Lie-Backlund symmetry reduction of nonlinear and non-evolution equations
The application of Lie-Backlund symmetry operators admitted by ordinary differential equations for reducing partial differential equations are studied. The ansatze for dependent variable are constructed by integrating ordinary differential equations. We show that the method is applicable for nonline...
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Інститут математики НАН України
2019
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552872707620864 |
|---|---|
| author | Tsyfra, I. Rzeszut, W. Цифра, І. Жешут, В. |
| author_facet | Tsyfra, I. Rzeszut, W. Цифра, І. Жешут, В. |
| author_institution_txt_mv | [
{
"author": "I. Tsyfra",
"institution": "AGH University of Science and Technology; Institute of Geophysics NAS of Ukraine"
},
{
"author": "W. Rzeszut",
"institution": "AGH University of Science and Technology"
}
] |
| author_sort | Tsyfra, I. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | The application of Lie-Backlund symmetry operators admitted by ordinary differential equations for reducing partial differential equations are studied. The ansatze for dependent variable are constructed by integrating ordinary differential equations. We show that the method is applicable for nonlinear evolution and non-evolution types equations. In the framework of the approach we construct the solution depending on arbitrary smooth function on one variable |
| first_indexed | 2026-08-04T01:06:20Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 174�180
ÓÄÊ 517.912:512.816
Lie�B�acklund symmetry reduction of
nonlinear and non-evolution equations
I.M. Tsyfra †‡, W. Rzeszut †
† AGH University of Science and Technology, Krakow, Poland
E-mail: tsyfra@agh.edu.pl, wojciech.rzeszut@gmail.com
‡ Institute of Geophysics NAS of Ukraine, Kyiv, Ukraine
Äîñëiäæåíî çàñòîñóâàííÿ îïåðàòîðiâ ñèìåòði¨ Ëi�Áåêëóíäà, ÿêi äîïó-
ñêàþòüñÿ çâè÷àéíèì äèôåðåíöiàëüíèì ðiâíÿííÿì, äëÿ ðåäóêöi¨ äè-
ôåðåíöiàëüíèõ ðiâíÿíü ç ÷àñòèííèìè ïîõiäíèìè. Àíçàöè äëÿ çàëåæ-
íî¨ çìiííî¨ ïîáóäîâàíî iíòåãðóâàííÿì çâè÷àéíèõ äèôåðåíöiàëüíèõ ðiâ-
íÿíü. Ïîêàçàíî, ùî ìåòîä ìîæíà çàñòîñîâóâàòè äëÿ ðiâíÿíü åâîëþöié-
íîãî i íååâîëþöiéíîãî òèïó. Ó ðàìêàõ öüîãî ïiäõîäó çíàéäåíî ðîâ'ÿçîê,
ùî çàëåæèòü âiä äîâiëüíî¨ ôóíêöi¨ îäíîãî àðãóìåíòó.
The application of Lie�B�acklund symmetry operators admitted by ordi-
nary di�erential equations for reducing partial di�erential equations are
studied. The ansatze for dependent variable are constructed by integrating
ordinary di�erential equations. We show that the method is applicable for
nonlinear evolution and non-evolution types equations. In the framework
of the approach we construct the solution depending on arbitrary smooth
function on one variable.
1. Introduction. It is a known fact that the symmetry groups
of nonlinear PDEs are being used for finding special solutions invariant
with respect to a certain subgroup of the complete symmetry group of
the equation. Invariant solutions are constructed by solving a reduced
equation with smaller number of independent variables than the initial
equation, an ODE in particular. Conditional symmetry is a generaliza-
tion of a classical Lie symmetry of differential equations and substan-
tially extends the possibilities of construction of solutions of nonlinear
PDEs. It must be noted, that the conditional symmetry method can be
effectively used both for integrable (in some sense) and non-integrable
equations. In [1, 5] concept of conditional Lie–Bäcklund symmetry of
evolution equation, which is a generalization of point conditional sym-
metry, is proposed. In the framework of this approach we obtain reduced
Lie–Bäcklund symmetry reduction 175
system of ODEs. The relationship of generalized conditional symmetry
of evolution equations to compatibility of systems of differential equa-
tions is studied in [2]. In [3] Svirshchevskii used Lie–Bäcklund symme-
try of linear homogeneous ODEs for reducing evolution equations to a
system of ODEs. To apply this method we have to solve the inverse
symmetry problem, namely to find linear homogeneous ODEs which ad-
mit given Lie–Bäcklund symmetry operator. We study the reduction of
nonlinear generalization of the heat equation and modified Korteweg–de
Vries equation by using Lie–Bäcklund symmetry property of linear and
nonlinear ODEs [4]. It allows us to construct solutions for equations of
evolution and non-evolution types.
2. Application of the symmetry reduction method. In this
section we discuss the relationship between the Lie–Bäcklund symmetry
of ordinary differential and reduction of generalized version of Korte-
weg–de Vries equation and nonlinear heat equation.
Example 1. We show how to apply the Lie–Bäcklund symmetry reduc-
tion using mKdV equation as an example. First step is finding an ODE
or ODEs invariant under the operator K[u] = K
(
x, u, ∂u∂x , . . . ,
∂pu
∂xp
)
, in
which case K[u] is the right-hand side of the mKdV equation. Let p be
a positive integer. Consider the ODE to be of the form uxx+g(u, ux) = 0,
where g is a differentiable function of u and ux. Invariance condition for
such ODE reads as
X∞
(
uxx + g(u, ux)
)∣∣
uxx+g(u,ux)=0
= 0, (1)
where X∞ is a prolongation of the vector field X = (uxxx + upux) ∂
∂u on
the jet space. After necessary substitutions equation (1) becomes
pup−1ux(guxux − 3g)− u3
xguuu
+ 3ux(uxguguux + uxgguuux + gguu)
− 3g(uxguxguux + uxguguxux + uxgguuxux + gguux)
+ g2(3guxguxux + gguxuxux) + p(p− 1)up−2u3
x = 0. (2)
The subscripts u and ux denote differentiation with respect to u and ux.
We assume that g(u, ux) =
k∑
i=j
λi(u)uix for some integers j and k. In that
case the left-hand side of the equation (2) becomes a power series of ux.
For every k ≥ 3 and j ≤ −1 the coefficients for the highest and lowest
176 I.M. Tsyfra, W. Rzeszut
powers of ux are 2k(k−1)(2k−1)λ4
k and 2j(j−1)(2j−1)λ4
j respectively,
which implies λi = 0 for i /∈ {0, 1, 2}, meaning g = λ2(u)u2
x + λ1(u)ux +
λ0(u). The six remaining coefficients in the now power series (2) become
the determining equations. They are
12λ4
2 − 30λ′2λ
2
2 + 6λ′22 + 9λ′′2λ2 − λ′′′2 = 0, (3)
30λ1λ
3
2 − 48λ′2λ1λ2 − 15λ′1λ
2
2
+ 9λ′2λ
′
1 + 9λ′′2λ1 + 6λ′′1λ2 − λ′′′1 = 0, (4)
24λ0λ
3
2 + 23λ2
1λ
2
2 − 42λ′2λ0λ2 − 18λ′2λ
2
1 − 21λ′1λ1λ2
− 6λ′0λ
2
2 + 6λ′2λ
′
0 + 3λ′21 − 9λ′′2λ0 + 6λ′′1λ1 + 3λ′′0λ2
− λ′′′0 − pup−1λ2 + p(p− 1)up−2 = 0, (5)
36λ0λ1λ
2
2 + 6λ3
1λ2 − 30λ′2λ0λ1 − 18λ′1λ0λ2 − 6λ′1λ
2
1
− 6λ′0λ1λ2 + 3λ′1λ
′
0 + 6λ′′1λ0 + 3λ′′0λ1 − 2pup−1λ1 = 0, (6)
12λ2
0λ
2
2 + 12λ0λ
2
1λ2 − 12λ′2λ
2
0 − 9λ1λ
′
1λ0 − 6λ′0λ0λ2
+ 3λ′′0λ0 − 3pup−1λ0 = 0, (7)
6λ2
0λ1λ2 − 3λ2
0λ
′
1 = 0. (8)
Based on equations (3)–(8) we will consider four cases:
Case (i): λ0 = 0, λ2 = ω
u , ω ∈
{
−1,− 1
2 , 0
}
. Because we restricted p
to be a nonzero natural number, any assumptions other than λ1 = λ2 =
0, p = 1 lead to contradictions, therefore a solution exists only for p = 1
and it is λi = 0, which means that the invariant equations is
uxx = 0. (9)
Case (ii): λ2 = 0, λ1 = κ = const. Here κ = 0, λ0 = 1
p+1u
p+1 +
α1u+ α2, αi ∈ R, therefore the invariant equation is
uxx + 1
p+1u
p+1 + α1u+ α2 = 0. (10)
Case (iii): λ2 = − 1
2u , λ1 = κ
u , κ = const. Here κ = 0, λ0 =
1
p+2u
p+1 + β1u+ β2
u , βi ∈ R, therefore the invariant equation is
uxx − u2
x
2u + 1
p+2u
p+1 + β1u+ β2
u = 0. (11)
Case (iv): λ2 = − 1
u , λ1 = κ
u2 , κ = const. Here κ = 0, λ0 =
p
(p+1)(p+2)u
p+1 + γ1 + γ2
u , γi ∈ R, therefore the invariant equation is
uxx − u2
x
u + p
(p+1)(p+2)u
p+1 + γ1 + γ2
u = 0. (12)
Lie–Bäcklund symmetry reduction 177
Second step is the variation of the parameters for the solution of the
ODE for time dependence. Outcome of this step is an ansatz for the
PDE, and in this case, the mKdV equation. Equation (9) is the only
linear one and its solution is a trivial ansatz u(x, t) = c1(t)x + c2(t).
Equations (10)–(12) however, are nonlinear and generate implicit an-
satzes (ε = ±1)
ε
∫ u(x,t)
0
da√
c1(t)− α1a2 − 2α2a− 2
(p+1)(p+2)a
p+2
= x+ c2(t),
ε
∫ u(x,t)
0
da√
c1(t)a− 2β1a2 + 2β2 − 2
(p+1)(p+2)a
p+2
= x+ c2(t),
ε
∫ u(x,t)
0
da√
c1(t)a2 + 2γ1a+ γ2 − 2
(p+1)(p+2)a
p+2
= x+ c2(t),
respectively. For certain parameters the ansatzes can be written in an
explicit form. For example when α1 = 0 and p = 1 the equation
uxx + 1
2u
2 + α2 = 0 (13)
produces an explicit ansatz
u(x, t) = −12℘
(
x+ c1(t),− 1
3α2, c2(t)
)
,
where ℘ denotes the Weierstrass function ℘(z, g2, g3).
Third step of the method would be substitution of the ansatz to the
equation we wish to reduce. Let us consider equation (13) with α2 = 0,
meaning
uxx + 1
2u
2 = 0. (14)
It is not linearizable and it admits trivial symmetries ut∂u, (uxxx +
uux)∂u. On the grounds of the aforementioned findings, the solution
to (14) provides an ansatz and a reduction for the KdV equation ut =
uxxx + uux. Before we proceed with the reduction, we can indulge in a
side challenge of finding some other PDEs sharing the same ansatz. For
this purpose we introduce independent variable z. It can identified with
the time variable t or viewed as a second space variable. One can show
the equation (14) is invariant under LBS operators(
u2uxuz + 2uxzu
2
x
)
∂u,
(
u2u2
z + 2uxzuxuz
)
∂u.
178 I.M. Tsyfra, W. Rzeszut
Since a linear combination of symmetry operators is itself a symmetry
operator, we can use the solution of equation (14) to reduce (1+2)-
dimensional equations
ut = uxxx + uux + u2uxuz + 2uxzu
2
x,
ut = uxxx + uux + u2u2
z + 2uxzuxuz,
or a non-evolution equation in one of the forms
εut + u2utux + 2utxu
2
x + uxxx + uux = 0, (15)
εut + u2u2
t + 2utxutux + uxxx + uux = 0, (16)
where ε is an arbitrary constant.
The solution of the ODE (14) is the Weierstrass elliptic function
u(x) = −12℘(x + c1, 0, c2), meaning the ansatz we substitute into the
presented PDEs for reduction is
u(x, t) = −12℘(x+ c1(t), 0, c2(t)).
After such substitution, equation (15) for example, reduces to a system
εc′2 = 0, εc′1 − 144c′2 = 0.
This means that for ε = 0 equation (15) has a class of solutions
u(x, t) = −12℘(x+ c1(t), 0, c2), c2 = const
and for ε 6= 0 there is only a stationary solution
u(x, t) = −12℘(x+ c1, 0, c2), c1, c2 = const.
For equation (16) the reduced equations are
c′1(144c′2 − ε) = 0, c′2(144c′2 − ε) = 0.
The solutions to this system are
c1 = const, c2 = const
or
c1(t) is arbitrary function, c2 = ε
144 t+ c0, c0 = const.
Lie–Bäcklund symmetry reduction 179
This means that the equation (16) has a class of solutions
u(x, t) = −12℘
(
x+ c1(t), 0, ε
144 t+ c0
)
, c0 = const
as well as a stationary solution
u(x, t) = −12℘(x+ c1, 0, c2), c1, c2 = const.
Example 2. Equations
ux = 1−u2
√
2
(17)
and
uxx + u− u3 = 0 (18)
share the same kink solution
u = tanh
(
x+c√
2
)
, c = const. (19)
It can be easily shown that both
(
uxx + u − u3
)
∂u and ut∂u are Lie–
Bäcklund symmetry operators of equation (17). It follows that the sub-
stitution
u = tanh
(x+c(t)√
2
)
(20)
reduces equation
ut = uxx + u
(
1− u2
)
(21)
to a first-order differential equation c′(t) = 0, which means that equa-
tion (21) admits a stationary solution
u(x, t) = u(x) = tanh
(
x+c√
2
)
.
The second-order ODE (18) is a differential consequence of equation (17),
therefore the right-hand side of the reduced equation vanishes on the
ansatz solution. To obtain non-stationary solutions using this particular
ansatz and this particular first-order ODE, we can add to the evolutio-
nary equation first-order terms corresponding to the contact symmetries
of (17). Contact symmetry of (17) in general form can be written as
f
(
ux
1−u2 ,
1−u
1+ue
√
2x
)
ux∂u,
180 I.M. Tsyfra, W. Rzeszut
where f is an arbitrary smooth function of two arguments. Substitution
of (20) into
ut = uxx + u
(
1− u2
)
+ f
(
ux
1−u2 ,
1−u
1+ue
√
2x
)
ux (22)
reduces this equation to a simple first-order ODE
c′(t) = f
(
1√
2
, e−
√
2c(t)
)
.
Equation (22) will have a kink solution if ∂f
∂c(t) = 0 and f 6= 0.
3. Conclusion. We show the application of the Lie–Bäcklund sym-
metry method for reducing the generalized version of Korteweg–de Vries
equation of and nonlinear heat equation. We construct the class of or-
dinary differential equations which admit given Lie–Bäcklund symmetry
operator and the corresponding ansatze reducing the equation under
study to the system of two ordinary differential equations. The method
enables us to find solutions which contain arbitrary functions on one
variable for the equations (15), (16) and the solution generalizing the
kink solution for nonlinear heat equation.
[1] Fokas A.S., Liu Q.M., Nonlinear interaction of traveling waves of non-integrable
equations, Phys. Rev. Lett. 72 (1994), 3293–3296.
[2] Kunzinger M., Popovych R.O., Generalized conditional symmetry of evolution
equations, J. Math. Anal. Appl. 379 (2011), 444–460.
[3] Svirshchevskii S.R., Lie–Bäcklund symmetries of linear ODEs and generalized
separation of variables in nonlinear equations, Phys. Lett. A 199 (1995), 344–
349.
[4] Tsyfra I.M., Symmetry reduction of nonlinear differential equations, Proc. Inst.
Math. 50 (2004), 266–270.
[5] Zhdanov R.Z., Conditional Lie–Bäcklund symmetry and reduction of evolution
equations, J. Phys. A: Math. Gen. 28 (1995), 3841–3850.
|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:20Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
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| spelling | oai:trim.imath.kiev.ua:article-3742020-08-13T08:52:53Z Lie-Backlund symmetry reduction of nonlinear and non-evolution equations Симетрійні редукції Лі-Беклунда нелінійних та не-еволюційних рівнянь Tsyfra, I. Rzeszut, W. Цифра, І. Жешут, В. The application of Lie-Backlund symmetry operators admitted by ordinary differential equations for reducing partial differential equations are studied. The ansatze for dependent variable are constructed by integrating ordinary differential equations. We show that the method is applicable for nonlinear evolution and non-evolution types equations. In the framework of the approach we construct the solution depending on arbitrary smooth function on one variable Досліджено застосування операторів симетрії Лі-Беклунда, які допускаються звичайним диференціальним рівнянням, для редукції диференціальних рівнянь з частинними похідними. Анзаци для залежної змінної побудовано інтегруванням звичайних диференціальних рівнянь. Показано, що метод можна застосовувати для рівнянь еволюційного і нееволюційного типу. У рамках цього підходу знайдено ров'язок, що залежить від довільної функції одного аргументу.   Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/374 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 174-180 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 174-180 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 174-180 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/374/371 Авторське право (c) 2019 І. Цифра, В. Жешут |
| spellingShingle | Tsyfra, I. Rzeszut, W. Цифра, І. Жешут, В. Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title | Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title_alt | Симетрійні редукції Лі-Беклунда нелінійних та не-еволюційних рівнянь |
| title_full | Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title_fullStr | Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title_full_unstemmed | Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title_short | Lie-Backlund symmetry reduction of nonlinear and non-evolution equations |
| title_sort | lie-backlund symmetry reduction of nonlinear and non-evolution equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/374 |
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