Darboux transformations for q-discretizations of 2D second order differential equations

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Darboux transformations for q-discretizations of 2Dsecond order differential equationsP Małkiewicz a b & M Nieszporski a c

a Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski , ul. Hoża 74, 00-682 ,Warszawa , Poland E-mail:b Sołtan Institute for Nuclear Studies , Hoża 69, 00-681 , Warszawa , Polandc Department of Applied Mathematics , University of Leeds , LS2 9JT , Leeds , UKPublished online: 21 Jan 2013.

To cite this article: P Małkiewicz & M Nieszporski (2005) Darboux transformations for q-discretizations of 2D second orderdifferential equations, Journal of Nonlinear Mathematical Physics, 12:sup2, 231-239, DOI: 10.2991/jnmp.2005.12.s2.17

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Journal of Nonlinear Mathematical Physics Volume 12, Supplement 2 (2005), 231–239 SIDE VI

Darboux transformations for q-discretizations of

2D second order differential equations

P MA LKIEWICZ a,b and M NIESZPORSKI a,c

a Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski ul. Hoza 74, 00-682Warszawa, Poland

E-mails: [email protected], [email protected] So ltan Institute for Nuclear Studies, Hoza 69, 00-681 Warszawa, Polandc Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

This article is a part of the special issue titled “Symmetries and Integrability of DifferenceEquations (SIDE VI)”

Abstract

We present q-discretizations of a second order differential equation in two independentvariables that not only go to the differential counterpart as q goes to 1 but admitMoutard-Darboux transformations as well.

1 Introduction

One of the important components of the theory of nonlinear S-integrable differential equa-tions [1] is Darboux transformations, the story of which starts with the paper by Moutard[2] and therefore, contrary to the common tendency, we will call them Moutard-Darbouxtransformations. An example of an equation which admits Moutard-Darboux transforma-tions is the second order linear differential equation in two independent variables

Lψ = 0L := a∂2

x + b∂2y + 2c∂x∂y + (a,x +c,y +w)∂x + (c,x +b,y +z)∂y − f

(1.1)

where ψ is dependent variable ψ : R2 ⊃ D ∋ (x, y) 7→ ψ(x, y) ∈ R while a, b, c, w, z and

w are given C1 functions of independent variables x and y (we use the standard notationa,x := ∂a

∂x, b,y := ∂b

∂yetc.). The goal of this paper is to introduce such q-discretizations of

the equation (1.1) that not only go to the equation (1.1) as q goes to 1 but admit Moutard-Darboux transformations as well. The literature concerning such discretizations is rich,see e.g. [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], but almost all of the mentioned papersconcern 4-point difference schemes. The main result of this paper is the Moutard-Darbouxtransformations introduced for a 7-point self-adjoint scheme [16] and a 6-point differencescheme [17] can easily be extended to q-difference equations.

The q-difference schemes we discuss here are the 6-point scheme

L6Ψ = 0L6 := aD

qxD

qx + bD

qyD

qy + cD

qxD

qy + gD

qx + hD

qy − f

(1.2)

Copyright c© 2005 by P Ma lkiewicz and M Nieszporski

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232 P Ma lkiewicz and M Nieszporski

and the self-adjoint 7-point schemes (a q-discretization of formally self-adjoint eq. (1.1)i.e. equation with w = 0 = z)

L7Ψ = 0

L7 := DqxaD

1

q

x +DqybD

1

q

y +DqxcD

1

q

y +DqycD

1

q

x − f(1.3)

where the q-derivative of a function f : R ∋ x 7→ f(x) ∈ R is defined by

Dqf : R ∋ x 7→ Dqf(x) :=f(qx) − f(x)

(q − 1)x

with R ∋ q 6= ±1, 0. We will express formulas mostly in terms of q-shift operators

T qxf(x, y) := f(qx, y) T qy f(x, y) := f(x, qy)

It is convenient to use the following notation for them

f1 := T1f := T qxf f12 := T12f := T qxTqy f

f−1 := T−1f := T1

q

x f etc.

The paper is organized as follows. In section 2 we derive Moutard-Darboux transfor-mations for the 2D second order differential equation (1.1). Applying the procedure fromthe continuous case we introduce Moutard-Darboux transformations for the q-difference6-point scheme in subsection 3.1 and for the q-difference 7-point scheme in subsection 3.2.We end the paper with concluding remarks (section 4) indicating differences between thediscrete and continuous cases and raising some open problems.

2 Moutard-Darboux Transformation for 2D second order

differential equation

The goal of this section is to derive Moutard-Darboux transformations for 2D generalsecond order differential equation

Lψ = 0L := a∂2


(2.1)

where a, b, c, w, z and w are given C1 functions of independent variables x and y suchthat ∀(x, y) ∈ D a2 +b2 +c2 6= 0. To do that we need a solution φ of the equation formallyadjoint to eq. (2.1)

L†φ = 0L† := ∂xa∂x + ∂yb∂y + ∂xc∂y + ∂yc∂x − ∂xw − ∂yz − f

(2.2)

Then the equation

φLψ − ψL†φ = 0 (2.3)

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Darboux transformations for q-discretizations of 2D second order differential equations233

which can be written explicitly in the form

[aφψ,x +cφψ,y +(wφ− aφ,x−cφ,y )ψ],x +[bφψ,y +cφψ,x +(zφ− bφ,y −cφ,x )ψ],y = 0

(2.4)

guarantees the existence of a potential r

r,x = bφψ,y +cφψ,x +(zφ− bφ,y −cφ,x )ψr,y = −aφψ,x−cφψ,y −(wφ− aφ,x−cφ,y )ψ

(2.5)

When we write the potential r in the form r = ψγ − φθpψθ

(the role of new functions ψ,γ, θ and p will become clear in a moment) then we obtain

(ψγ),x = θ[(p+ c)φ(ψθ

),x +bφ(ψθ

),y ] + [(φθp),x−θ2(b(φ

θ),y +c(φ

θ),x−z

φθ)]ψθ

(ψγ),y = θ[(p− c)φ(ψθ

),y −aφ(ψθ

),x ] + [(φθp),y +θ2(a(φθ),x +c(φ

θ),y −w

φθ)]ψθ

(2.6)

The crucial point is to assure that the map ψ 7→ ψ is an invertible map between twoequations of the second order. We demand that the factor multiplying ψ

θin eq.(2.6)

vanish i.e.

(φθp),x = θ2[b(φθ),y +c(φ

θ),x−z

φθ]

(φθp),y = −θ2[a(φθ),x +c(φ

θ),y −w

φθ]

(2.7)

and φ2θ2(p2 − c2 + ab) does not vanish in the domain. In virtue of (2.2) compatibilityconditions of eqs. (2.7) reads

Lθ = 0 (2.8)

So finally we have theorem

Theorem 1. Assume that Lθ = 0, L†φ = 0, p is defined by eqs. (2.7), d := φθ(p2 − c2 +ab) 6= 0 in the domain and ψ is the kernel of the operator L then formulas

[

(γψ),x(γψ),y

]

= φθ

[

c+ p b

−a p− c

]

(

ψθ

)

,x(

ψθ

)

,y

(2.9)

constitute transformation ψ 7→ ψ from solution space of the equation (2.1) to the solutionspace of the equation

Lψ = 0L := a∂2


(2.10)

Where the coefficients of (2.10) are given by

a = aγδd

b = bγδd

c = cγδd

w = (aγx+cγy

d− (p

d)yγ)δ − aγ

dδx −

cγdδy z = (

bγy+cγx

d+ (p

d)xγ)δ − bγ

dδy −

cγdδx

f = {−aγxx+bγyy+2cγxy

d− [(a

d)x + ( c−p

d)y]γx − [( b

d)y + ( c+p

d)x]γy}δ

(2.11)

where γ and δ are arbitrary (of class C2) functions.

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3 Moutard-Darboux transformation for q-difference schemes

3.1 The 6-point case

We consider the following q-difference equation

L6Ψ = 0L6 := AT11 +BT22 + 2CT12 +GT1 +HT2 − F

(3.1)

which relates the points of the rectangular lattice shown on the Figure 1.

t

t

t

t

t

t

1

2 12

11

22

Figure 1. The 6-point scheme

Similarly like in theorem 1 we think of Ψ as the kernel of the operator L6 while Θ andΦ are given elements of the kernels of L6 and L

†6 respectively i.e.

L6Θ = 0 (3.2)

L†6Φ = 0

L†6 := 1

q2A−1−1T−1−1 + 1

q2B−2−2T−2−2

+2 1q2C−1−2T−1−2 + 1

qG−1T−1 + 1

qH−2T−2 − F

(3.3)

One can consider the operator L†6 as the operator formally adjoint to the operator L6 since

the Green’s identity fL6g − gL†6f = D

qxM(f, g) + D

qyN(f, g) holds. From the above we

conclude that there exists a function S such that

DqyS = x{1

q(GΘ1 +AΘ11 + CΘ12)−1Φ−1 + 1

q2A−1−1ΘΦ−1−1 + 1

q2C−1−2ΘΦ−1−2}

DqxS = −y{1

q(HΘ2 +BΘ22 + CΘ12)−2Φ−2 + 1

q2B−2−2ΘΦ−2−2 + 1

q2C−1−2ΘΦ−1−2}

Next we define

P :=1

xy(q − 1)

S12

Θ12Φ(3.4)

Starting with the equation

ΦL6Ψ − ΨL†6Φ = 0 (3.5)

one obtains

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Theorem 2. Let Ψ be the kernel of the operator L6 and Θ and Φ be given solutionsof the equations L6Θ = 0 and L

†6Φ = 0 respectively, P is given by (3.4) and D :=

x2y2Φ−1Φ−2Θ1Θ2[A−1B−2 − (C + P )−2(C − P )−1] 6= 0 in a domain, then the equation

[

Dqx(ΨΓ)

Dqy(ΨΓ)

]

=

[

xy(C + P )−2Φ−2Θ1 y2B−2Φ−2Θ2

−x2A−1Φ−1Θ1 xy(P − C)−1Φ−1Θ2

] [

Dqx(Ψ

Θ)

Dqy(

ΨΘ)

]

defines a map Ψ 7→ Ψ and Ψ is the kernel of bared L6 operator i.e.

AΨ11 + BΨ22 + 2CΨ12 + GΨ1 + HΨ2 − F Ψ = 0

with the new potentials

A = ∆Γ11Θ11ΦAD1

B = ∆Γ22Θ22ΦBD2

C =(

Θ11Φ1−2(C+P )1−2

D1+ Θ22Φ

−12(C−P )−12

D2

)

∆Γ12

2

G =(

−Θ11Φ1−2(C+P )1−2+Θ11ΦAD1

− Θ2Φ−1(C−P )

−1+Θ1Φ−1A−1

D

)

∆Γ1

H =(

−Θ22Φ−12(C−P )

−12+Θ22ΦBD2

− Θ1Φ−2(C+P )

−2+Θ2Φ−2B−2

D

)

∆Γ2

FΓ = A

Γ11+ B

Γ22+ 2 C

Γ12+ G

Γ1+ H

Γ2

where ∆ and Γ are arbitrary functions.

3.2 The 7-point self-adjoint case

A similar procedure can be applied to the following 7-point scheme

L7Ψ = 0L7 := AT1 + 1

qA−1T−1 + BT2 + 1

qB−2T−2 + C1T1−2 + C2T−12 −F

(3.6)

which relates the points of a rectangular lattice shown on the Figure 2. We take only one

tt

t

t

t t

t

1-2

-12

-1

2

1

-2

Figure 2. The 7-point scheme

function satisfying L7Φ = 0, instead of two functions. Actually, we may define the second

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function, for example: L7Θ = 0, but the reasoning from the 6-point case can be appliedhere if Θ ≡ Φ. The continuous limit of the equation is Moutard (self-adjoint) reduction[16] of the general Moutard-Darboux transformation we have presented in section 2.

Starting with the equation

ΦL7Ψ − ΨL7Φ = 0 (3.7)

one obtains

Theorem 3. Let Ψ be the kernel of the operator L7, Φ be a particular solution of L7Φ = 0and function D = x2y2ΦA−1B−2 + x2y2Φ−1CA−1 + x2y2Φ−2CB−2 do not vanish in adomain then the equation

[

Dqx(ΨΦ)

Dqy(ΨΦ)

]

=

[

xyCΦ−1Φ−2 −y2(B−2ΦΦ−2 + CΦ−1Φ−2)x2(A−1ΦΦ−1 + CΦ−1Φ−2) −xyCΦ−1Φ−2

]

D1

q

x (ΨΦ )

D1

q

y (ΨΦ )

defines a map

Ψ 7→ Ψ

where Ψ is the kernel of bared L7 operator i.e.

AΨ1 +1

qA−1Ψ−1 + BΨ2 +

1

qB−2Ψ−2 + C1Ψ1−2 + C2Ψ−12 − FΨ = 0

with the new potentials

A = ΦΦ1A−1

qΦ−2D

B = ΦΦ2B−2

qΦ−1D

C = C−1−2Φ

−1Φ−2Φ

−1−2D−1−2

F = Φ(A 1Φ1

+ 1qA−1

1Φ

−1+ B 1

Φ2+ 1

qB−2

1Φ

−2+ C1

1Φ1−2

+ C21

Φ−12

)

4 Concluding remarks

The q-discretizations of Moutard-Darboux transformations we have shown in this paperdo not differ essentially from discretizations of Moutard-Darboux transformations. Butthe world of both discretizations is essentially different from their continuous counterpart.We will end this paper with a brief review of the differences indicating open problems.

The Moutard-Darboux transformations we have just presented can be reduced or spec-ified i.e. one can impose such constraints on transformations that allow the preservationof particular forms of the equations. In the continuous case one can:

• specify the gauge e.g. one can choose the affine gauge i.e. put f = 0 in the eq.(1.1) and demand that f in eq. (2.11) to be equal zero as well. It can be achievedby imposing constraints on function γ. The simplest way (but not the only one) isto put γ = 1

• specify the operator; due to simple transformation rules for coefficients a, b andc one can put e.g. a = 0 = b or c = 0

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• reduce the transformation; in the case of a reduction it is necessary to relatefunctions ψ and θ e.g. in the Moutard reduction p = 0, w = 0 = z and ψ = θ

Both for 6-point scheme and for 7-point scheme specifications to affine gauge (F = A +B + 2C + G + H and F = A + A−1 + B + B−2 + C1 + C2 respectively) are admissible.For the 6-point scheme there is a specification of Moutard-Darboux transformation to4-point scheme (A = 0 = B) and C = 0 does not yield specification while for the 7-point scheme there exist specification to 5-point scheme (C = 0) and one can not put(A = 0 = B). The situation is less investigated as far as reductions are concerned.The self-adjoint 7-point scheme obviously can not be a reduction of the 6-point scheme.A Moutard reduction for 6-point scheme is not known (there exists Moutard reductionfor 4-point scheme specification [8] and it is not self-adjoint reduction anymore [18]). Ageneralization of the 7-point scheme to a scheme that goes to the general 2D second orderdifferential equation is not known as well. We finally observe that 6-point scheme isappropriate to solve quite different boundary problems (Figure 3.) than 7-point schemedoes (Figure 4). It makes the problems of existence of Moutard reduction of the 6-pointscheme and generalizations of 7-point scheme especially interesting.

Figure 3.An initial-boundary problem for the 6-point scheme.

Having given the initial conditions at dark blue (black) points on two solid lines andboundary conditions at yellow (grey) points one can propagate the solution inside thedomain (white points). Wave fronts are drawn with the doted lines

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Figure 4.A Dirichlet problem for the 7-point scheme.

Having given boundary conditions for the hexagonal lattice at dark blue (black) pointsone can find unique solution at all internal (white) points.

Acknowledgments. One of us (M.N.) would like to express his thankfulness to organizersof the SIDE VI conference for the support that enable him to take part in the fruitfulconference. The initial and main stage of the work was supported by KBN grant 1 P03B017 28 while at the final stage M.N. was supported by the European Community under aMarie Curie Intra-European Fellowship, contract no MEIF-CT-2005-011228.

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Darboux transformations for q-discretizations of 2D second order differential equations