Conservation Laws of Discrete Korteweg-de Vries Equation
All three-point and five-point conservation laws for the discrete Korteweg-de Vries equations are found. These conservation laws satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. Our method uses computer algebra intensively, because the deter...
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| Cite this: | Conservation Laws of Discrete Korteweg-de Vries Equation / O.G. Rasin, P.E. Hydon // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 3 назв. — англ. |
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| author_facet | Rasin, O.G. Hydon, P.E. |
| citation_txt | Conservation Laws of Discrete Korteweg-de Vries Equation / O.G. Rasin, P.E. Hydon // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 3 назв. — англ. |
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| description | All three-point and five-point conservation laws for the discrete Korteweg-de Vries equations are found. These conservation laws satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. Our method uses computer algebra intensively, because the determining functional equation is quite complicated.
|
| first_indexed | 2025-11-27T05:14:58Z |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 1 (2005), Paper 026, 6 pages
Conservation Laws
of Discrete Korteweg–de Vries Equation
Olexandr G. RASIN and Peter E. HYDON
Department of Mathematics and Statistics, University of Surrey,
Guildford, Surrey GU2 7XH, UK
E-mail: o.rasin@surrey.ac.uk, p.hydon@surrey.ac.uk
Received October 21, 2005, in final form December 06, 2005; Published online December 09, 2005
Original article is available at http://www.emis.de/journals/SIGMA/2005/Paper026/
Abstract. All three-point and five-point conservation laws for the discrete Korteweg–de
Vries equations are found. These conservation laws satisfy a functional equation, which we
solve by reducing it to a system of partial differential equations. Our method uses computer
algebra intensively, because the determining functional equation is quite complicated.
Key words: conservation laws; discrete equations; quad-graph
2000 Mathematics Subject Classification: 70H33; 37K10; 39A05
1 Introduction
A direct method for calculation of conservation laws for partial difference equations (P∆E’s)
was recently introduced by Hydon [3]. This method, which does not use Noether’s Theorem,
has been used to calculate some low-order conservation laws of various integrable difference
equations that are defined on the quad-graph shown in Fig. 1. (For a classification of integrable
quad-graph equations, see [1].)
In the current paper, we present a modified version of Hydon’s direct method, and use it to
derive conservation laws of the discrete Korteweg–de Vries equation [2]:(
p + q + vl+1
k+1 − vl
k
)(
q − p + vl
k+1 − vl+1
k
)
= q2 − p2, (1)
which is an integrable quad-graph equation. Here p, q are parameters and p2 6= q2. To simply
matters, we use the transformation
vl
k = ul
k
√
q2 − p2 − qk − pl
to reduce (1) to(
ul+1
k+1 − ul
k
)(
ul
k+1 − ul+1
k
)
= 1.
We shall call this equation dKdV. As with all integrable quad-graph equations, dKdV may be
solved to write any one of ul
k, ul
k+1, ul+1
k , ul+1
k+1 in terms of the other three. In particular, we
will write dKdV as either
ul+1
k+1 = ω, where ω =
1
ul
k+1 − ul+1
k
+ ul
k,
or
ul
k+1 = Ω, where Ω =
1
ul+1
k+1 − ul
k
+ ul+1
k .
mailto:o.rasin@surrey.ac.uk
mailto:p.hydon@surrey.ac.uk
http://www.emis.de/journals/SIGMA/2005/Paper026/
2 O.G. Rasin and P.E. Hydon
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l
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v
l
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l+1
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v
l+1
k+1
Figure 1. Quad-graph.
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F
G
u
l
k
u
l
k+1
u
l+1
k
u
l+1
k+1
Figure 2. Three-point conservation law.
A conservation law for a partial difference equation (P∆E) on Z2 is an expression of the form
(Sk − id)F + (Sl − id)G = 0
that is satisfied on all solutions of the equation. Here id is the identity mapping and Sk, Sl are
forward shifts of the coordinates k and l respectively:
Sk :
(
k, l, ul
k
)
→
(
k + 1, l, ul
k+1
)
, Sl :
(
k, l, ul
k
)
→
(
k, l + 1, ul+1
k
)
.
A conservation law is trivial if it holds identically (not just on solutions of the P∆E), or if F
and G both vanish on all solutions of the equation. We search for nontrivial conservation laws.
2 Three-point conservation laws
In this section we consider conservation laws that lie on the quad-graph. This means that the
functions F , G, SkF and SlG must depend upon only k, l, ul
k, ul
k+1, ul+1
k and ul+1
k+1. Consequently
the most general form of F and G is:
F = F
(
k, l, ul
k, u
l+1
k
)
, G = G
(
k, l, ul
k, u
l
k+1
)
.
The dependence of F and G upon the continuous variables uj
i is illustrated in Fig. 2; together,
these functions lie on three points of the quad-graph. For this reason, we call such conservation
laws three-point conservation laws.
The three-point conservation laws can be determined directly by substituting dKdV into
F
(
k + 1, l, ul
k+1, u
l+1
k+1
)
− F
(
k, l, ul
k, u
l+1
k
)
+ G
(
k, l + 1, ul+1
k , ul+1
k+1
)
−G
(
k, l, ul
k, u
l
k+1
)
= 0, (2)
and solving the resulting functional equation. The substitution ul+1
k+1 = ω yields
F
(
k + 1, l, ul
k+1, ω
)
− F
(
k, l, ul
k, u
l+1
k
)
+ G
(
k, l + 1, ul+1
k , ω
)
−G
(
k, l, ul
k, u
l
k+1
)
= 0. (3)
In order to solve this functional equation we have to reduce it to a system of partial differential
equations. To do this, first eliminate terms that contain ω, by applying each of the following
(commuting) differential operators to (3):
L1 =
∂
∂ul+1
k
−
ωul+1
k
ωul
k
∂
∂ul
k
, L2 =
∂
∂ul
k+1
−
ωul
k+1
ωul
k
∂
∂ul
k
.
The operators L1, L2 differentiate with respect to ul
k+1, ul+1
k respectively, keeping ω fixed. This
procedure does not depend upon the form of ω; it can be applied equally to any quad-graph
equation. In particular, for dKdV, (3) is reduced to
Ful
kul
k
+ Gul
kul
k
−
(
ul
k+1 − ul+1
k
)2(
Ful
kul+1
k
−Gul
kul
k+1
)
Conservation Laws of Discrete KdV Equation 3
− 2
(
ul
k+1 − ul+1
k
)(
Ful
k
+ Gul
k
)
= 0, (4)
where F = F (k, l, ul
k, u
l+1
k ) and G = G(k, l, ul
k, u
l
k+1). Differentiating (4) three times with
respect to ul+1
k eliminates G and its derivatives, leaving the necessary condition
F
ul
k
2
ul+1
k
3 −
(
ul
k+1 − ul+1
k
)2
F
ul
kul+1
k
4 + 4
(
ul
k+1 − ul+1
k
)
F
ul
kul+1
k
3 = 0. (5)
This equation can be split into an overdetermined system by equating powers of ul
k+1. Further
information about F may be found by substituting ul
k+1 = Ω into (2). Differentiating
F
(
k + 1, l, Ω, ul+1
k+1
)
− F
(
k, l, ul
k, u
l+1
k
)
+ G
(
k, l + 1, ul+1
k , ul+1
k+1
)
−G
(
k, l, ul
k,Ω
)
= 0
with respect to ul
k, ul+1
k+1 and keeping Ω fixed yields
Ful+1
k ul+1
k
− G̃ul+1
k ul+1
k
−
(
ul+1
k+1 − ul
k
)2(
Ful
kul+1
k
+ G̃ul+1
k ul+1
k+1
)
− 2
(
ul+1
k+1 − ul
k
)(
Ful
k
− G̃ul+1
k
)
= 0,
where G̃ = G(k, l + 1, ul
k, u
l+1
k+1). The function G̃ and its derivatives are eliminated by differen-
tiating three times with respect to ul+1
k , which yields
F
ul
k
3
ul+1
k
2 −
(
ul+1
k+1 − ul
k
)2
F
ul
k
4
ul+1
k
+ 4
(
ul+1
k+1 − ul
k
)
F
ul
k
3
ul+1
k
= 0. (6)
The overdetermined system of partial differential equations (5), (6) is easily solved to obtain
F = C1u
l
ku
l+1
k + C2u
l
k
2
ul+1
k + C3u
l
ku
l+1
k
2
+ C4u
l
k
2
ul+1
k
2
+ F1 + F2,
where each Ci is an arbitrary function of k, l, and F1 = F1(k, l, ul+1
k ), F2 = F2(k, l, ul
k) are
arbitrary functions. The term F2(k, l, ul
k) can be removed (without loss of generality) by adding
the trivial conservation law
FT = (Sl − id)F2, GT = −(Sk − id)F2,
to F and G respectively.
So far, we have differentiated the determining equations for a conservation law five times; this
has created a hierarchy of functional differential equations that every three-point conservation
law must satisfy. The unknown functions Ci, F1 and G are found by going up the hierarchy, a step
at a time, to determine the constraints these equations place on the unknown functions. As the
constraints are solved sequentially, more and more information is gained about the functions.
At the highest stage, the determining equations are satisfied, and the only remaining unknowns
are the constants that multiply each conservation law. This is a simple but lengthy process; for
brevity, we omit the details.
By this technique we have found all independent nontrivial three-point conservation laws for
the dKdV equation; they are as follows1:
1. F = ul
k
(
ul+1
k
)2 − (ul
k
)2
ul+1
k + ul
k − ul+1
k ,
G =
(
ul
k
)2
ul
k+1 − ul
k
(
ul
k+1
)2
,
2. F = (−1)k+l+1
{
ul
k
(
ul+1
k
)2 +
(
ul
k
)2
ul+1
k − ul
k − ul+1
k
}
,
1Note that the conservation laws 1 and 2 are connected to each other by a discrete symmetry of the form
ul
k 7→ (−1)k+liul
k; we are grateful to a referee for this observation.
4 O.G. Rasin and P.E. Hydon
G = (−1)k+l
{(
ul
k
)2
ul
k+1 + ul
k
(
ul
k+1
)2}
,
3. F = (−1)k+l+1
{(
ul
ku
l+1
k
)2 − 2ul
ku
l+1
k +
1
2
}
,
G = (−1)k+l
{(
ul
ku
l
k+1
)2}
,
4. F = (−1)k+l+1
{
ul
ku
l+1
k − 1
2
}
,
G = (−1)k+l
{
ul
ku
l
k+1
}
.
Three of these conservation laws depend on k and l explicitly. If we had chosen functions F
and G that depended only upon uj
i on the quad-graph (and not also upon k and l), we would
have found only the first of four the three-point conservation laws.
3 Five-point conservation laws
Higher conservation laws can be found by the approach described above, but the complexity of
the calculations increases rapidly with the number of variables uj
i on which F and G depend. The
simplest higher conservation laws are defined on five points, as shown in Fig. 3. The functions F
and G are of the form
F = F
(
k, l, ul+1
k−1, u
l
k−1, u
l
k, u
l−1
k
)
, G = G
(
k, l, ul
k−1, u
l
k, u
l−1
k , ul−1
k+1
)
.
The points lie in a pair of quad-graphs with a single common point. We seek conservation laws
that contain points from both quad-graphs and cannot be reduced to the conservation laws from
the previous section (modulo trivial conservation laws).
The determining equation for the five-point conservation laws is
F (k + 1, l, ul+1
k , ul
k, u
l
k+1, u
l−1
k+1)− F (k, l, ul+1
k−1, u
l
k−1, u
l
k, u
l−1
k )
+ G(k, l + 1, ul+1
k−1, u
l+1
k , ul
k, u
l
k+1)−G(k, l, ul
k−1, u
l
k, u
l−1
k , ul−1
k+1) = 0.
Shifted versions of the dKdV equation are used to eliminate ul
k+1, ul+1
k in favour of the variab-
les uj
i at the five points that lie on the ‘step’ shown in bold in Fig. 3. The determining equation
can be solved by the same technique which is described in the previous section. This is a very
lengthy calculation, so we state the results only.
The extra five-point conservation laws of dKdV equation are
5. F = ln
(
ul
k − ul+1
k−1
)
,
G = ln
(
1
ul−1
k+1 − ul
k
− ul
k−1 + ul−1
k
)
,
6. F = ln
(
1
ul−1
k − ul
k−1
− ul
k + ul+1
k−1
)
,
G = ln
(
ul
k−1 − ul−1
k
)
,
7. F = l ln
(
ul
k − ul+1
k−1 +
1
ul
k−1 − ul−1
k
)
− (k − 1) ln
(
ul+1
k−1 − ul
k
)
,
G = l ln
(
ul
k−1 − ul−1
k
)
− k ln
(
1
ul−1
k+1 − ul
k
− ul
k−1 + ul−1
k
)
.
Conservation Laws of Discrete KdV Equation 5
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F
G
u
l
k
u
l
k+1
u
l+1
k
u
l−1
k
u
l
k−1
u
l−1
k+1
u
l+1
k−1
Figure 3. Five-point CL.
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F
G
u
l
k
u
l
k+1
u
l+1
k
u
l−1
k
u
l
k−1
Figure 4. Five-point CL.
These conservation laws are irrational, so it is clear that they cannot be reduced to the
polynomial conservation laws in the previous section. The five-point conservation laws can be
simplified somewhat with the substitutions
ul+1
k−1 = ul
k +
1
ul
k−1 − ul+1
k
, ul−1
k+1 = ul
k +
1
ul−1
k − ul
k+1
.
These substitutions move the five-point conservation laws onto the cross shown in Fig. 4. The
functions F and G are now of the form
F = F ′(k, l, ul
k−1, u
l−1
k , ul
k, u
l+1
k
)
, G = G′(k, l, ul
k−1, u
l−1
k , ul
k, u
l
k+1
)
.
Specifically, the five-point conservation laws have the following components:
5. F ′ = − ln
(
ul+1
k − ul
k−1
)
,
G′ = ln
(
ul
k+1 − ul
k−1
)
,
6. F ′ = ln
(
ul−1
k − ul+1
k(
ul+1
k − ul
k−1
)(
ul
k−1 − ul−1
k
)) ,
G′ = ln
(
ul
k−1 − ul−1
k
)
,
7. F ′ = l ln
(
ul+1
k − ul−1
k(
ul+1
k − ul
k−1
)(
ul
k−1 − ul−1
k
))+ (k − 1) ln
(
ul
k−1 − ul+1
k
)
,
G′ = l ln
(
ul
k−1 − ul−1
k
)
− k ln
(
ul
k+1 − ul
k−1
)
.
Surprisingly, none of them do depend upon ul
k. We will describe the circumstances under which
this occurs for other quad-graph equations in a separate paper.
4 Conclusion and outlook
The main results of this paper are as follows.
• New conservation laws for the dKdV equation have been found.
• These include higher-order conservation laws, which are irrational.
• We have improved the effectiveness of Hydon’s direct method for constructing conservation
laws [3].
6 O.G. Rasin and P.E. Hydon
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�
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�
F
G
Figure 5. Seven-point CL.
In principle, the same method can be used to construct conservation laws with seven or more
points (Fig. 5). However, the calculations become extremely complex, placing heavy demands
on even the most sophisticated computer algebra systems. We are currently working to improve
the efficiency of the method still further, but it is unlikely that it will be possible to determine
conservation laws of very high order directly. However, we are developing ways of combining
direct and indirect methods to achieve this aim, as will be reported elsewhere.
[1] Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency
approach, Comm. Math. Phys., 2003, V.233, 513–543; nlin.SI/0202024.
[2] Hirota R., Nonlinear partial difference equations. I. A difference analog of the Korteweg–de Vries equation,
J. Phys. Soc. Japan, 1977, V.43, 1423–1433.
[3] Hydon P.E., Conservation laws of partial difference equations with two independent variables, J. Phys. A:
Math. Gen., 2001, V.34, 10347–10355.
1 Introduction
2 Three-point conservation laws
3 Five-point conservation laws
4 Conclusion and outlook
|
| id | nasplib_isofts_kiev_ua-123456789-209334 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T05:14:58Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rasin, O.G. Hydon, P.E. 2025-11-19T12:18:53Z 2005 Conservation Laws of Discrete Korteweg-de Vries Equation / O.G. Rasin, P.E. Hydon // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 3 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 70H33; 37K10; 39A05 https://nasplib.isofts.kiev.ua/handle/123456789/209334 https://doi.org/10.3842/SIGMA.2005.026 All three-point and five-point conservation laws for the discrete Korteweg-de Vries equations are found. These conservation laws satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. Our method uses computer algebra intensively, because the determining functional equation is quite complicated. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Conservation Laws of Discrete Korteweg-de Vries Equation Article published earlier |
| spellingShingle | Conservation Laws of Discrete Korteweg-de Vries Equation Rasin, O.G. Hydon, P.E. |
| title | Conservation Laws of Discrete Korteweg-de Vries Equation |
| title_full | Conservation Laws of Discrete Korteweg-de Vries Equation |
| title_fullStr | Conservation Laws of Discrete Korteweg-de Vries Equation |
| title_full_unstemmed | Conservation Laws of Discrete Korteweg-de Vries Equation |
| title_short | Conservation Laws of Discrete Korteweg-de Vries Equation |
| title_sort | conservation laws of discrete korteweg-de vries equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209334 |
| work_keys_str_mv | AT rasinog conservationlawsofdiscretekortewegdevriesequation AT hydonpe conservationlawsofdiscretekortewegdevriesequation |