Classification of discrete equations linearizable by point transformation on a square lattice

0.11 MB·English

by  Christian Scimiterna

Classification of discrete equations linearizable by point
transformation on a square lattice
Christian Scimiterna and Decio Levi
Dipartimento di Ingegneria Elettronica
Universit`a degli Studi Roma Tre and INFN Sezione di Roma Tre
Via della Vasca Navale 84, 00146 Roma, Italy
E-mail: scimiterna@fis.uniroma3.it;[email protected]
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0 Abstract
2
Weprovideacompletesetoflinearizabilityconditionsfornonlinearpartialdifferenceequationsde-
n
finedonfourpointsand,usingthem,weclassifyalllinearizablemultilinearpartialdifferenceequations
a
definedon four points up to a M¨obious transformation.
J

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PACS numbers:
1
Mathematics Subject Classification:

]
I
S
. 1 Introduction
n
i
l
n In a series of papers [1–4] one has provided necessary conditions for the linearizability of real dispersive
[ multilinear difference equations on a quad–graph (see Fig. 1). and on three points (Fig. 2).

1
v un,m+1 un+1,m+1
6 • •
2
4
2
.
1
0
3
1
:
v
i • •
X
u u
n,m n+1,m
r
a
Figure 1: The quad–graphwhere a partial difference equation is defined
These conditions, obtained by considering the existence of point transformationsand symmetries have
been sufficient to classify the multilinear equations defined on three points [2] but not those defined on
four points. In [4] we considered the problem from the point of view of the symmetries, both point and
nonlocal. In this way we get a different set of conditions with respect to those obtained before which,
however, are not yet sufficient to classify the multilinear equations defined on four points. So here, using
the experience of[2] and[4]we constructthe largestpossible set oflinearizability conditions and, through
them, we classify the multilinear equations on a square lattice. We assume a partial difference equation
on a quad–graph to be given by
∂E
E =E(u ,u ,u ,u )=0, 6=0, i,j =0,1, (1)
n,m n,m+1 n+1,m n+1,m+1
∂u
n+i,m+j
1
r
3
@ r
2
@
@su
n,m+1
@
@
@
@
@
@
@
@
s s r1
@
u u
n,m n@+1,m
@
@
@
Figure 2: Points related by an equation defined on three points.
for a field u which linearizes into a linear autonomous equation for u
n,m n,m
aun,m+bun+1,m+cun,m+1+dun+1,m+1+ee=0 (2)
with a, b, c, d and e beinge(n,m)–iendependentearbitraryenon zero complex coefficients. The choice that
(2) be autonomous is a restriction but it is also a natural simplifying ansatz when one is dealing with
autonomous equations. Moreover,as (1, 2) are taken to be autonomous equations, i.e. they have no n,m
dependent coefficients, they are translationally invariant under shifts in n and m. So we can with no loss
of generality choose as reference point n = 0 and m = 0. This will also be assumed to be true for the
linearizing point tranformation.
By a linearizing point transformation we mean a transformation
u =f(u ) (3)
0,0 0,0
between(2)and(1)characterizedbyafunctieondependingjustfromthefunctionu0,0andonsomeconstant
parameters. ItwillbeaLiepointtransformationiff =f satisfiesallLiegroupaxiomsand,inparticular,
0,0
thecompositionlaw. Inthefollowingwewillrequiremuchless,i.e. wewillonlyassumethedifferentiability
of the function f up to at least second order.
In Section 2 we discuss point transformations, present the linearizability conditions which ensure that
the given equation is linearizable and the differential equations which define the transformation f. In
Section3weclassifyallmultilinearequationswhichbelongtotheclass(1)uptoaMo¨bioustransformation
while in the final Section we present some conclusive remarks and open problems.
2 Discrete equations defined on a square linearizable by a point
transformation.
.
In the autonomous case a generic partialdifference equation (1) for the complex function u =u can
n,m 0,0
be rewritten as
∂E
E(u ,u ,u ,u )=0, 6=0, i,j =0,1, (4)
0,0 1,0 0,1 1,1
∂u
i,j
We will assume that we can solve (4) with respect to each one of the four variables in its argument
u = F(u ,u ,u ), F 6=0 F 6=0, F 6=0, (5a)
1,1 0,0 1,0 0,1 ,u0,0 ,u1,0 ,u0,1
u = G(u ,u ,u ), G 6=0, G 6=0, G 6=0, (5b)
1,0 0,0 0,1 1,1 ,u0,0 ,u0,1 ,u1,1
u = S(u ,u ,u ), S 6=0, S 6=0 S 6=0, (5c)
0,1 0,0 1,0 1,1 ,u0,0 ,u1,0 ,u1,1
u = T (u ,u ,u ), T 6=0, T 6=0 T 6=0, (5d)
0,0 1,0 0,1 1,1 ,u1,0 ,u0,1 ,u1,1
2
and that (4) can be linearized by the linearizing autonomous point transformation (3) into the linear
.
equation (2) for the complex function u =u , i.e
n,m 0,0
au +bu +cu +du +e=0. (6)
0,0 1,0 0,1 1,1
Hence, assuming we can solve (4)ewith reespect teo u , wee can choose as independent variables u , u
1,1 0,0 1,0
and u and we will have that
0,1
af +bf +cf +df | +e=0, (7)
0,0 1,0 0,1 1,1 u1,1=F
must be identically satisfied for any u , u and u . Differentiating (7) with respect to u , u or
0,0 1,0 0,1 0,0 1,0
u , we obtain
0,1
df df
0,0 1,1
a +d | F =0, (8a)
du du u1,1=F ,u0,0
0,0 1,1
df df
1,0 1,1
b +d | F =0, (8b)
du du u1,1=F ,u1,0
1,0 1,1
df df
0,1 1,1
c +d | F =0, (8c)
du du u1,1=F ,u0,1
0,1 1,1
which have to be identically satisfied for any u , u and u . From them, considering that df(x) 6= 0,
0,0 1,0 0,1 dx
we derive that d 6= 0, otherwise a = b = c = e = 0. As a consequence, considering that also F 6= 0,
,u0,0
F 6= 0 and F 6= 0, we have a 6= 0, b 6= 0 and c 6= 0. Then in all generality we can divide (6) by d
,u1,0 ,u0,1 . . . .
and, introducing the new parameters α = a/d 6= 0, β = b/d 6= 0, γ = c/d 6= 0 and ǫ = e/a, (6) can be
rewritten as
αu +βu +γu +u +ǫ=0. (9)
0,0 1,0 0,1 1,1
Defining df(x) =. H(x), from (8) wee obtaine e e
dx
F αH(u )
,u0,0 = 0,0 , (10a)
F βH(u )
,u1,0 1,0
F αH(u )
,u0,0 = 0,0 . (10b)
F γH(u )
,u0,1 0,1
From (10) we get the following linearizability conditions
A(x,u )=. F,u0,0| = α, ∀x, u , (11a)
0,1 F u0,0=u1,0=x β 0,1
,u1,0
B(x,u )=. F,u0,0| = α, ∀x, u , (11b)
1,0 F u0,0=u0,1=x γ 1,0
,u0,1
C(x,u )=. F,u0,1| = γ, ∀x, u , (11c)
0,0 F u1,0=u0,1=x β 0,0
,u1,0
∂ F,u0,0 =0, ∀u , u , u , (11d)
0,0 1,0 0,1
∂u F
0,1 ,u1,0
∂ F
,u0,0 =0, ∀u , u , u , (11e)
0,0 1,0 0,1
∂u F
1,0 ,u0,1
∂ F
,u0,1 =0, ∀u , u , u . (11f)
0,0 1,0 0,1
∂u F
0,0 ,u1,0
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Alternatively the conditions (11a-11c) can be substituted by the following ones
d
A(x,u )=0, ∀x, u , (12a)
0,1 0,1
dx
d
B(x,u )=0, ∀x, u , (12b)
1,0 1,0
dx
d
C(x,u )=0, ∀x, u . (12c)
0,0 0,0
dx
Taking the (principal value of the) logarithm of (8a), we have
df df F
log 0,0 −log 1,1| =log − ,u0,0 (mod2πi). (13)
du du u1,1=F (cid:18) α (cid:19)
0,0 1,1
Then, let us introduce the linear operator B
B =. ∂ − F,u0,0 ∂ , (14)
∂u F ∂u
0,0 ,u1,0 1,0
such that and Bφ(F (u ,u ,u )) = 0, where φ is an arbitrary functions of its argument. When we
0,0 1,0 0,1
apply (14) to (13), we obtain an ordinary differential equation describing the evolution of the linearizing
transformation
d df 1
0,0
log = W F ;F , (15)
du du F F (u0,0) ,u1,0 ,u0,0
0,0 0,0 ,u0,0 ,u1,0 (cid:2) (cid:3)
.
where W [f;g] =fg −gf stands for the Wronskian of the functions f and g. Let’s remark that the
(x) ,x ,x
linearizabilityconditions(11d-11f)implythattherighthandmemberoftheequation(15)doesnotdepend
onu andu . Theseconditions wereconsideredin[3]andhadnotbeen sufficientto classify (1). Other
1,0 0,1
similarconditionscanbeobtainedstartingfrom(8b)or(8c). Thelinearizabilityconditionspresentedhere
have been obtained starting from (5a). Similar results could be obtained starting from (5b), (5c) or (5d).
However these results would not have provided any really new linearizability condition. So, here, in the
next Section we start the classifying process from the more basic equations (11) as we did in the case of
equations depending on just three points [4].
3 Classification of complex autonomous multilinear partial dif-
ference equations defined on four points linearizable by a point
transformation.
Let E(u ,u ,u ,u )=0 be the complex multilinear equation
0,0 1,0 0,1 1,1
a u + a u +a u +a u +a u u +a u u + (16)
1 0,0 2 1,0 3 0,1 4 1,1 5 0,0 1,0 6 0,0 0,1
+ a u u +a u u +a u u +a u u +
7 0,0 1,1 8 1,0 0,1 9 1,0 1,1 10 0,1 1,1
+ a u u u +a u u u +a u u u +
11 0,0 1,0 0,1 12 0,0 1,0 1,1 13 0,0 0,1 1,1
+ a u u u +a u u u u +a =0,
14 1,0 0,1 1,1 15 0,0 1,0 0,1 1,1 16
wherea ,j =1,...,16,arearbitrarycomplexfreeparameters. ThisequationisinvariantunderaMo¨bious
j
transformation of the dependent variable
. b1v0,0+b2
u = , (17)
0,0
b v +b
3 0,0 4
where b , k =1,..., 4 are four arbitrary complex parameters such that b b −b b 6=0. As we operate in
k 1 4 2 3
thefieldofcomplexnumbersandweclassifyuptoMo¨bioustransformations,usinginversions,dilationsand
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translations we can always simplify (16) by setting either 1) a = a = 0 or 2) 4 a = 10 a =
15 16 k=1 k k=5 k
14 a = a = 0, a = 1. Let’s now apply to these two cases the six nPecessary linPearizability
k=11 k 15 16
cPonditions (11). This amounts to solving a systemof96 algebraicin generalnonlinear equationsinvolving
thecoefficientsa ,j =1,...,14. Theirsolutionimplies,throughtheintegrationofthedifferentialequation
j
(15), that the function f(x) appearing in the linearizing transformation (2) can only be of the following
two types:
1. The fractional linear function
c x+c
1 2
f(x)= , c c −c c 6=0, (18)
1 4 2 3
c x+c
3 4
which, as the classification is up to Mo¨bious transformations of u , can always be reduced to be
0,0
f(x)=1/x;
2. The (principal branch) of the logarithmic function
d x+d
2 3
f(x)=d log +d d 6=0, d d −d d 6=0, (19)
1 6 1 2 5 3 4
(cid:18)d x+d (cid:19)
4 5
which,astheclassificationisuptoMo¨bioustransformations,canalwaysbereducedtof(x)=log(x).
Moreoverin this case the ratios α/β and α/γ are alwaysreal and of modulus 1, so that the possible
linear equations can be only of the following four types:
α(u +u +u )+u +ǫ=0; (20a)
0,0 1,0 0,1 1,1
α(u +u −u )+u +ǫ=0; (20b)
e0,0 e1,0 e0,1 e1,1
α(u −u +u )+u +ǫ=0; (20c)
e0,0 e1,0 e0,1 e1,1
α(u −u −u )+u +ǫ=0. (20d)
e0,0 e1,0 e0,1 e1,1
.
It is easy to prove that, if the traensformeationue =loeg(u ) has to produce a multilinear equation
0,0 0,0
for u , we must have α = ±1. In fact, as F (u ,u ,u ) should be a fractional linear function
0,0 0,0 1,0 0,1
of u with coefficients depending on u andeu , we have that the relations
0,0 1,0 0,1
uα (e u +e )=e u +e , e =e (u ,u ), j =1,...,4,
0,0 1 0,0 2 3 0,0 4 j j 1,0 0,1
wheree e −e e isnotidenticallyzeroforallu andu ,mustbe identicallysatisfiedforallu .
1 4 2 3 1,0 0,1 0,0
Differentiating (21) twice with respect to u , we get e (α+1)u +e (α−1)= 0 identically for
0,0 1 0,0 2
all u , so that
0,0
e (α+1)=0, e (α−1)=0.
1 2
Consideringthate ande cannotbe simultaneouslyidentically zero,itfollowsthatα=±1. Inthis
1 2
wayweobtainasetofeightlinearequationscorrespondingtoeightlinearizablenonlinearequations.
Hence we can summarize the results obtained in the following Theorem:
Theorem 1 Apart from theclass of equations linearizable by aM¨obious transformation, which can berep-
resented up to a M¨obious transformation of the dependent variable (eventually composed with an exchange
of the independent variables n↔m) by the equation
w w (w +βw )+w w (γw +δw )+ǫw w w w =0, ǫ=0,1, β,γ,δ 6=0, (21)
0,1 1,1 1,0 0,0 0,0 1,0 1,1 0,1 0,0 1,0 0,1 1,1
linearizable by the inversion u =1/w to the equation
0,0 0,0
e u +βu +γu +δu +ǫ=0, (22)
0,0 1,0 0,1 1,1
e e e e
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the only other linearizable equations are up to a M¨obious transformation of the dependent variable (even-
tually composed with an exchange of the independent variables n ↔ m), represented by the following six
nonlinear equations
w w w w −1 =0, (23a)
0,0 1,0 0,1 1,1
w −w w w =0, (23b)
0,0 1,0 0,1 1,1
w −w w w =0, (23c)
1,0 0,0 0,1 1,1
w −w w w =0, (23d)
1,1 0,0 1,0 0,1
w w −θw w =0, (23e)
0,1 1,1 0,0 1,0
w w −θw w =0, (23f)
0,0 1,1 1,0 0,1
where θ 6= 0 is an otherwise arbitrary complex parameter. They are linearizable by the transformation
u =logw to the equations
0,0 0,0
e u0,0+u1,0+u0,1+u1,1 =2πiz, (23g)
−u +u +u +u =2πiz, (23h)
e 0,0 e 1,0 e 0,1 e 1,1
u −u +u +u =2πiz, (23i)
e0,0 e1,0 e0,1 e1,1
−u −u −u +u =2πiz, (23j)
e 0,0 e 1,0 e 0,1 e 1,1
−u −u +u +u =logθ+2πiz, (23k)
e0,0 e1,0 e0,1 e1,1
u −u −u +u =logθ+2πiz, (23l)
e0,0 e1,0 e0,1 e1,1
where log always stands for thepriencipalebrancheof theecomplex logarithmic function andwhere logθ stands
for the principal branch of the complex logarithmic function of the parameter θ.
Let’s note that, given a solution w of eqs. (23a-23f), the choice of the principal branch of the complex
0,0
logarithm function in the transformation u =logw reflects in the inhomogeneous terms of the linear
0,0 0,0
eqs. (23g-23l) and is always |z| ≤ 2. This can be easily seen considering that we must have 2π|z| ≤
|Im(u )|+|Im(u )|+|Im(u )|+e|Im(u )| ≤ 4π (with a slight difference in (23k) and (23l)).
0,0 1,0 0,1 1,1
Through the translation u = v +πiz/2 the linear equation (23g) can be made homogeneous. As a
0,0 0,0
consequence in (23a) w0,0 =ev0,0eπiz/2, so that the integer parameter z can take the values
z =−1, 0, 1, 2. Hence weeconclude that the term 2πiz in (23g) takes into account the discrete symmetry
of (23a) given by w → ζw , where ζ stands for one of the four roots of unity ζ = 1, −1, i and −i.
0,0 0,0
The same happens when applying the translation u = v + πiz to the linear equations (23h, 23i)
0,0 0,0
and u0,0 = v0,0 −πiz to (23j). In (23b, 23c, 23d) we have w0,0 = ev0,0e±πiz so that, without any loss
of generality, we can restrict the values of z to z =e 0, 1. The term 2πiz in (23h, 23i, 23j) takes into
accouentthe discretesymmetryof(23b, 23c, 23d)givenbyw →ζw ,whereζ stands foroneofthe two
0,0 0,0
roots of unity ζ =1 and −1. The same happens for the linear equations (23k) when we consider the non
autonomous translation u0,0 = v0,0+(logθ+2πiz)m/2. In (23e) w0,0 = ev0,0eπimzθm/2 so that, without
any loss of generality, we can also restrict the values z to z =0, 1. Hence we conclude that the term 2πiz
in(23k)takesintoaccounetthediscretesymmetryof(23e)givenbyw →(−1)mw . Thesamehappens
0,0 0,0
for the linear equation (23l) with the non autonomous translation u = v +(logθ+2πiz)nm. Now,
0,0 0,0
as e2πinmz = 1 for any z, in (23f) w0,0 = ev0,0θnm so that, without any loss of generality, we can choose
z =0. e
4 Concluding remarks and outlook
In this paper we have classified all multilinear partial difference equations which can be linearized by a
point transformation. The resulting linearizableequations arepresentedin Theorem1 together with their
linearized counterparts.
It seems interesting at this point to try to analyze the case of more complicate classes of equations
involving more lattice points and which could provide in the continuous case parabolic or elliptic partial
differential equations. Work on this is in progress.
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Acknowledgements
DL and CS have been partly supported by the Italian Ministry of Education and Research, 2010 PRIN
“Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps”.
References
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[2] D.LeviaandC.Scimiterna,LinearizabilityofNonlinearEquationsonaQuad-GraphbyaPoint,Two
PointsandGeneralizedHopf-ColeTransformations,SIGMA7(2011),079,24pages,arXiv:1108.3648,
http://dx.doi.org/10.3842/SIGMA.2011.079.
[3] D. Levi and C. Scimiterna, Linearization through symmetries for discrete equations, submitted to
SIGMA 2012.
[4] C.ScimiternaandD.Levi,Threepointspartialdifferenceequationslinearizablebylocalandnonlocal
transformations,submitted to J. Phys. A, 2012.
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