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The discretized Boussinesq equation and its conditional symmetry
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The discretized Boussinesq equation and its

conditional symmetry reduction

Decio Levi∗, Miguel A. Rodŕıguez†, Zora Thomova+

∗ INFN, Sezione Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy.
† Departamento de F́ısica Teórica, Universidad Complutense, 28040 Madrid, Spain
+ SUNY Polytechnic Institute, 100 Seymour Road, Utica, NY 13502, USA

E-mail: levi@roma.infn.it, rodrigue@ucm.es, zora.thomova@sunypoly.edu

Abstract. In this article we show that we can carry out the symmetry preserving

discretization of the Boussinesq equation with respect to three of its more significant

conditional symmetries. We perform the symmetry reduction of the obtained nonlinear

discrete schemes with respect to the conditional symmetries and obtain the reduced

discrete equations which unlike in the continuous case, are not always reducible to

second order difference equations. A numerical comparison with the exact continuous

solution given by Weierstrass elliptic functions is carried out.

1. Introduction

Nonlinear Partial Differential Equations (PDEs), which are found everywhere in modern

physics [47], are often difficult to solve. One of the approaches to find analytical solutions

is the symmetry reduction method based on the work by Sophus Lie and described in

many books, see for example [6,42]. Not all PDEs have Lie point symmetries, and many

times the symmetries do not provide the type of solutions we are looking for. So one

looks for extensions or modifications of the construction which could overcome some of

these problems. A technique not satisfying all the properties of a Lie group but just

providing solutions by symmetry reduction was introduced by Bluman and Cole with the

name non-classical method [5]. It consists of adding a well defined “condition” to the set

of determining equation. Conditional symmetries transform a subset of solutions into

solutions. They do not form a group but they can be used to obtain additional reductions

leading to explicit solutions. Conditional symmetries and the reductions provided

by them have been applied to obtain explicit (new, physically relevant) solutions for

problems arising in non-newtonian fluid flow models [1], in magnetohydrodynamics [20],

in reaction-diffusion-convection equations [12], in equations of heat and acoustics [18],

etc. .

This research is part of a general program the aim of which is to make full use

of the theory of Lie group to study the solution space of discrete equations and in

particular to solve difference equations [21,24,25,27,29,30,34–36]. One of the questions

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The discretized Boussinesq equation and its conditional symmetry reduction 2

is how to do the symmetry preserving discretization, i.e. how to construct discretized

equations which possess the same symmetries or a subgroup of the symmetry group

of a given PDE [2–4, 7, 10, 11, 15–17, 24–26, 43–45, 49]. The construction of the Partial

Difference Equation (P∆E) is obtained through the computation of discrete invariants

of the symmetry group of the PDE or one of its subgroups looking for a function of

these invariants which yields the PDE in the continuous limit.

In this article we will be interested in showing that we can discretize the Boussinesq

equation preserving the conditional symmetries.

In Section 2 we recall the construction of the Boussinesq equation starting from

the conditional symmetries and its symmetry reductions while in Section 3 we provide

the conditionally invariant discretizations of the Boussinesq equation together with

its invariant lattices, i.e. the discrete Boussinesq scheme. Section 4 is devoted to

carrying out the reduction of the so obtained discretized Boussinesq schemes to ordinary

difference schemes as well as a numerical comparison with the exact result of the

continuous case. In Section 5 we present a summary of the results, some concluding

remarks and prospects for future works.

2. Construction of nonlinear PDEs with conditional symmetries: the

conditional symmetries of the Boussinesq equation

Consider a PDE E = 0 in Rn that is invariant under a continuous group of Lie point

transformations. The corresponding infinitesimal symmetry generator is given as

X̂ = ξ(x, u) · ∇+ φ(x, u)∂u, (1)

To construct conditional symmetries using the non-classical method one adds an

auxiliary first-order equation to E = 0, build up in terms of the coefficients of the

infinitesimal generator X̂, namely

C = C(x, u,∇u) = ξ(x, u) · ∇u− φ(x, u) = 0, (2)

the infinitesimal symmetry generator (1) written in characteristic form [42] set equal to

zero. Equation (2) will be determined together with the vector field X̂, as it involves

the same functions ξ and φ. In fact we look for the simultaneous symmetry group of

the overdetermined system of equations E = 0 and C = 0. It is easy to prove that (2) is

weakly invariant under the first prolongation of (1)

prX̂C = −(ξu · ∇u− φu) C, (3)

without imposing any conditions on the functions ξ and φ. Consequently, we need just

to apply the following invariance condition

prX̂E
∣∣∣
(E=0, C=0)

= 0. (4)

where the expression C = 0 represents the condition and all of its differential

consequences up to the order of the equation, see [12, 18] and the references therein

for more results. Equation (4) gives nonlinear determining equations for ξ and φ which

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The discretized Boussinesq equation and its conditional symmetry reduction 3

provide at the same time the classical and non–classical symmetries. As C = 0 appears

in (4) as a condition imposed on the determining equations one has called the resulting

symmetries conditional symmetries [6, 18,19,33,41].

It has been shown in [31, 32] that a PDE E = 0 can be written in terms of the

invariants Ij of its conditional symmetries X̂ as

E({Ij})
∣∣∣
C=0

= 0, j = 0, 1, . . . . (5)

Let us recall here the conditional symmetries of the Boussinesq equation and

how one re-constructs the Boussinesq equations from the invariants of the conditional

symmetries.

The Boussinesq equation

uyy + uuxx + u2x + uxxxx = 0, (6)

was introduced in 1871 by Boussinesq to describe the propagation of long waves

in shallow water [8, 9] and it is of considerable physical and mathematical interest

[46,48,50,51].

The Lie point symmetries are given by [40]

D = x∂x + 2y∂y − 2u∂u, P1 = ∂x, P2 = ∂y. (7)

The conditional symmetries are divided in two classes. If η 6= 0 in the vector field

X̂ = ξ(x, y, u)∂x + η(x, y, u)∂y + φ(x, y, u)∂u (8)

we can always put η = 1. If η = 0 and ξ 6= 0, we can put ξ = 1. The conditional

symmetries of the Boussinesq equation for η 6= 0 were obtained by Levi and Winternitz

in [33], and by Clarkson and Kruskal in [14] by non group techniques. The singular case

η = 0, vector fields X̂6 and X̂7, can be found in a review by Clarkson [13]:

X̂1 = ∂y + y∂x − 2y∂u, (9)

X̂2 = ∂y −
x

y
∂x +

(
2

y
u+

6

y3
x2
)
∂u, (10)

X̂3 = ∂y +

(
−x
y

+ y4
)
∂x +

(
2

y
u+

6

y3
x2 − 2y2x− 4y7

)
∂u, (11)

X̂4 = ∂y +

(
x

2y
+ y

)
∂x −

1

y
(u+ 2x+ 4y2)∂u, (12)

X̂5 = ∂y +
℘̇

2℘
(x+ βW )∂x −

[
℘̇

℘
u+ 3x2℘̇+

β

2

(
1

℘
+ 12℘̇W

)
+

β2

2
W

(
1

℘
+ 6℘̇W

)]
∂u, W (y) =

∫ y

0

℘(s)

℘̇(s)2
ds, (13)

X̂6 = ∂x +

(
2

x
u+

48

x3

)
∂u, (14)

X̂7 = ∂x +

(
−2xQ+ c1Q+ c2Q

∫ y

0

ds

Q(s)2

)
∂u, (15)

where Q = ℘(y + c3; 0, g3) and ℘ is a Weierstrass elliptic function.

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The discretized Boussinesq equation and its conditional symmetry reduction 4

From the way they are constructed it follows that the generators X̂1, · · · , X̂7 are

defined in (9)–(15) up to an arbitrary multiplicative function.

In the following we present an overview of the results obtained earlier [31, 32],

namely the construction of the Boussinesq equation from the invariants of three chosen

conditional symmetries and the reduction with respect to these conditional symmetries.

These are the cases we shall discretize in Section 3.

2.1. Conditional invariant Boussinesq equation associated to X̂1

The invariants of X̂1 = ∂y + y∂x − 2y∂u which will be of interest here are:

I0 = x− 1

2
y2, I1 = 2x+ u, I2 = ux, I3 = 2y + yux + uy,

I4 = uxx, I5 = uyy + 2yuxy + y2uxx, I6 = uxxxx.
(16)

and the condition is given by I3 = 0 i.e. C = 2y + yux + uy = 0. Then the Boussinesq

equation, written in terms of the invariants is:

(I0 + I1)I4 + I22 + I5 + I6

∣∣∣
Cx=0

= 0. (17)

As Cx = uxy + yuxx, we have:

uyy + 2y(uxy + yuxx) + uuxx + uxxxx + u2x

∣∣∣
Cx=0

= uyy + uuxx + uxxxx + u2x = 0. (18)

Symmetry variables associated to X̂1 are:

z = x− 1

2
y2, u(x, y) = w(z)− y2. (19)

and the equation in these new variables is (w′ = dw/dz):

wiv + ww′′ + w′2 − w′ − 2 = 0. (20)

After a quadrature (20) becomes:

w′′′ + ww′ − w − 2z + A = 0, (21)

where A is an integration constant. Equation (21) can be integrated to a second order

differential equation and solved in terms of PII transcendents (see [14]).

2.2. Conditional invariant Boussinesq equation associated to X̂2

The invariants of interest in the case of X̂2 = ∂y − x
y
∂x +

(
2
y
u+ 6x2

y3

)
∂u are

I0 = xy, I1 = x2u+
x4

y2
, I2 = x3ux +

2x4

y2
,

I3 = x2
(
uy −

x

y
ux −

2u

y
− 6x2

y3

)
, I4 = x4uxx +

2x4

y2
, I5 = x6uxxxx.

(22)

The condition is given by I3 = 0, or by any other expression C = f(x, y)I3 = 0. We

choose

C = yuy − xux − 2u− 6x2/y2 = 0. (23)

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The discretized Boussinesq equation and its conditional symmetry reduction 5

Then the Boussinesq equation written in terms of the invariants is:

I1I4 + I5 + I22 +Q
∣∣∣
{C=0}

= x6
(
uyy + uuxx + uxxxx + u2x

)
, (24)

where by {C = 0} we mean the substitution of C = 0 and all its differential

consequences. In this case the function of C to be added is Q = x6(xCx + y Cy +C)/y2.

The symmetry variables associated to X̂2 are

z = xy, u(x, y) = y2w(z)− x2

y2
. (25)

The Boussinesq equation in the symmetry variables (25) is:

wiv + ww′′ + w′2 = 0 (26)

and after two integration we get:

w′′ +
1

2
w2 = c1z + c0, (27)

whose solution is:

(i) If c1 = 0, an elliptic function

(ii) If c1 6= 0, a Painlevé I function

2.3. Conditionally invariant Boussinesq equation associated to X̂6

To construct the Boussinesq equation we need the following invariants of X̂6 =

∂x +
(

2
x
u+ 48

x3

)
∂u:

I0 = y, I1 =
1

x4

(
12 + x2u

)
, I2 =

1

x5

(
x3ux − 2x2u− 48

)
, I3 =

1

x2
uyy. (28)

We will use the characteristic of X̂6, proportional to I2, in the construction of the

condition for the Boussinesq equation:

C = x2I2 = ux −
2u

x
− 48

x3
= 0. (29)

Then the Boussinesq equation, written in terms of the invariants is:

6I21 + I3 +Q
∣∣∣
{C=0}

=
1

x2
(
uyy + uuxx + uxxxx + u2x

)
, (30)

where the function Q, written in terms of the condition C is:

Q =
1

x2
Cxxx +

2

x3
Cxx +

1

x2

(
u− 2

x2

)
Cx +

6

x3

(
u+

16

x2

)
C +

1

x2
C2. (31)

The symmetry variables associated to X̂6 are:

z = y, u(x, y) = x2w(z)− 12

x2
. (32)

The Boussinesq equation in the symmetry variables (32) is:

w′′ + 6w2 = 0, (33)

with a solution given by w = −℘(z + c1; 0, c2).

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The discretized Boussinesq equation and its conditional symmetry reduction 6

3. The conditionally discretized Boussinesq equations

The aim of this section is to discretize the Boussinesq equation preserving given

conditional symmetries. We include the necessary detailed description and formulas

of the discretization formalism used here in the Appendix A, for the convenience of the

reader.

Symmetry preserving discretization of the partial differential equation requires that

the P∆E in the continuous limit goes to the original PDE, and that it is invariant

under the given symmetry generator prolonged appropriately. Additionally, one must

introduce a symmetry invariant lattice, since the fixed nontransformable lattice allows

only dilation symmetries of the dependent variable.

Thus we proceed in the following way:

(i) Construct a discrete lattice that is invariant under the given conditional symmetry.

(ii) Discretize the PDE over the obtained lattice, using the same invariants, mutatis

mutandis, of the conditional symmetry as introduced in the continuous case (see

Section 2).

(iii) Go over to the symmetry variables as in most of the cases the resulting equations

and lattices are simpler.

We will present only the cases X̂1 and X̂2 (where η = 1) and the case X̂6 (where

η = 0) to show that conditional discretization can be carried out. The discussion of the

other cases follows the same lines.

3.1. X̂1 preserving discretization of the Boussinesq equation

When applied to dependent and independent variables living on a lattice, the X̂1 reads:

X̂1
nm = ynm∂xnm + ∂ynm − 2ynm∂unm (34)

thus ξnm = ynm, ηnm = 1 and φnm = −2ynm. To define the lattice we construct the

prolongation of X̂1
nm on points (n,m), (n+ 1,m), (n,m+ 1) and (n+ 1,m+ 1) as given

in (A.14):

ζ
(x)
n+i,m+j = yn+i+1,m+j − yn+i,m+j = σ

(y)
n+i,m+j,

ζ
(y)
n+i,m+j = 0,

χ
(x)
n+i,m+j = yn+i,m+j+1 − yn+i,m+j = h

(y)
n+i,m+j,

χ
(y)
n+i,m+j = 0, (i, j)→ (0, 1).

(35)

The lattice invariants are obtained solving the characteristic equations

dxnm
ynm

=
dynm

1
=
dhxn+i,m+j

σyn+i,m+j

=
dσxn+i,m+j

hyn+i,m+j

. (36)

We get that hynm and σynm are invariants ∀ m,n. We simplify the lattice further by

choosing

hynm = k, σynm = σ, ∀ n,m, (37)

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The discretized Boussinesq equation and its conditional symmetry reduction 7

where k and σ are constants. Furthermore, from (36) we get

xnm −
1

2
y2nm = J 0

nm, (38)

hxn+i,m+j − ynmσ = J 1
ij, (39)

σxn+i,m+j − ynmk = J 2
ij, (40)

where J 0
nm, J 1

ij, J 2
ij are invariants obtained from (36) ∀ m,n, i, j. Choosing σ = 0 and

J 2
ij = 0 we obtain hxn+i,m+j = h with h constant and σxn+i,m+j = kynm ∀ i, j. Thus the

invariant lattice is given by:

hyn+i,m+j = k, σyn+i,m+j = 0, hxn+i,m+j = h, σxn+i,m+j = kynm (41)

where h and k are the constant invariant spacings in the two independent directions

(see Figures A1 and A2 in the Appendix). In the following we will use the notation

(if there is no risk of confusion), xnm = x, ynm = y and unm = u. The lattice is non

orthogonal since σx depends on y but with constant spacing and Schwarzian i.e. such

that the second order partial differences commute.

On this lattice the two discrete derivatives (A.7) are:

Dx =
1

h
∆n, Dy =

1

k
∆m − yDx. (42)

The relevant discrete invariants are:

I(0) = J 0 = x− 1

2
y2, I(1) = 2x+ u, I(2) = Dxu, I(3) = 2y + yDxu+Dyu,

I(4) = D2
xu, I(5) = D2

yu+ y2D2
xu+ 2yDxDyu, I(6) = D4

xu.
(43)

They can also be obtained by taking into account that I(i) = I(i)nm, ∀n,m. For example:

I(1)nm = 2xnm + unm. (44)

Then from the invariants

I(1)nm = 2xnm + unm, I(1)n+1,m = 2xn+1,m + un+1,m (45)

by substraction and division by the invariant J 1
ij = h we easily get using (41) and (42)

that I(2) given by (43) is an invariant. A similar construction will provide all other

higher invariants.

The condition, like in the continuous case, is just C = I(3) = 0 and by discrete

differentiation we get

DxC = DxDyu+ yD2
xu = 0. (46)

The conditional symmetry X̂1 preserving discretization of the Boussinesq equation is

then given by:

(I(1) + I(0))I(4) +
(
I(2)
)2

+ I(5) + I(6)
∣∣∣
DxC=0

= (y2 + u)D2
xu+D2

yu+ y2D2
xu+ 2yDxDyu+D4

xu+ (Dxu)2
∣∣∣
DxDyu+yD2

xu=0

= D2
yu+ uD2

xu+D4
xu+ (Dxu)2 = 0, (47)

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The discretized Boussinesq equation and its conditional symmetry reduction 8

which, as a difference equation, reads

1

k2
(un,m+2 − 2un,m+1 + unm)− 2y

hk
(un+1,m+1 − un+1,m − un,m+1 + unm)

+
y2

h2
(un+2,m − 2un+1,m + unm) +

unm
h2

(un+2,m − 2un+1,m + unm)

+
1

h4
(un+4,m − 4un+3,m + 6un+2,m − 4un+1,m + unm)

+
1

h2
(un+1,m − unm)2 − 1

h
(un+1,m+1 − un,m+1) = 0. (48)

From the invariants I(0) and I(1) of X̂1
nm we can introduce a new dependent variable

wnm(xnm, ynm) = w given by

w = u+ y2. (49)

From (47) the equation satisfied by w is:

D4
xw +D2

yw + (Dxw)2 + (w − y2)D2
xw − 2 = 0 (50)

or, explicitly,

1

h4
(wn+4,m − 4wn+3,m + 6wn+2,m − 4wn+1,m + wnm)

+
1

h2
(w2

n+1,m + wnm(wn+2,m − 4wn+1,m + 2wnm))

− 2y

hk
(wn+1,m+1 − wn+1,m − wn,m+1 + w)− 1

h
(wn+1,m+1 − wn,m+1)

+
1

k2
(wn,m+2 − 2wn,m+1 + wnm)− 2 = 0. (51)

By going from the independent variable x to the new independent invariant variable

znm = z given by

z = x− 1

2
y2, (52)

we get that the following relations turn out to be true:

zn+1,m − znm =hznm = xn+1,m − xnm −
1

2
(y2n+1,m − y2nm) = h,

zn,m+1 − znm =σznm = xn,m+1 − xnm −
1

2
(y2n,m+1 − y2nm) = −1

2
k2.

(53)

The fact that in the symmetry variables the lattice spacing is constant, even if non

orthogonal, is not an apriori obvious result. However it derives from the fact that in the

symmetry variables the infinitesimal generator reduces to X̂1 = ∂ynm .

Then the differences of the function w(x, y) = v(z, t) (where tnm = ynm and

we changed the name of the dependent variable from w to v to stress the change of

variables), are:

Dzv =
vn+1,m − vnm

h
,

Dtv =k
vn+1,m − vnm

2h
+
vn,m+1 − vnm

k

(54)

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The discretized Boussinesq equation and its conditional symmetry reduction 9

so that

vn+1,m =vnm + hDzv,

vn,m+1 =vnm + kDtv −
k2

2
Dzv.

(55)

Then we can rewrite the differences with respect to x, y in terms of z, t as:

Dxw =
1

h
(vn+1,m − vnm) = Dzv,

Dyw =
1

k
(vn,m+1 − vnm)− y

h
(vn+1,m − vnm) = Dtv −

(
y +

k

2

)
Dzv.

(56)

The coefficients of the discrete derivatives provide the Jacobian of the transformation

(49, 52). Any other higher order difference can be equally computed. Those we will

need for our work are:

D2
xv = D2

zv, D4
xv = D4

zv,

D2
yv =

(
y2 + ky +

3k2

4

)
D2
zv − 2(y + k)DzDtv +D2

t v −Dzv.
(57)

Substituting (57) in (50) we get the Boussinesq equation in the invariant discrete variable

D4
zv +

(
v + kt+

3k2

4

)
D2
zv − 2(t+ k)DzDtv +D2

t v −Dzv + (Dzv)2 − 2 = 0 (58)

where v = vnm, t = tnm, z = znm.

This will be the starting point for the symmetry reduction which will be carried

out in the following Section.

3.2. X̂2 preserving discretization of the Boussinesq equation

In this case we have

X̂2
nm = ∂ynm −

x

y
∂xnm +

(
2

ynm
unm +

6

y3nm
x2nm

)
∂unm . (59)

Repeating the detailed calculation we made in the previous subsection we find the

invariant lattice given by:

hynm = k, σynm = 0, hxn+i,m+j =
h

ynm
, σxn+i,m+j = − kxnm

yn,m+1

∀ i, j,m, n (60)

where h and k are two constants related to the invariant spacing in the two independent

directions (see Figures A1 and A2 in the Appendix) and we will set as before, when

possible, xnm = x, ynm = y and unm = u. The lattice is non orthogonal, with non

constant spacing in the x direction and non Schwarzian.

On this lattice the two discrete derivatives (A.7) are:

Dx =
y

h
∆n, Dy =

1

k
∆m +

x

y + k
Dx. (61)

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The discretized Boussinesq equation and its conditional symmetry reduction 10

The invariants we need for our construction are:

I(0) = xy, I(1) = x2u+
x4

y2
, I(2) = x3Dxu+

2x4

y2
+
x3h

y3
,

I(3) =
x2y2

(y + k)2
Dyu−

x3y2

(y + k)3
Dxu−

x2(k + 2y)

(y + k)2
u

−x
4(k + 2y)(k2 + ky + y2)(k2 + 3ky + 3y2)

y2(y + k)6
,

I(4) = x4D2
xu+

2x4

y2
, I(5) = x6D4

xu.

(62)

It is easy to check that in the continuous limit (h, k)→ (0, 0)):

I(i) → Ii i = 1, . . . , 5 (63)

where Ii are the invariants of the continuous case (22). Then the expression I(1)I(4) +

I(5) +
(
I(2)
)2

has the appropriate continuous limit too when (h, k)→ (0, 0):

I(1)I(4) + I(5) +
(
I(2)
)2 → I1I4 + I5 + I22 (64)

Regarding the condition, we choose, (for the sake of simplicity)

C =
(y + k)2

x2 y
I(3)

= yDyu−
xyDxu
y + k

− 2y + k

y
u− x2 (k + 2 y) (k2 + ky + y2) (k2 + 3 ky + 3 y2)

y3 (y + k)4
. (65)

with C → C (23) when (h, k)→ (0, 0). We define:

Q =
x6

y2
(yDyC + xDxC + C) . (66)

The explicit expression of I(1)I(4) + I(5) + (I(2))2 +Q, after dividing by x6, is:

D4
xu+ (Dxu)2 + uD2

xu+D2
yu

+
k

y
D2
yu−

k2x

y(y + k)(y + 2k)
Dxyu

+
2h(y + k)2(y + 2k) + k2x (y2 − 3ky − 2k2)

y3(y + k)2(y + 2k)
Dxu−

k

y(y + k)
Dyu

+
kx2 (y2 + 4ky + 2k2)− hx (y2 + 3ky + k2)

y2(y + k)2(y + 2k)
D2
xu−

k2

y3(y + k)
u

+
1

y6(y + k)5(y + 2k)4

(
k10(−16hx− 48x2y) + k9(16h2 − 96hxy − 352x2y2)

+k8(112h2y − 264hxy2 − 1192x2y3) + k7(344h2y2 − 488hxy3 − 2488x2y4)

+k6(608h2y3 − 761hxy4 − 3571x2y5) + k5(681h2y4 − 1036hxy5 − 3664x2y6)

+k4(501h2y5 − 1087hxy6 − 2677x2y7) + k3(242h2y6 − 770hxy7 − 1312x2y8)

+k2(74h2y7 − 335hxy8 − 379x2y9) + k(13h2y8 − 80hxy9 − 48x2y10)

+h2y9 − 8hxy10
)

= 0, (67)

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The discretized Boussinesq equation and its conditional symmetry reduction 11

with lattice parameters (60).

Let us introduce the symmetry dependent variable of X̂2
nm wnm(x, y) ≡ w given by

w =
u

y2
+
x2

y4
. (68)

Although (67) written in terms of w is very complicated the equation greatly simplifies

when we go to the independent symmetry variables znm = z, tnm = t

z = x y, t = y. (69)

In this case we have the following relations:

zn+1,m − z =hznm = xn+1,m yn+1,m − x y = h,

zn,m+1 − z =σznm = xn,m+1 yn,m+1 − x y = 0,

tn+1,m − t = σtnm = yn+1,m − y = 0,

tn,m+1 − t = htnm = yn,m+1 − y = k,

(70)

which imply, as it should be, that in the new variables the lattice is orthogonal

and uniform in both directions. The variations of the dependent function w(x, y) =

vnm(z, t) = v are:

Dzv =
vn+1,m − vnm

h
,

Dtv =
vn,m+1 − vnm

k

, (71)

and then we obtain the following expressions for the shifted dependent variable:

vn+1,m =vnm + hDzv,
vn,m+1 =vnm + kDtv.

(72)

Then the differences with respect to x, y becomes:

Dxw =y
wn+1,m − wnm

h
= yDzv = tDzv,

Dyw =
wn,m+1 − wnm

k
+

x y

y + k

wn+1,m − wnm
h

=
x y

y + k
Dzv +Dtv =

tz

t+ k
Dzv +Dtv.

(73)

Any other higher order difference can be equally computed. Then (67) reduces to the

much simpler expression

t6D4
zv + t6(Dzv)2 + t6vD2

zv +
(t+ k)(t+ 2k)2

t
D2
t v +

(4t2 + 9kt+ 5k2)

t
Dtv

+
z (2t2 + 3kt+ k2)

t
Dztv = 0. (74)

3.3. X̂6 preserving discretization of the Boussinesq equation

In this case,

X̂6
nm = ∂xnm +

(
2

xnm
unm +

48

x3nm

)
∂unm , (75)

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The discretized Boussinesq equation and its conditional symmetry reduction 12

and repeating the calculations outlined earlier, the invariant lattice is given by:

hynm = k, σynm = 0, hxnm = h, σxnm = 0, (76)

where h and k are two constant invariant spacing in the two independent directions and,

as before, we set, whenever possible, xnm = x, ynm = y and unm = u. The lattice is

orthogonal with constant spacing and Schwarzian.

On this lattice the two discrete derivatives (A.7) are:

Dx =
1

h
∆n, Dy =

1

k
∆m. (77)

The relevant discrete invariants are:

I(0) = y, I(1) =
u

x2
+

12

x4
,

I(2) =
1

(x+ h)2
Dxu−

2x+ h

x2 (x+ h)2
u− 12 (h+ 2x) (h2 + 2xh+ 2x2)

x4 (x+ h)4
, I(3) =

1

x2
D2
yu.
(78)

The discretized Boussinesq equation is obtained considering the invariants:

6
(
I(1)
)2

+ I(3) =
6

x4
u2 +

144

x6
u+

864

x8
+

1

x2
D2
yu, (79)

when the condition is taken from I(2) to be

C = (x+ h)2I(2) = Dxu−
2x+ h

x2
u− 12 (h+ 2x) (h2 + 2xh+ 2x2)

x4 (x+ h)2
. (80)

We get the discretized Boussinesq equation as in the continuous case, by adding to

(79) the following combination of the condition C and its discrete derivatives.

Q =
1

x2
D3
xC +

2

x3
D2
xC +

1

x2

(
u− 2

x2

)
DxC +

6

x3

(
u+

16

x2

)
C +

1

x2
C2. (81)

The explicit equation is:

D4
xu+ (Dxu)2 + uD2

xu+D2
yu

+
(4h (−216h6 − 360h5x+ 264h4x2 + 849h3x3 + 633h2x4 + 194hx5 + 21x6)

x4(x+ h)2(x+ 2h)2(x+ 3h)2

−h (2h2 + 2hx+ x2)

x2(x+ h)2
u
)
Dxu−

h (72h3 + 210h2x+ 119hx2 + 18x3)

x2(x+ 2h)2(x+ 3h)2
D2
xu

+
h(5x+ 18h)

x(x+ 3h)2
D3
xu+

h (h3 − 2x2(x+ 2h))

x4(x+ h)2
u2

+
2hu

x6(x+ h)4(x+ 2h)2(x+ 3h)2

(
432h9 + 1152h8x− 2544h7x2 − 16236h6x3

−33421h5x4 − 37700h4x5 − 25384h3x6 − 10062h2x7 − 2139hx8 − 186x9
)

+
24h

x8(x+ h)4(x+ 2h)4(x+ 3h)4(x+ 4h)2

(
124416h15 + 476928h14x− 946080h13x2

−11056320h12x3 − 38053920h11x4 − 78993456h10x5 − 113245354h9x6

−117950624h8x7 − 91213065h7x8 − 52724836h6x9 − 22680916h5x10

−7147120h4x11 − 1599493h3x12 − 240292h2x13 − 21692hx14 − 888x15
)

= 0. (82)

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The discretized Boussinesq equation and its conditional symmetry reduction 13

We introduce the invariant dependent variable

w(xnm, ynm) ≡ w =
u

x2
+

12

x4
, (83)

and independent variables znm = ynm = z, tnm = xnm = t. Rewriting the equation

(82) in terms of w(x, y) = vnm(z, t) and their differences we obtain the much simpler

equation:

(t+ 4h)2D4
t v +

(
18h2

t
+ 8t+ 33h

)
D3
t v +

(
t2(t+ 2h)2v − 4h(9t+ 14h)

t2

)
D2
t v

+(t+ h)4(Dtv)2 +

(
t
(
15ht+ 8t2 + 6h2

)
v +

6h(t+ 4h)

t3

)
Dtv + t2D2

zv + 6t2v2 = 0. (84)

4. Symmetry reduction of the conditional discretization of the Boussinessq

equation

To be able to carry out the symmetry reduction we start from the conditionally

discretized Boussinesq equations (47, 74, 84) rewritten in terms of the corresponding

symmetry variables (z, t, v).

4.1. Case X̂1.

Starting from (58) we can easily see that we can make the restriction

vnm = vn = v. (85)

Then,

Dzv =
vn+1 − vn

h
,

Dtv =k
vn+1 − vn

2h
=
k

2
Dzv.

(86)

The other differences are:

D2
t v =

k2

4
D2
zv, Dztv =

k

2
D2
zv, (87)

Equation (58) becomes:

D4
zv +

(
v + kt+

3k2

4

)
D2
zv − 2

k

2
(t+ k)D2

zv +
k2

4
D2
zv −Dzv + (Dzv)2 − 2 = 0, (88)

which, when k is set equal to zero, becomes the difference equation

D4
zv + vD2

zv −Dzv + (Dzv)2 − 2 = 0. (89)

Equation (89) is the reduction of the discretized Boussinesq equation corresponding to

the conditional symmetry X̂1
nm.

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The discretized Boussinesq equation and its conditional symmetry reduction 14

4.2. Case X̂2.

In (74) we can make the same ansatz as in the X̂1 case, vnm = vn. Then,

Dtv = 0, D2
t v = 0, Dztv = 0, (90)

where znm = zn = z, tnm = tm = t. Equation (74) becomes the difference equation:

(Dzv)2 + vD2
zv +D4

zv = 0. (91)

In the continuous limit we were able to integrate it twice. Using results contained

in [38, 39] on the variational derivative of a discrete expression like (91) we can show

that (91) can not be integrated. This equation, goes in the continuous limit to PI or

to a differential equation whose solution is given by an elliptic function. A discrete PI
equation [28] is given by

vn(vn+1 + vn + vn−1 − 6) + nα + 3 = 0, (92)

and it is easy to see that (92) is not a reduction of (91).

4.3. Case X̂6.

Assuming in (84), vnm = vm = v, we get

Dtv = 0, D2
t v = 0, D3

t v = 0, D4
t v = 0. (93)

Equation (84) becomes the discrete difference equation:

D2
zv + 6v2 = 0, (94)

which has the correct continuous limit and whose solution will go in the continuous limit

to an elliptic function.

4.4. Numerical computation

In the X̂1 case we have not at our disposal an exact solution to compare with a numerical

approach. So we will discuss the numerical aspects for cases X̂2 and X̂6 where the

exact solutions correspond to ℘-Weierstrass functions. The resulting solution can be

transformed into solutions of the Boussinesq by writing u(x, y) in terms of v(z) according

to (25) and (32).

4.4.1. X̂2 conditional symmetry We will consider here the equation we got as reduction

of the Boussinesq equation under the conditional symmetry X̂2 (27.) In the simple case

of the Weierstrass equation with g2 = 0, g3 = 2 it is:

v = −12℘(z; 0, 2). (95)

We will consider the interval (1, 2.5) where the solution is regular (the closest

singularities are in z = 0 and z ≈ 2.72).

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The discretized Boussinesq equation and its conditional symmetry reduction 15

We can use the discrete scheme we found in Section 4.2, (91), with a Cartesian

lattice:

zn+1 = zn + h, D4
zv + vD2

zv + (Dzv)2 = 0, (96)

or, in an explicit form:

vn = 4vn−1 − 6vn−2 + 4vn−3 − vn−4 − h2(v2n−3 + 2v2n−4 − 4vn−4vn−3 + vn−4vn−2). (97)

Taking h = 0.0001, with the initial conditions given by the exact solution (that is,

value of the solution at z = 1, vn−3 = 1.0001, vn−2 = 1.0002, vn−1 = 1.0003) we get the

graphics plotted in Figure 1 ( dots) superimposed on the solid curve representing the

exact solution.

1.2 1.4 1.6 1.8 2.0 2.2 2.4-120

-100

-80

-60

-40

-20

Figure 1. The numeric solution of the discrete reduced Boussinesq equation with the

conditional symmetry X̂2 (dots over the exact solution, solid curve).

The χ estimator, defined as:

χ =

√∑N
i=1(Vi − vi)2∑N

i=1 V
2
i

, (98)

where Vi stands for the exact value at a point zi and vi for the approximated value at

the same point, is:

χ = 3.62737× 10−3 (99)

in the given interval (1, 2.5).

4.4.2. X̂6 conditional symmetry Using the conditional symmetry X̂6, the Boussinesq

equation is directly reduced to the Weierstrass differential equation for the function v(z)

(33), as in the previous example. We will consider the solution

v(z) = −℘(z; 0, 2) (100)

in the interval (0.50, 2.25) (again with singularities at z = 0 and z ≈ 2.72).

Although the continuous equation and corresponding solution are, apart from a

numerical factor, equal to those studied in the previous subsection, we use in this case

the discrete scheme presented in Section 4.3, with a cartesian lattice (94):

zm+1 = zm + k, vm = 2vm−1 − vm−2 − 6k2v2m−2. (101)

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1.0 1.5 2.0

-4

-3

-2

-1

Figure 2. Solutions of the Boussinesq equation corresponding to the reduction using

X̂6. The solid curve and surface are exact solutions of the reduced equation (33) and

the Boussinesq equation (6), respectively. The dots superimposed to the curve and to

the surface are computed using our invariant scheme.

Taking k = 10−5 and initial data from the exact solution above, we represent the

approximate result with the exact solution in Figure 2.

The value of the error estimator χ is in this case, in the interval (0.50, 2.25)

χ = 1.22× 10−12. (102)

As an example we plot in this case in Figure 2 also the solution of the Boussinesq

equation as obtained both from the continuous and the discretized reduction.

5. Conclusions

In this article we showed that we can carry out conditional symmetry preserving

discretization. We consider three conditional symmetries of the Boussinesq equation

which give rise to three different lattices, an orthogonal lattice, a lattice depending non

trivially on the independent variable and non orthogonal but Schwarzian and a non

orthogonal non Schwarzian lattice.

The discrete equations one obtain (47, 67, 82) are almost always very complicated

with many terms depending on the lattice spacing h and k of the two lattice directions

which in the continuous limit go to zero providing the continuous Boussinesq equation.

In all three cases we are able to rewrite the discrete equation in terms of the symmetry

variables of the conditional symmetries. The obtained discrete equations (58, 74, 84) are

greatly simplified, and the three lattices on the plane have constant angles and spacing.

The symmetry reduction can then be carried out and we get three equations which

are a simple discretization of the continuous case which however cannot be integrated

to second order difference equations as was the case in the corresponding continuous

reductions. On these reductions we carried out some numerical calculations to verify

the behaviour with respect to the exact solution, when existing. As the lattice spacing

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is always constant, away from the singularity the discrete scheme provides a good

approximation of the continuous solution.

An important point on which we are presently working is a more detailed numerical

study of the discretized equation we obtained (47, 67, 82). A second point is the

construction of a different discrete derivative as the approximation of the continuous

derivative used here is not symmetric with respect to the point (n,m), property which

seems to be essential if we expect the discrete equations to be integrable [37], as should

be the case of any discretization of the Boussinesq equation. Work on all these problems

is in progress.

Acknowledgments

DL has been supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics.

DL thanks the Departamento de F́ısica Teórica of Universidad Complutense de Madrid

(Spain) for its hospitality. MAR was supported by the Spanish MINECO under project

FIS2015-63966-P and thanks the INFN Sezione Roma Tre (Italy) for its hospitality. DL

and ZT also thanks the hospitality of CRM, Montreal (Canada).

Appendix A. Description of two dimentional lattices and P∆Es

Symmetry preserving discretization is the construction of discrete equations which in

the continuous limit go over to the given PDE and preserve its geometric structure.

As discrete equations defined on a fixed non transformable lattice have only dilation

symmetries as their Lie point symmetries, a discrete equation with more general Lie

symmetries will have a transformable lattice, will thus be described by what is called

a discrete scheme.

In the following we will provide the basis for the description of a difference scheme

with a given symmetry. For the sake of completeness we present here results which will

be necessary for the calculations performed in Section 3, contained in ref. [29, 30].

Appendix A.1. Geometry of 2-dimensional quadrilateral lattices

A sequence of points P (x, y) in R2 will be characterized by two indices laying on a

2-dimensional quadrilateral lattice.

Four points (xnm, ynm), (xn+1,m, yn+1,m), (xn,m+1, yn,m+1) and (xn+1,m+1, yn+1,m+1),

will define a quadrilateral in R2 (see figure A1). We can define alternative variables,

geometrically more relevant, given by the differences,

xn+1,m − xnm = hxnm, xn,m+1 − xnm = σxnm,

yn+1,m − ynm = σynm, yn,m+1 − ynm = hynm,
(A.1)

where hxnm, hynm, σxnm and σynm are defined graphically in Fig. A2.

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The discretized Boussinesq equation and its conditional symmetry reduction 18

n
n n+ 1

m

m

m+ 1

x

y

n
n+ 1

m m+ 1

Figure A1. A 2-dimensional lattice of points, on the left the computational and on

the right the physical coordinates.

hxn,m+1

hynm

hxnm

hyn+1,m

σxnm

σxn+1,mσynm

σyn,m+1

(xnm, ynm)

(xn,m+1, yn.m+1)

(xn+1,m+1, yn+1,m+1)

(xn+1,m, yn+1,m)

Figure A2. An elementary cell.

As the point (xn+1,m+1, yn+1,m+1) can be reached following two routes we have the

consistency conditions,

hxn,m+1 − hxnm = σxn+1,m − σxnm, hyn+1,m − hynm = σyn,m+1 − σynm. (A.2)

The vectors defining two adjoint sides of the quadrilateral starting at the point of

indices (n,m) must not be parallel,

hxnmh
y
nm − σxnmσynm 6= 0. (A.3)

Moreover the expression (A.3) must preserve the sign to avoid the folding of the lattice

itself.

The quadrilateral will be a parallelogram if the sides are parallel:

hxn,m+1

σyn,m+1

=
hxnm
σynm

= an,
hyn+1,m

σxn+1,m

=
hynm
σxnm

= bm. (A.4)

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The discretized Boussinesq equation and its conditional symmetry reduction 19

Appendix A.2. Differential and difference operators

In the lowest order terms of the Taylor series expansion of un+1,m and un,m+1:

un+1,m = unm + (xn+1,m − xnm)ux + (yn+1,m − ynm)uy +O(h2)

un,m+1 = unm + (xn,m+1 − xnm)ux + (yn,m+1 − ynm)uy +O(h2),
(A.5)

truncating it at the first order in h, we can substitute the first order partial derivatives

ux and uy by the approximate discrete first order partial derivatives Dxu and Dyu and

(A.5) reads

un+1,m = unm + hxnmDxu+ σynmDyu
un,m+1 = unm + σxnmDxu+ hynmDyu.

(A.6)

Inverting (A.6) we can approximate the partial differential equations on the lattice by

introducing the following discrete partial differential derivatives in the x and y direction

Dx =
1

hxnmh
y
nm − σxnmσ

y
nm

(hynm∆n − σynm∆m)

Dy =
1

hxnmh
y
nm − σxnmσ

y
nm

(−σxnm∆n + hxnm∆m)
(A.7)

with Tnfnm = fn+1,m, ∆n = Tn − 1, and Tmfnm = fn,m+1, ∆m = Tm − 1. The functions

hxnm, hynm, σxnm and σynm are such that (A.2) and (A.3) are satisfied.

Proposition 1 Any partial differential equation in two dependent variables can be

constructed in terms of the partial difference operators (A.7) in the directions x and

y, provided the conditions (A.2, A.3) are satisfied.

Appendix A.3. Schemes for partial difference equations and symmetries

Here, as an example, we limit ourselves to a scheme in R2 of six points (n,m), (n+1,m),

(n,m+1), (n+2,m), (n,m+2), (n+1,m+1), the minimum number of points necessary

to get all partial second derivatives as first order approximations. The variables x, y and

u(x, y) in all points correspond to 18 data, 12 related to the independent variables and

6 to the dependent one. Having 12 data for the independent variables we can construct

from them 10 differences

x00, y00, h
x
0,0 = x10 − x00, hy00 = y01 − y00, σx00 = x01 − x00, σy00 = y10 − y00,

hx10 = x20 − x10, hy01 = y02 − y01, σx01 = x02 − x01, σy10 = y20 − y10,
hx01 = x11 − x01, hy10 = y11 − y10,

(A.8)

From the values of the dependent variables in the 6 points we can calculate the 6

quantities

u00, Dxu00, Dyu00, D2
xu00, D2

yu00, DxDyu00. (A.9)

DyDxu00 is not independent from the 6 quantities (A.9). It can be written in terms of

(A.8) and (A.9). For a generic lattice we have:

DyDxu00 = −DxDxu00
(hx00 − hx01) (hx00 − σx00)

hx01h
y
00 + hx00σ

y
00 − hx01σ

y
00 − σx00σ

y
00

+

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The discretized Boussinesq equation and its conditional symmetry reduction 20

+DyDyu00
(hy00 − h

y
10) (hy00 − σ

y
00)

hx01h
y
00 + hx00σ

y
00 − hx01σ

y
00 − σx00σ

y
00

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+DxDyu00
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x
00 − h

y
1,0σ

x
00 − σx00σ

y
00

hx01h
y
00 + hx00σ

y
00 − hx01σ

y
00 − σx00σ

y
00

(A.10)

Formulas (A.8–A.10) can be simplified if we require the validity of the Clairaut–

Schwarz–Young theorem i.e. DxDyu00 = DyDxu00, i.e.

σxnm = σxn+1,m ≡ σxm, hxnm = hxn,m+1 ≡ hxn,

σynm = σyn,m+1 ≡ σyn, hynm = hyn+1,m ≡ hym.
(A.11)

Using the operators Dx and Dy given in (A.7) and the new independent variables

(A.8) we can transform the standard discrete prolongation

prX̂nm = X̂nm + X̂n+1,m + X̂n+2,m + X̂n,m+1 + X̂n,m+2 + X̂n+1,m+1 (A.12)

of the vector field

X̂nm = ξnm∂xnm + ηnm∂ynm + φnm∂unm (A.13)

to the new set of independent variables (A.8, A.9). We get

prX̂nm = X̂nm

+
∑

i,j∈{0,1}

(
ζ
(x)
n+i,m+j∂hxn+i,m+j

+ χ
(x)
n+i,m+j∂σx

n+i,m+j
+ ζ

(y)
n+i,m+j∂hyn+i,m+j

+ χ
(y)
n+i,m+j∂σy

n+i,m+j

)
+φ(1,x)

nm ∂Dxunm + φ(1,y)
nm ∂Dyunm + φ(2,xx)

nm ∂D2
xunm

+ φ(2,xy)
nm ∂DxDyunm + φ(2,yy)

nm ∂D2
yunm

, (A.14)

where

ζ
(x)
n+i,m+j = ξn+1+i,m+j − ξn+i,m+j, ζ

(y)
n+i,m+j = ηn+i,m+1+j − ηn+i,m+j,

χ
(x)
n+i,m+j = ξn+i,m+1+j − ξn+i,m+j, χ

(y)
n+i,m+j = ηn+1+i,m+j − ηn+i,m+j,

φ(1,x)
nm = Dxφnm −DxunmDxξnm −DyunmDxηnm,
φ(1,y)
nm = Dyφnm −DxunmDyξnm −DyunmDyηnm,
φ(2,xx)
nm = Dxφ(1,x)

nm −D2
xunmDxξnm −DyDxunmDxηnm,

φ(2,xy)
nm = Dxφ(1,y)

nm −DxDyunmDxξnm −D2
yunmDxηnm,

φ(2,yy)
nm = Dyφ(1,y)

nm −DxDyunmDyξnm −D2
yunmDyηnm.

(A.15)

In the continuous limit, when hxn+i,m+j, h
y
n+i,m+j, σ

x
n+i,m+j and σyn+i,m+j go to 0, ζ

(x)
n+i,m+j,

ζ
(y)
n+i,m+j, χ

(x)
n+i,m+j and χ

(y)
n+i,m+j go also to 0 while φ

(1,x)
nm , φ

(1,y)
nm , φ

(2,xx)
nm , φ

(2,xy)
nm and φ

(2,yy)
nm

go to the corresponding continuous prolongations.

Applying the prolongation (A.14) of the infinitesimal generator X̂nm onto the

Schwarzian condition we get that both functions ξnm and ηnm must satisfy the discrete

wave equations

ξn,m+1 − ξnm − ξn+1,m+1 + ξn+1,m = 0,

ηn,m+1 − ηnm − ηn+1,m+1 + ηn+1,m = 0.

This is a constraint for the symmetry coefficients if the Clairaut–Schwarz–Young

theorem is to be satisfied.

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