AB − S AB − S AB − S 2





 F  = Θ A (1/Θ),x ,x + B  (1/Θ),y ,y + S  (1/Θ),x ,y + S  (1/Θ),y ,x .

(50)

(51)

(52)

A =

Again Θ  = 1/Θ is a solution of (51).

(53)

250

M. Nieszporski et al. / Physics Letters A 323 (2004) 241–250

Acknowledgements One of us (P.M.S.) acknowledges useful comments of M.J. Ablowitz on the numerical schemes used to discretize the 2D Laplace operator. This work was supported by the cultural and scientific agreement between the University of Roma “La Sapienza” and the University of Warsaw, and by the University of Warmia and Mazury in Olsztyn under the grant 522-1307-0201 and partially supported by KBN grant 2 P03B 126 22.

References [1] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, in: Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. [2] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [3] S.V. Manakov, S.P. Novikov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York, 1984. [4] Y.B. Suris, R-matrices and integrable discretization, in: Discrete Integrable Geometry and Physics, Clarendon Press, Oxford, 1999. [5] S.P. Novikov, Russian Math. Surveys 52 (1997) 226. [6] S.P. Novikov, I.A. Dynnikov, Russian Math. Surveys 52 (1997) 1057. [7] A.P. Veselov, S.P. Novikov, Soviet Math. Dokl. 30 (1984) 588. [8] C.W. Misner, Phys. Rev. D 18 (1978) 4510. [9] F.J. Ernst, Phys. Rev. 167 (1968) 1175. [10] M. Toda, Theory of Nonlinear Lattices, in: Springer Series in Solid-State Sciences, vol. 20, Springer, Berlin, 1981. [11] H. Jonas, Berlin Sitzungsber. XIV (1915) 96. [12] L.P. Eisenhart, Transformation of Surfaces, Princeton Univ. Press, Princeton, NJ, 1923. [13] L.V. Bogdanov, B.G. Konopelchenko, J. Phys. A: Math. Gen. 28 (1995) L173. [14] A. Doliwa, Phys. Lett. A 234 (1997) 187. [15] A. Doliwa, P.M. Santini, Phys. Lett. A 233 (1997) 365. [16] M. Manas, A. Doliwa, P.M. Santini, Phys. Lett. A 232 (1997) 99. [17] B.G. Konopelchenko, W.K. Schief, Proc. R. Soc. London A 454 (1998) 3075. [18] A. Doliwa, P.M. Santini, M. Manas, J. Math. Phys. 41 (2000) 944. [19] V.E. Adler, S.Ya. Startsev, Theor. Math. Phys. 121 (1999) 1484. [20] Th.-F. Moutard, J. Ec. Pol. 45 (1878) 1. [21] J.J.C. Nimmo, W.K. Schief, Proc. R. Soc. London A 453 (1997) 255. [22] M. Nieszporski, Theor. Math. Phys. 133 (2002) 1576. [23] M. Grundland, D. Levi, L. Martina, J. Phys. A: Math. Gen. 36 (2003) 4599. [24] F.B. Hildebrand, Finite Difference Equations and Simulations, Prentice–Hall, Englewood Cliffs, 1968. [25] I.A. Dynnikov, S.P. Novikov, math-ph/0208041. [26] H. Flaschka, Prog. Theor. Phys. 51 (1974) 703. [27] S.V. Manakov, Soviet Phys. JETP 40 (1975) 269. [28] W.K. Schief, Appl. Math. Lett. 10 (1997) 13. [29] A. Shabat, in: M. Boiti, L. Martina, F. Pempinelli, B. Prinari, G. Soliani (Eds.), Nonlinearity, Integrability and All That. Twenty Years After NEEDS’79, World Scientific, Singapore, 2000, p. 331.