Conservation laws and symmetry analysis of (1+1)-dimensional Sawada-Kotera equation Текст научной статьи по специальности «Математика»

Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 3(19). C. 10-19. ISSN 2079-6641

DOI: 10.18454/2079-6641-2017-19-3-10-19 MSC 35R01+76M60+17B66

CONSERVATION LAWS AND SYMMETRY ANALYSIS OF (1+1)-DIMENSIONAL SAWADA-KOTERA

EQUATION

S. R. Hejazi, E. Lashkarian

Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.

E-mail: ra.hejazi@gmail.com, lashkarianelham@yahoo.com

The paper addresses an extended (1+1)-dimensional Sawada-Kotera (SK) equation. The Lie symmetry analysis leads to many plethora of solutions to the equation. The non-linear self-adjointness condition for the SK equation established and subsequently used to construct simplified independent conserved vectors. In particular, we also get conservation laws of the equation with the corresponding Lie symmetry.

Key words: Fluid mechanics, Lie symmetry, Partial differential equation, Shear stress, Optimal system, Partial differential equation; KdV equation, Lie symmetry; Conservation Laws

(c) Hejazi S.R., Lashkarian E., 2017

УДК 548.71

ЗАКОНЫ СОХРАНЕНИЯ И АНАЛИЗ СИММЕТРИИ (1 + 1)-МЕРНОГО

УРАВНЕНИЯ САВАДА-КОТЕРА

С. Р. Хеязи, Е. Лашкариан

Отдел математики, Иранский университет науки и техники, Тегеран, Иран E-mail: ra.hejazi@gmail.com, lashkarianelham@yahoo.com

В работе рассматривается расширенное (1 + 1)-мерное уравнение Савада-Котера (СК). Анализ симметрии Ли приводит к множеству решений уравнения. Условие нелинейной самосопряженности для уравнения СК, установленное и впоследствии используемое для построения упрощенных независимых консервативных векторов. В частности, мы также получаем законы сохранения уравнения с соответствующей симметрией Ли.

Ключевые слова: механика жидкости, симметрия Ли, уравнение с частными производными, напряжение сдвига, оптимальная система, уравнение с частными производными; уравнение КдФ, симметрия Ли, законы сохранения

(с) Хеязи С. Р, Лашкариан Е., 2017

Introduction

In this paper, we study the following equation

Ask := U — 20U2Ux — 25wxwxx — 10UUxxx — wxxxxx = 0, (1)

which is a 5-th order PDE or a type of 5-th order KdV-equation found first by Sawada and Kotera and then by Caunderey, Dodd and Gibbon, which describes long waves in water of relatively shallow depth [2].

The symmetry group of a system of differential equations (DEs) transform solution of the system to other solutions of the system. For constructing the solutions of non-linear PDEs, Lie symmetry group theory can be regarded as one of the most powerful methods in the theory on non-linear PDEs [8, 9, 10, 12, 18, 19].

In the study of DEs, conservation laws play signeficant roles not only in obtaining in-depth understanding of physical properties of various systems, but also in constructing of their exact solutions [1, 3, 4, 7, 11, 14, 15]. They described physical conserved quantities sucha s mass, energy, momentum and angular momentum, as well as charge and other constant of motion. They are important for investigating integribility and linearization mapping and for stablishing exsistence and uniqeness of solutions. They are also used in the analysis of stability and global behaviour of solutions. In addition they play an essential role in the developement of numerical methods and provide an essential starting point for find non-locally related systems and potential variables. Moreover, the structure of conservation laws is coordinate-free, as a point or contact transformation maps a conservation laws into a conservation laws. A systematic way of constructing the conservation laws of a sytem of DEs that admits a variational principle is via Noether's theorem. Its application allows physicists to gain powerful insights into any general theory in physics just by analyzing the various transformations that would make bthe form of the laws involved invariant. For instance, the invariant of physical systems with respect to spatial translation, rotation and time translation respectively give rise to the well known conservation laws of linear momentum, angular momentum and energy. Among the generalization of Noether's theorem an Ibragimov's theorem, based on the self-adjontness concepy of DEs allows to find independent conservation laws for a system of PDEs. Thus, in this paper the adjointness of the SK equation is first established, then Lie point symmetries are applied to find non-trivial conservation laws for the equation (1).

Adjoint equation

In accordance to [13], the formal Lagrangian of the equation (1) is given by:

L = v(ut — 20 U2Ux — 25 UxUxx — 10 UUxxx — Uxxxxx), (2)

where v in a new dependent variable. The adjoint equation for SK equation is

8 L

Ak = f, (3)

where

Su du + S=1Dil Dis dutv..is' (4)

is the variational derivative with respect to u.

Taking into account Eq. (3) and (4) we obtain the following adjoint equation for SK equation:

A£K = Vxxxxx - 20UxxVx + 5UxVxx + 10uVxxx + 25UxVx + 20u2Vx - Vt.

This equation will be used for the derivation of non-linearity self-adjoint condition later on.

Non-linear self-adjointness

This concept has significantly expanded the notion of adjointness with respect to construction of canservation laws. It incorporates all the previous concepts of adjointness and thus enables more consereved vectors for DEs to be constructed. According to the definition of non-linearly self-adjointness the Eq. (1) is non-linear self-adjoint if the equation obtained from the adjoint A£K after the substituition v = ç(x,t, u) is identical with original Eq. (1) where ç is an arbitrary function. That is, if

A*

— Z (x, t, u)Ask , (5)

v—$ (x,t ,u)

for some indeterminate variable coefficient Z.

Take the substituition v written together with the necessary derivatives

Vx = Çx + ÇuUx, V; = Ç + çuU, Vxx = Çxx + 2ÇxUUx + çmm"2 + ÇuUxx,

vxxx = Çxxx + (3ÇxxU + 2çxuu)ux xx

2 3

+ ÇxuuUx + ÇuuuUx + 3ÇuuU

xuxx + 2ÇxuU xxx

xxxxx — tyxxxxx + (5^xxxxu + 2^xxWMM)ux + 8$xxxuuu +-----+ 2^xuuU xuxxxx + 2Çxuuxxxxx + 2$:

x

'xxxuu,

to Eq. (5) we conclude that v is just a constant non-zero function.

Conservation laws

In this section we use both Noether's method and direct method to construct canservation laws for Eq. (1).

Conservation laws constructed by symmetries

Recall thet the following general result on construction on conserved vectors associated with Lie point symmetries of any system of DEs holds [13, 17].

Theorem. Any symmetry (Lie point symmnetry, Lie-Backlund symmetry, non-local symmetry)

d d V = ^ '(x, u, d u, + (x, u, d u,...)

of a system of q—differential equations

Ao (x, u, d u,..., d su) = 0, o = 1,..., q, (6)

with p—independent variables x = (x1,...,xp) and q—dependent variables u = (u1,...,uq) is inherited by the adjoint system. Specifically the operator

V * = % ^ +

+

(7)

dx* ' rudu° rad

with appropriately chosen coefficients fâ, is admitted by the system of equations consisting of Eq. (6) and adjoint equation

A^(x, u, v, du,..., dsu, dsv) =

_ 5 (vaAa)

5 ua

= 0, a = 1,..., q.

(8)

Furthermore, the combined system (6) and (8) has the conservation law D^C = 0, where

d L

a = % lL+qct( d^a - D

d L \ Id L

+ D j^ dj

d uaj

, dL t dL

+Dj(QaH ^ra - Dd — 1 +

d ua

+D j Dk(Qa )

/ d L

ld uj

d u*jK

+

with the characteristics

Qa = 0a - %kua, a = 1,...,q,

and the formal Lagrangian

L = vaAa(x, u, du,..., dsu). Now the theorem can be applied as follows. Let us

f-1 d ,.2 d , d

V = k V + k V + ,

d x d t d u

(9)

(10)

(11)

(12)

be an arbitrary symmetry of the SK, then the generator gives the conservation law

Dx(C1) + Dt (C2)|(i)= 0, (13)

with the conserved vector components

c1=%il+Q( dL -dJ dL

d L

+Dx(QM ^ - Dx

+d2(q)

d ux. d L

d uxxx

+ D2

d uxx

d L

d uxxx

d L

d uxxxx

+ x( d uxx) +

4

x 1 d u

d L

-D3

d L

d uxxxx

+d3(q) -D

d L

d uxxxx:

+D4

d L

d u

C2 = % 2L + Q

xxxxx

d L d u '

(14)

(15)

where L is as given in Eq. (2) and the characteristic (10) become

Q = 0 - %1 Ux - % 2ut. (16)

The explicit form for the infinitesimal (12) provide the following 3-dimensional Lie algebra for the Eq. (1) spanned by the perators

V d v d VX d +1 d2u d (17)

d x' d t' 5 d x d t 5 d u

In what follows, we use each of the Lie algebra (17) and the general non-linearity self -adjoint conditions of above theorem to find non-trivial conservation laws for the Eq. (1).

Conservation laws through Vi

For this symmetry operator the, the characteristic is

Q = - Ux.

Effecting this value into the vector components (14) and (15), yields

C1 = Ut + 10Ux(Ux - Uxx), C2 = - Ux.

Conservation laws through V2

For this symmetry operator the, the characteristic is

Q = -Ut.

Effecting this value into the vector components (14) and (15) as well as above, yields

1 2 2 2

C — Mxxxxt + WxWxxxx + 20u Wx + 15WxUö + 25wxMxx + lÖMMxxt + UxU

-x^-xxxxx;

C - Uxxxxx 10uuxxx 25uxUxx 20u ux.

Conservation laws through V3

For this symmetry operator the, the characteristic is

2u x

Q = —5 - tut - 5Ux. Effecting this value into the vector components (14) and (15), yields

C1 - 8u3 + + 2uuxx + 4(1 + 10iw2)w2wx + 2(2x + 100iw2)wwxxx

+5 (3t + 5x + 100iw2)wxMxx + 2 Uxxxx + tUxxxxt + (20tu2 + x) Uxxxxx 5

C2 = T + 5 Ux - 20tu2ux - 25tUxUxxx - 10tUUxxx - tUxxxxx.

Conservation laws constructed by direct method

As the basic definition of conservation laws, if one of the independent variables of the system (6) is time t the conservation laws takes the form

Df$(x, u, d u,..., d1 u) + Dt ¥f(x, u, d u,..., d £u) = 0, (18)

where ¥ is reffered to as a density and Q1 is spatial fluxes of the conservation law (18).

Consider the system of differential equations defined in (6). A set of differentiable function {ACT(x,u,du,...,d£u)}|= called local multipliers yields a divergence expression for the PDE system (6) if the identity

(x, u, du,..., d£u)Aa = (x, u, du,..., d£u),

on the solutions of the system. The following theorem connects local multipliers and conservation laws [4]. A set on non-singular local multipliers {ACT(x,u, du,..., d£u)} yields a conservation law for the system (6) if and only if the set of identities

(ACT (x, u, d u,..., d £u)ACT) = 0, (19)

8 u°

holds for arbitrary functions u(x).

For instance for investigating the zeroth order set of multipliers of the Eq. (1), we should take the function A(x,t, u) in the theorem which satisfies the determining equation (19) in the form of

S

JU (A(x, t, w)Ask ) = 0,

for the variational derivative

S d„d„d „2 d

S u d u x d ux t d ut + x d ux

+... + D^-—d . (20)

duxxxxx

Expanding the left hand side of the equation (20), yields the following determining equation:

At + Axxxxx + 20u2Ax + 2AuUxxxxx +-----+ 20uAuUxxx + 135AuUxUxx = 0. (21)

Solving the equation (21) with respect to A and its derivatives shows that A is just a constant number. Thus, the density and the flux are

$(x, t, u) = F (x, t), ¥(x, t, u) = F (x, t )dt + G(x).

Similarly, if we take the multiplier A as the first order multiplier A(x, t, u, ux, ut), we obtain that it is just a constant number too. In this case the density and flux are:

/u

Fx(x, t, u)dw + G(x, t), ¥(x,t,u,ux,ut) = —F(x,t,u)ux — J Fx(x,t,u)dw — JGx(x,t)dt + H(x).

Doing as well as the above case for zeroth, first and etc. order density and flux, we can set the following list of conservation laws in the form of (18) in the more simple polynomial form. Thus, the set of density and fluxes of the SK equation up to fourth order in the polynomial form is comming in table 1:

Таблица 1. Fluxes and densities of the KS Eq.

Flux

x Uxxxx 10uuxx 2~U2 3~U Ut

— — Uxtt — tUxttt + Uttt — xUxttt

— UxxxUtttt — UtttUxxxt — Uxttt

— UxxxUtttt — UtttUxxxt UxxxUxttt UxttUxxxt

UxxtUxxxt UxxxUxxxt UxUxxxt + UxxUxxt UxtUxxx

— Uxxx — tUxxxt + Uxxt — xUxxxt — U«t — UttUtttt

— UxtUtttt — UxttUttt — Uxxtt

— UxxUtttt — UxxtUttt — — UttUttt

— UxUtttt — UxtUttt — — UtUttt

— UxUxttt — UxtUxttt UxxUxttt UxxtUxtt

— UxUxttt — UxtUxttt — UUxttt — UtUxtt

2

— — — — xUttt

— UxtUxxtt — UxxtUxtt

2

— UxxUxxtt — Uxxt — Uxtt — UxtUxtt UxUxxtt UxtUxxt

Uxxtt UtUxxt UUxxt + UxUxt UtUxx Utt — xux« — Uxt — tUxtt

— UUxxxt — UtUxxx

2

— — xuxxt + Uxt — Uxx — tUxxt UUxttt — UxUttt + UtUxtt — UxtUtt

— MMjtt — MjUtt — MjUxtt — UxtUtt

2

UUxxtt — UxUxtt + UtUxxt — uxi

— UUxtt — UtUxt — UtUtt

—— — + UUxtt — UxUtt — Uxxt

uuxxxt uxuxxt + utuxxx

UUxtt — UxUtt — tUxt — Ux — xUxt + Ut

— UUxt — UxUt — UUtt — UUtt

Density

t + U + Uxx + Ux

Uxttt + tUxxtt + xUxxtt

Uxxtt + xUxxxt

UxxxUxttt + UtttUxxxx

UxxxUxxtt + UxttUxxxx

uxxxuxxxt + uxxtuxxxx + uxxxuxxxx

+^xuxxxx

tUxxxx + xuxxx + UttUxttt + UxttUttt

UxtUxttt + UxxtUttt + Uxxxt

UxxUxttt + UxxxUttt + UtUxttt + UxtUttt

UxUxttt + UxxUttt + UUxttt + UxUttt 2

UX« + UttUxxtt

UxxUxxtt + UxxxUxtt

UxUxxtt + UxxUxtt + UUxxtt + UxUxtt

tUxtt + tUxtt

2

UxtUxxxt + uxxt

UxxUxxtt + uxxxuxxt + Uxxt + UxtUxxt

uxuxxxt + uxxuxxt

uuxxx + uuxxxt + uxuxxt

tuxxt + xuxxt

uuxxxx + uxuxxx

utuxtt + uxtutt + xuxxx + tuxxx

— uuxxtt + uxxutt

2

UUxtt + MxUtt + MtUxxi + Uxt

uuxxxt + uxxuxt uuxxt + uxuxt + utuxt

Ut + xUt + tUxt uuxxt + UtUxx + Uxxx 2

— UUxxxx + Uxx

tUxx + xUxx UUxxt + UtUxx UUxx + U2 + UUxt + UxUt

20 U3 + 10UUxx + 15 U2 — xUt + Uxxxx — Utt — 2xt xUx + Uxt + t2

Similarity reduction

In this section we make some discussion on the SK equation based on the symmetries (17), [5, 6, 16, 17].

(I) Vi

For the generator V1, we have

u = 9 (r, s), (22)

where r = x,s = t are the group-invariants. Substituting (22) in (1), one can get

9'"" + 1099'" + 2599" + 2092 9/ = 0, where 9/,9//,... are derivatives of 9 with respect to r.

(II) V2

For the generator V2 we get

u = 9 (r, s), (23)

where r = t,x = s are the group-invariants. Substituting (23) in (1), we reduce it to the following ODE

9/ = 0.

(II) V3

For the generator V2 we get

u = expf — 5sj 9(r, s), (24)

where

2

r = - 5 +12, s = - ln^/5(>/5x - 5t)

are the group-invariants. Substituting (24) in (1), we reduce it to the following so complicated equation!

exp (s(V5 — 5)) ^ 15625000exp ^ — —0 9/// + 187500000exp ( — 5s(V5 — 1)^9//

- 25000exp ^-2s(3^5- 2)r^ yffff + 250ex^-4s(2^5 - 1)^

+---- 17187500 exp ^—5s(2^5 — 1)^ r99// — 234375000^5 exp Q^ r29//

+460937500^5 exp ^4^ r39/// + 9765625exp ^4^ r59/w/j = 0.

(IV) V1 + V2

The symmetry V1 + V2 yields the following invariants

r = t — s, , s = x, u = 9 (r, s).

Treating 9 as the new dependent variable and r as new independent variable, the KS equation transforms to

9///// +1099/// + 2599// + (2092 + 1)9/ = 0.

Conclusion

In this paper, by using the Lie symmetry groups, we studied the symmetry properties and similarity reduction forms of the (1+1)-dimensional SK equation. Moreover, we also derived the non-linear self-adjonitness of Eq. (1), by virtue of this fact, some conservation laws through Lie symmetries are given. The direct method is used to find many new conservation laws for the Eq. (1) in the polynomial form.

References

[1] Alexandrova A. A., Ibragimov N.H., Imamutdinova K. V., Lukashchuk V. O., "Local and nonlocal conserved vectors for the nonlinear filtration equation", Ufa Math J., . 9:4 (2012), 179-85.

[2] Kerishnan E. V., "On Sawada-Kotera equations", II Nuovo Cimento B, 92:1, 23-26.

[3] Bluman G.W., Cheviakov A.F., Anco C., "Construction of Conservation Laws: How the Direct Method Generalizes Noether's Theorem", Proceeding of 4th Workshop "Group Analysis of Differential Equations & Integribility", 2009, 1-23.

[4] Bluman G. W., Cheviakov A. F., Anco C., Application of Symmetry Methods to Partial Differential Equations, Springer, New York, 2000.

[5] Bluman G.W., Cole J.D., "The general similarity solution of the heat equation", J. Math. Mech, 18 (1969), 1025-1042.

[6] Fushchych W. I., Popovych R. O., "Symmetry reduction and exact solutions of the Navier-Stokes equations", J. Nonlinear Math. Phys, 1:75-113 (1994), 156-188.

[7] Hejazi S. R., "Lie group analaysis, Hamiltonian equations and conservation laws of Born-Infeld equation", Asian-European Journal of Mathematics, 7:3 (2014), 1450040.

[8] Hydon P. E., Stmmetry Method for Differential Equations, Cambridge University Press, UK, Cambridge, 2000.

[9] Ibragimov N. H., Transformation group applied to mathematical physics, Riedel, Dordrecht, 1985.

[10] Ibragimov N.H., Aksenov AV., Baikov V. A., Chugunov V. A. , Gazizov R. K. and Meshkov A. G., CRC handbook of Lie group analysis of differential equations. Applications in engineering and physical sciences. V. 2, CRC Press, Boca Raton, 1995.

[11] Ibragimov N.H., "Nonlinear self-adjointness in constructing conservation laws", Arch ALGA, 7/8 (2010-2011), 1-99.

[12] Ibragimov N. H., Anderson R. L., "Lie theory of differential equations", Lie group analysis of differential equations. Symmetries, exact solutions and conservation laws.. V. 1, CRC Press, Boca Raton, 1994, 7-14.

[13] Ibragimov N.H., N Arch ALGA 2010-2011, 7/8, 1-99.

[14] Li J.B., Wu J.H. and Zhu H.P., "Travelling Waves for an Integrable Higher Order KdV Type Wave Equation", Int. J. Bifur Chaos Appl. Sci. Eng., 2006, 2235-2260.

[15] Nadjafikhah M., Hejazi S. R., "Symmetry analysis of cylindrical Laplace equation", Balkan journal of geometry and applications, 2009.

[16] Olver P. J., Equivalence, Invariant and Symmetry, Cambridge, Cambridge University Press, 1995.

[17] Ovsiannikov L. V., Group Analysis of Differential Equations, Academic Press, New York, 1982.

[18] Zwillinger D., Handbook of Differential Equations, Academic Press, Boston, 1997, 132 pp.

References (GOST)

[1] Alexandrova A. A., Ibragimov N.H., Imamutdinova K. V., Lukashchuk V. O. Local and nonlocal conserved vectors for the nonlinear filtration equation // Ufa Math J. 2012. vol. 9. issue 4. pp. 179-85.

[2] Kerishnan E. V. On Sawada-Kotera equations // II Nuovo Cimento B. vol. 92. issue 1, pp 23-26.

[3] Bluman G.W., Cheviakov A. F., Anco C. Construction of Conservation Laws: How the Direct Method Generalizes Noether's Theorem // Proceeding of 4th Workshop "Group Analysis of Differential Equations & Integribility". 2009. pp. 1-23.

[4] Bluman G.W., Cheviakov A.F., Anco C. Application of Symmetry Methods to Partial Differential Equations. New York: Springer, 2000.

[5] Bluman G.W., Cole J. D. The general similarity solution of the heat equation // J. Math. Mech. 1969. vol. 18. pp. 1025-1042.

[6] Fushchych W. I., Popovych R. O. Symmetry reduction and exact solutions of the Navier-Stokes equations // J. Nonlinear Math. Phys. 1994. vol. 1. no. 75-113. pp. 156-188.

[7] Hejazi S. R. Lie group analaysis, Hamiltonian equations and conservation laws of Born-Infeld equation // Asian-European Journal of Mathematics. 2014. vol. 7. no. 3. 1450040.

[8] Hydon P. E. Stmmetry Method for Differential Equations. UK. Cambridge: Cambridge University Press, 2000

[9] Ibragimov N. H. Transformation group applied to mathematical physics. Riedel, Dordrecht, 1985.

[10] Ibragimov N.H., Aksenov AV., Baikov V. A., Chugunov V. A. , Gazizov R. K. and Meshkov A. G. CRC handbook of Lie group analysis of differential equations. In: Ibragimov NH, editor. Applications in engineering and physical sciences, vol. 2. Boca Raton: CRC Press; 1995.

[11] Ibragimov N. H. Nonlinear self-adjointness in constructing conservation laws. Arch ALGA 2010-2011. 7/8. pp. 1-99.

[12] Ibragimov N.H., Anderson R. L. Lie theory of differential equations. In: Ibragimov NH, editor. Lie group analysis of differential equations, vol 1, Symmetries, exact solutions and conservation laws. Boca Raton: CRC Press; 1994. pp. 7-14.

[13] Ibragimov N.H., N Arch ALGA 2010-2011. 7/8. pp. 1-99.

[14] Li J.B., Wu J. H. and Zhu H. P. Travelling Waves for an Integrable Higher Order KdV Type Wave Equation // Int. J. Bifur Chaos Appl. Sci. Eng. 2006. pp. 2235-2260.

[15] Nadjafikhah M., Hejazi S. R. Symmetry analysis of cylindrical Laplace equation // Balkan journal of geometry and applications. 2009.

[16] Olver P.J. Equivalence, Invariant and Symmetry. Cambridge: Cambridge University Press, 1995.

[17] Olver P.J. Applications of Lie Groups to Differential equations. Second Edition. GTM, vol. 107. New York: Springer-Verlage, 1993.

[18] Ovsiannikov L. V. Group Analysis of Differential Equations. New York: Academic Press, 1982.

[19] Zwillinger D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997. p. 132.

Для цитирования: Hejazi S. R., Lashkarian E. Conservation laws and symmetry analysis

of (1+1)-dimensional Sawada-Kotera equation // Вестник КРАУНЦ. Физ.-мат. науки. 2017.

№ 3(19). C. 10-19. DOI: 10.18454/2079-6641-2017-19-3-10-19

For citation: Hejazi S. R., Lashkarian E. Conservation laws and symmetry analysis of

(1+1)-dimensional Sawada-Kotera equation, Vestnik KRAUNC. Fiz.-mat. nauki. 2017, 19: 3, 1019. DOI: 10.18454/2079-6641-2017-19-3-10-19

Поступила в редакцию / Original article submitted: 26.09.2017