TEMA (São Carlos) vol.14 número1

Uma Publiação daSoiedade BrasileiradeMatemátiaApliadaeComputaional.

A review on some results on loal onservation

laws for ertain evolution equations 1

I.L. Freire 2

, CMCC - Centro de Matemátia, Computação e Cognição, UFABC

-Universidade Federal do ABC,Rua Santa Adélia, 166, BairroBangu, 09.210-170

SantoAndré,SP,Brasil

J.C.S. Sampaio 3

, IMECC - Departamento de Matemátia Apliada, UNICAMP

-UniversidadeEstadualdeCampinas,RuaSérgioBuarquedeHolanda,651,Cidade

UniversitáriaZeferinoVaz,BarãoGeraldo,13083-859Campinas,SP,Brasil.

Abstrat. Inthis workwe revisitsome reentresults ononservationlaws for a

lassoffth-orderevolutionequationsuptofth-order.

Key-words. Ibragimov'stheorem,onservationlaws,fth-orderKdVequations.

1. Introdution

Thisworkorrespondstothetalk[6℄,givenbyJ.C.S.Sampaio,duringtheXXXIV

CNMAC -XXXIV CongressoNaional deMatemátiaApliadaeComputaional

-September,17-212012,ÁguasdeLindóia-SP,Brasil,where itwaspresentedand

disussed somereent results[5℄ with respet to how to obtainonservation laws

foranequationwithoutalassialLagrangian.

Itis well known that forequationsarisingfrom theEuler-Lagrangeequations,

theelebratedNoethertheoremprovidesanelegantwayforndingonserved

quan-tities for suh equation, see [1, 17, 20℄. However,when it is onsidered equations

withoutvariationalstruture, it is impossibleto apply Noether's approah. Then

thefollowingquestionarises: howanonendaonservedvetorforanequation

withoutLagrangians?

Arossthelastentury,manyapproaheshavebeenproposedin orderto

over-ame this problem. The interested readeris direted to thereferene [19℄, where

someof these developmentsweredisussed. Inthis paperweuse themost reent

approah proposed in [12℄, where it was shown ageneral result onneting

sym-metries and onservation laws, in order to nd onserved vetorsfor the general

equation

u

t

+

αuu

x

+

βu

2

u

x

+

γu

xxx

+

µu

xxxxx

= 0.

(1.1)

1

ThisworkwassupportedbyFAPESP,grantsno2011/23538-0and2011/19089-6.

2

igor.freireufab.edu.brandigor.leite.freiregmail.om.

3

Following[2℄, wewould liketo allthis new formulationfor nding onserved

vetorsasIbragimov'stheoremononservationlaws.

Inthis paperitisonsidered theequation(1.1)withthefollowingrestritions:

(α, β)

6

= (0,

0)

and

(γ, µ)

6

= (0,

0)

. Infat,in theasewhenever

(γ, µ) = (0,

0)

itis

obtainedapartiular ase ofthe invisidBurgers equation, whih wasonsidered

in [2, 4℄. Whenever

α

=

β

= 0

it is obtained a linear equation and here we are interestedinnonlinearphenomena.

Apartfromthepreviousmentionedequations,equation(1.1)inludes

•

TheelebratedKdVequation

u

t

+

αuu

x

+

γu

xxx

= 0;

(1.2)

•

GeneralKawaharaequation

u

t

+

αuu

x

+

γu

xxx

+

µu

xxxxx

= 0;

(1.3)

•

SimpliedmodiedKawaharaequation

u

t

+

βu

2

u

x

+

µu

xxxxx

= 0;

(1.4)

•

SimpliedKawaharaequation

u

t

+

αuu

x

+

µu

xxxxx

= 0

(1.5)

and

•

Gardner equation

u

t

+ 6(u

+

u

2

)u

x

+

u

xxx

= 0.

(1.6)

Allof theseequationsis omingfromthe elebratedKdV equation,whihwas

deduedbyKortewegandhisstudentde-Vriesformodelingshallowwaveequations.

These equations are employed in Mathematial Physisin orderto desribea lot

of dispersivephenomena, suh asplasmaphenomena, while equationsof thetype

(1.2)arealsousedtomodeltheGreatRedSpot ofJupiter,see[21℄.

Althoughour main purpose is to revisit someof our previousresults given in

[5, 21℄, in thepresentwork wealso presentsomenew onservation lawsfor

equa-tions of the lass (1.1). Namely, the new onserved vetors derived by using the

developments[12,15,16℄are

C

0

₌

_u

2

_,

C

1

₌

2

3

αu

3

_{+ 2µuu}

xxxx

−

2µu

x

u

xxx

+

µu

2

xx

,

(1.7)

forthesimpliedKawaharaequation(1.5),and

C

0

₌

₋

_u

₋

_u

2

_,

C

1

_{= 3u}

4

_{+ 6u}

3

_{+ 3u}

2

_{+ 2uu}

xx

+

u

xx

−

u

2

x

,

fortheGardnerequation(1.6).

The remaining of this paper is as the follows. In the next setion we revisit

theIbragimov'stheorem on onservationand the neessarytoolsto employ itfor

nding loal onservedvetors. Next,in the setion3, weshow that the lass of

equations(1.1)isnonlinearlyself-adjoint. Thisallowsustondloalonservation

lawsfor theseequations, whihwill bedonein thefollowingsetions. Some parts

ofthisreviewloselyfollowourreferenes[5,21℄.

2. Ibragimov's theory

Inthissetionwerevisitthebasitoolsaboutthereentdevelopmentsstartedwith

thefruitfulwork[12℄,whereIbragimovprovedanewonservationtheorem. Nextwe

introduethe reentonept of nonlinearself-adjointness, proposed by Ibragimov

in[15,16℄.

Thiseld, nowadays,is arih branh in the eld of groupanalysis and ithas

been attrating the attention of a big number of researhers. Many of them are

interested in nding nonlinearly self-adjoint properties of equations, as it an be

seenin[2,3,4,5,7,8,9,10,11, 13,14, 22℄andreferenestherein.

2.1. Adjoint equations and nonlinear self-adjointness

Let

x

= (x

1

_,

_{· · ·}

_{, x}

n

₎

be

n

independentvariables,

u

=

u(x)

beadependentvariable. Thesetof

k

thorderderivativesof

u

isdenotedby

u

(k)

,where

k

inapositiveinteger number. Considerthesetoffuntions depending on

x, u

and

u

derivativesupto a niteorder.

Aloallyanalytifuntion ofanitenumberofthevariables

x, u

and

u

deriva-tivesisalled adierentialfuntion. Thehighestorder ofderivativesappearingin

thedierentialfuntionisalledtheorderofthisfuntion. Thevetorspaeofall

dierentialfuntionsofniteorderis denotedby

A

.

Theformalsum

δ

δu

=

∂

∂u

+

∞

X

j=1

(

−

1)

j

_D

i

₁

· · ·

D

i

j

∂

∂u

i

₁

···

i

_j

(2.9)

isthewellknownEuler-Lagrangeoperator.

Let

F

∈ A

be a dierential funtion. From this funtion we an obtain a dierentialequation

F

(x, u,

· · ·

, u

(s)

) = 0

(2.10)

andthefollowingnewdierentialfuntion,alled formalLagrangian,givenby

L

=

vF

, where

v

=

v(x)

isanotherdependentvariable.

Equation

(2.10)

is said to be nonlinearly self-adjoint if the equation obtained fromtheadjointequation

F

∗

(x, u, v,

· · ·

, u

(s)

, v

(s)

) :=

δ

L

bythesubstitution

v

=

φ(x, u)

withaertainfuntion

φ(x, u)

6

= 0

isidentialwith theoriginalequation

(2.10)

,that is,

F

∗

(x, u, v, u

(1)

, v

(1)

,

· · ·

, u

(s)

, v

(s)

)

_v=φ(x,u)

= 0.

(2.12)

Whenever

(2.12)

holdsforaertaindierentialfuntion

φ

suhthat

φ

u

6

= 0

and

φ

x

6

= 0

,equation

(2.11)

isalled weakself-adjoint.

In other words: equation (2.11) is said to be nonlinearly self-adjoint if there

existsafuntion

φ

=

φ(x, u)

suhthat

F

∗

|

_v=φ

=

λ(x, u,

· · ·

)F,

(2.13)

forsomedierentialfuntion

λ

=

λ(x, u,

· · ·

)

.

2.2. Ibragimov's theorem on onservation laws

Thefollowingresultwasprovedin[12℄.

Theorem2.1 (Ibragimov'stheoremononservationlaws). Let

X

=

ξ

i

∂

∂x

i

+

η

∂

∂u

be any symmetry (Lie point, Lie-Bäklund, nonloal symmetry) of equation

(2.10)

and

(2.11)

be the adjoint equation toequation

(2.10)

. The ombined system

(2.10)

and

(2.11)

hasthe onservation law

D

i

C

i

_{= 0}

,where

C

i

₌

_ξ

i

_L

₊

_W

_∂

_L

∂u

i

−

D

j

_∂

_L

∂u

ij

+

D

j

D

k

∂

L

∂u

ijk

− · · ·

+D

j

(W

)

_∂

_L

∂u

ij

−

D

k

_∂

_L

∂u

ijk

+

· · ·

+D

j

D

k

(W

)

_∂

_L

∂u

ijk

− · · ·

+

· · ·

(2.14)

and

W

=

η

−

ξ

i

_u

i

.

2.3. Algorithm

Ibragimov'stheorem on onservation laws an be resumed by the following

algo-rithm(see[12,2℄forfurther details): givenaPDE

F

=

F

(x, u, u

(1)

· · ·

, u

(n)

) = 0,

•

weonstrutaLagrangian

L

=

vF

.

•

FromtheEuler-Lagrangeequations,thefollowingsystemisobtained:

F

(x, u, u

(1)

· · ·

, u

(n)

) = 0,

(2.15)

F

∗

(x, u, v,

· · ·

, u

(s)

, v

(s)

) = 0.

(2.16)

3. Nonlinearlyself-adjoint lassiation ofthe

equa-tion (1.1)

Thefollowingtheoremwasprovedin [5℄ (seealso[21℄).

Theorem 3.2. Equation

(1.1)

, with

(γ, µ)

= 0

6

and

(α, β)

6

= (0,

0)

, is nonlinearly

self-adjoint.

Remarks: Equation(1.1)isalsononlinearlyself-adjointwithouttherequested

hypothesis in the Theorem 3.2. The mentionedhypothesis only reets the

non-linearityoftheequationandthefat thatweareonsideringdispersiveequations.

However,astraightforwardalulationshowsthatweanremovethem. Weleave

thedetailstotheinterestedreader. Itanalsobeuseful,onegoinginthisdiretion,

tosee [2,3,4℄.

Proof. Letusdenotetheleftsideof(1.1)by

F

=

u

t

+

αuu

x

+

γu

xxx

+

µu

xxxxx

.

(3.17)

Then,theadjointequationto

F

= 0

is

F

∗

:=

−

v

t

−

(αu

+

βu

2

)v

x

−

γv

xxx

−

µv

xxxxx

= 0.

(3.18)

Nowweonlymustanalyzethefollowingtwoases. Below,

a

1

, a

2

and

a

3

are arbi-traryonstants.

•

Assume

β

= 0

in (1.1). Thenthesubstitution

φ(x, t, u) =

a

1

(x

−

αtu) +

a

2

u

+

a

3

in (3.18)makestheadjointequationequivalentto theoriginalequation.

•

Nowsuppose

β

6

= 0

in(1.1). Substituting

φ(x, t, u) =

a

1

u

+

a

2

insteadof

v

in (3.18), weobtainamultipleof(1.1).

Thisprovesourstatement.

4. Non-loal onservation laws

Here we nd nonloal onservation laws for the equation (1.1). We follow the

algorithmpreviouslydisussedinSetion2.3.

Atually,thersttwostepsofsuhamentionedalgorithmhavejustbeendone

beausetheformalLagrangianhadalreadybeenobtainedwhiletheadjointequation

Withrespetto theseond step,theomponentsofthevetoraregivenby

C

0

₌

_τ

_L

₊

_{W v,}

C

1

₌

_ξ

_L

₊

_W

v(αu

+

βu

2

_{) +}

_γv

xx

+

µv

xxxx

−

D

x

(W

)(γv

x

+

µv

xxx

) +

D

x

2

(W

)(γv

+

µv

xx

)

−

D

3

x

(W

)(µv

x

) +

D

x

4

(W

)(µv),

(4.19)

where

L

=

vF

,

W

=

η

−

τ u

t

−

ξu

x

and

X

=

τ(x, t, u)

∂

∂t

+

ξ(x, t, u)

∂

∂x

+

η(x, t, u)

∂

∂u

isanyLiepointsymmetryof(1.1).

Theobtainedonservedvetordependson

v

beauseitisaonservedvetorto thesystem







u

t

+

αuu

x

+

βu

2

u

x

+

γu

xxx

+

µu

xxxxx

= 0,

−

v

t

−

(αu

+

βu

2

)v

x

−

γv

xxx

−

µv

xxxxx

= 0.

Suhaonservedvetor

C

= (C

0

_{, C}

1

₎

is, then,anonloalonservedvetor.

5. Loal onservation laws

AordingtoIbragimov'stheoremononservationlaws(seealsotheorresponding

algorithm), a onserved vetor to (1.1) and (3.18) is

C

= (C

0

_{, C}

1

₎

, whose the

omponents are given by (4.19) and, as it was already pointed out, a nonloal

onservedvetortotheoriginalequation.

Inordertondloal onservedvetors,weusetheLiepointsymmetriesfound

in [18, 21℄ and the fat that under the substitutions given by Theorem 3.2, the

nonloalonservedvetorsbeomealoalonservedvetor. Weillustrate thisfat

at the same time that weproeed ouralulation for establishingthe elds (1.7)

and(1.8).

5.1. Conservation law for the simplied Kawahara equation

ConerningthesimpliedKawaharaequation(1.5),byusingthedilational

symme-try

X

=

x

∂

∂x

+ 5t

∂

∂t

−

4u

∂

foundin[18℄, from(4.19)weobtain

C

0

₌

₋

_4vu

₋

_xvu

x

+ 5αtvuu

x

+ 5µtvu

xxxxx

,

C

1

₌

_xvu

t

−

4αvu

2

−

4µuv

xxxx

−

µxu

x

v

xxxx

−

5αtvuu

t

−

5µtu

t

v

xxxx

+ 5µu

x

v

xxx

+

µxu

xx

v

xxx

+ 5µtu

xt

v

xxx

−

6µv

xx

u

xx

−

µxv

xx

u

xxx

−

5µtv

xx

u

xxt

+ 7µv

x

u

xxx

+µxv

x

u

xxxx

+ 5µtv

x

u

xxxt

−

8µvu

xxxx

−

5µtvu

xxxxt

.

(5.21)

Substituting

v

=

x

−

αtu

intotheomponentsabove,wearrivedat

C

0

=

−

2xu

+

αtu

2

+

D

x

−

x

2

u

+ 3αxtu

2

−

5

3

α

2

_t

2

_u

3

+D

x

5µxtu

xxxx

−

5µtu

xxx

−

5αµt

2

uu

xxxx

+ 5αµt

2

u

x

u

xxx

−

5

2

αµt

2

_u

2

xx

,

C

1

₌

₋

_αxu

2

₊

2

3

α

2

_tu

3

₋

_2µxu

xxxx

+ 2µu

xxx

+ 2αµtuu

xxxx

−

2αµtu

x

u

xxx

+

αµtu

2

xx

+

D

t

x

2

u

−

3αxtu

2

+

5

3

α

2

_t

2

_u

3

D

t

−

5µxtu

xxxx

+ 5µtu

xxx

+ 5αµt

2

uu

xxxx

−

5αµt

2

u

x

u

xxx

+

5

2

αµt

2

_u

2

xx

Onetransferredtheterm

D

x

(

· · ·

)

from

C

0

to

C

1

,wendtheonservedvetor

C

= (C

0

_{, C}

1

₎

,where

C

0

₌

₋

_2xu

₊

_αtu

2

_,

C

1

₌

₋

_αxu

2

₊

2

3

α

2

_tu

3

₋

_2µxu

xxxx

+ 2µu

xxx

+ 2αµtuu

xxxx

−

2αµtu

x

u

xxx

+

αµtu

2

xx

forthesimpliedKahawaraequation.

Now,substituting

v

=

u

into(5.21), itisobtained

C

0

₌

₋

_4u

2

₋

_xuu

x

+ 5αtu

2

u

x

+ 5µtuu

xxxxx

,

C

1

₌

_xuu

t

−

4αu

3

−

12µuu

xxxx

−

5αtu

2

u

t

−

5µtu

t

u

xxxx

+ 12µu

x

u

xxx

+

andthen,

C

0

₌

₋

7

2

u

2

₊

_D

x

−

xu

2

2

+

5

3

αtu

3

_{+ 5µtuu}

xxxx

−

5µtu

x

u

xxx

+

5

2

µtu

2

xx

,

C

1

₌

₋

7

3

αu

3

₋

_7µuu

xxxx

+ 7µu

x

u

xxx

−

7

2

µu

2

xx

−

D

t

−

xu

2

2

+

5

3

αtu

3

_{+ 5µtuu}

xxxx

−

5µtu

x

u

xxx

+

5

2

µtu

2

xx

.

Afterastraightforwardalulationand multiplyingthenalresultby

−

7/2

,it

isobtainedthevetor(1.7).

5.2. Conservation law for the Gardner equation

Inorder to establishtheonservedvetor(1.8)for theGardner equation, werst

substitutetheomponentsofthefollowingLiepointsymmetrygenerator

X

3

= (2x

+ 6t)

∂

∂x

+ 6t

∂

∂t

−

(2u

+ 1)

∂

∂u

intotheEq. (4.19)with

µ

= 0, α

=

β

= 6

and

γ

= 1

. Thus

C

0

₌

₋

_6vt(6u

_{+ 6u}

2

_)u

x

−

6vtu

xxx

−

(2u

+ 1)v

−

(2x

+ 6t)vu

x

,

C

1

₌

_(2x

_{+ 6t)vu}

t

−

(2x

+ 6t)vu

xxx

+(2u

+ 1)(6u

+ 6u

2

_)v

_{+ (2u}

_{+ 1)v}

xx

+ (2x

+ 6t)v

xx

u

x

+6t(6u

+ 6u

2

_)vu

t

+ 6tv

xx

u

t

−

4v

x

u

x

−

(2x

+ 6t)v

x

u

xx

−

6tv

x

u

xt

+ 6vu

xx

+ (2x

+ 6t)vu

xxx

+ 6tvu

xxt

.

(5.22)

Now,setting

v

=

u

into

(5.22)

,itisobtained

C

0

₌

₋

_36tu

2

_u

x

−

36tu

3

u

x

−

6tuu

xxx

−

2u

2

−

u

−

2xuu

x

−

6tuu

x

,

C

1

₌

_2xuu

t

+ 6tuu

t

+ 18u

3

+ 12u

4

+ 6u

2

+ 2uu

xx

+

u

xx

+ 36tu

2

u

t

+36tu

3

_u

t

+ 6tu

t

u

xx

−

4(u

x

)

2

−

6tu

x

u

xt

+ 6uu

xx

+ 6tuu

xxt

,

whihareequivalentto

C

0

₌

₋

_u

₋

_u

2

₋

_D

x

(xu

2

+ 3tu

2

+ 6tuu

xx

−

3t(u

x

)

2

+ 9tu

4

+ 12tu

3

),

C

1

₌

_3u

4

_{+ 6u}

3

_{+ 3u}

2

_{+ 2uu}

xx

+

u

xx

−

(u

x

)

2

+D

t

(xu

2

+ 3tu

2

+ 6tuu

xx

−

3t(u

x

)

2

+ 9tu

4

+ 12tu

3

).

6. Conlusion

Inthis review paperwehaverevisited somereent resultsonnetingLie

symme-triesandonservationlawsforequationsnotneessarilybeingfromEuler-Lagrange

equations. Moreover,wehaveillustratedourdisussionwithsomenewonservation

lawsobtainedviaIbragimov'sapproah. Suhonservationlawsareestablished,at

leastusingsuhanewdevelopment,forthersttimein thepresentreview.

Theinterested readeran foundmore loal onservation laws forequations of

the type (1.1), where we established onservation lawsfor the generalKawahara

equation(1.3)andmodiedKawaharaequation(1.5). Moreover,someonservation

lawsfortheKdVequationanalsobeestablishedusingthoseobtainedresults,see

also [16℄. Morereently,somenewlassesof evolutionequations upto fthorder

havebeendisovered,see[7℄.

Resumo. Nestetrabalhorevisitamosalgunsresultadosreentessobreleisde

on-servaçãodeumalassedeequaçõesevolutivasatéquintaordem.

Palavras-have. Teorema de Ibragimov, leis deonservação, equaçõesKdV de

quintaordem.

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