Conservation Law s for CKdV and BSSK System s

KANG Zhouzheng

CollegeofM athematics,InnerM ongoliaUniversity forNationalities,

Tongliao028043,China.

Received 26 June 2015;A ccep ted 1 August 2015

Conservation Law s for CKdV and BSSK System s

KANG Zhouzheng?

CollegeofM athematics,InnerM ongoliaUniversity forNationalities,

Tongliao028043,China.

Received 26 June 2015;A ccep ted 1 August 2015

.In cu rrent paper,the coup led KdV(CKdV)system and Bosonized Supersymm etric Saw ada-Kotera(BSSK)system are considered.Som e linearly independent conservation law s for the tw o system s are derived via the fi rst hom otopy app roach and sym bolic com putation.

Coup led KdV system;Bosonized Supersymm etric Saw ada-Kotera system;conservation law s.

1 In troduction

It is w ell know n that one o f the im portant p roblem s in the fi elds o fm athem atics and physics is to construct asm any as possible conservation law s of nonlinear d ifferential system(NLDS).Conservation law s describeessentialphysicalp ropertiesof them odeled process.And in the study of d ifferential system,especially integrable system and soliton theory,conservation law s p lay a key role.Conservation law s have som e app lications in find ing exact solu tions,analyzing various characteristics of solu tions,and d iscussing the qualitative p ropertiessuch as thebi-or tri-Ham iltonian structures,Liouville integrability, recu rsion operators and so forth.A sequence ofm ethods,such as N ¨oether’s theorem [1],m ultip lierm ethod[2],the fi rst hom otopy m ethod[3,4],Lax pair m ethod[5]and others,have been w ell established and used.How ever,am ong thesem ethods,the fi rst hom otopym ethod has itsadvantage in dealing w ith com p licated form sofm ultip liersor equationsby using hom otopy operatorsarising from d ifferen tialgeom etry.

We begin in Section 2 by listing som e definitions and theorem s related w ith the fi rst hom otopy m ethod.In Section 3 and Section 4,w e app ly the fi rst hom otopy m ethodto construct som e linearly independen t conservation law s of CKdV and BSSK system s respectively.Finally,a shortsumm ary isgiven.

2 Prelim inaries

Conservation law s for a NLDSoforder k,

w ith independentvariables x ≡ (x1,x2,···,xn)and dependentvariables u ≡ (u1,u2,···, um),arew ritten by scalar d ivergence exp ressions

w here Φxi(i=1,2, ···,n)rep resen t fl uxes and totalderivative operators

In order to determ ine som e conservation law s for system(2.1),one fi rstly needs to find m ultip liers?Λα= Λα(x,U, ?U, ···, ?lU)?Nα=1,such thata linear com bination ofequations isa d ivergence exp ression,i.e.,

w h ich ho lds identically for arbitrary functions U(x).The fo llow ing defin itions and theorem s(see[3,4])are necessary.

Theorem 2.1.A set ofnon-singularmultipliers?Λα= Λα(x,U,?U,···,?lU)?Nα=1yields a local conservation law ofsystem(2.1)ifand only ifthe identities

hold forarbitrary functionsU(x).

Definition 2.1.Then-dimensional Euleroperatorw ith respect to U(x1,x2, ···,xn)isgiven by

w here U(k1+k2+···+kn)= ?k1+k2+···+knU/?k1x1 ?k2x2 ···?knxn.

Definition 2.2.The n-dimensional homotopy operator corresponding to independent variables xiisdefined by

where U(x)=?U1,U2,···,Um?,f isan exact differential function and for j=1,···,m,

Theorem 2.2.Assume that f is expressed asa divergence form

Ifexpressions(2.2)converge,then Φxi=H(xi)U(f)(i=1,2, ···,n)are the corresponding fluxes ofa conservation law.

3 CKdV system

First,w e study the CKdV system w hich is used to describe tw o-layer fl uids in d ifferent d ispersion relations.The requ irem ent b2=5 yielded that this system is a Painlev′e in tegrablem odelw ith d ifferen t linear d ispersion relations[6].Eqs.(3.1)and(3.2)w ere p roved to be also Lax integrable by a prolongation technique,and M iu ra transform ation and m od ified KdV equation associated w ith this system w ere p resen ted[7].In add ition,the sim ilarity so lu tions and reduction equationsw ere given via Clarkson and Kruskal’s d irectm ethod[8].

Herew e suppose thatm u ltip liers Λ1and Λ2are functionsw ith respect to(t,x,U,V, Ux,Vx).With theaid ofGeM[4],the determ ining equations for{Λ1,Λ2}are

The solu tionsare then obtained by solving theabove equations

w here C1and C2are arbitrary constants.Exp licitly,accord ing to the free constants,tw o cases can begiven as follow s.Herew eonly list the resu lts instead of derivations.

Case 1:By letting C1=1,C2=0,w ehave

For this case,the density and fl ux are

nam ely,a conservation law isgiven by

Case 2:By letting C2=1,C1=0,w ehave

For this case,the correspond ing density and fl ux are

Hence,the second conservation law isgiven by

4 BSSK system

The BSSK system reads

w hich w as derived through app lying bosonization m ethod to a SSK system.Moreover, Eq.(4.1)is the usualSK equation.Eq.(4.2)is linear nonhom ogeneous in v.Eqs.(4.3)and (4.4)are linear hom ogeneous in p and q respectively.In[9],the symm etry analysisw as perform ed to illustrate that thissystem is invariantunder scaling transform ations,spacetim e translations and Galilean boosts.And a seriesof reduction equationsand sim ilarity so lu tionsw ere p roposed.

In investigating the conservation law s for the BSSK system,w e assum e thatm ultip liers Λ1,Λ2,Λ3,and Λ4depend on(t,x,U,V,P,Q,Ux,Vx,Px,Qx).Then,the determ ining equations for the setofm u ltip liers{Λ1,Λ2,Λ3,Λ4}are

Solving the resu lting equations leads to w here C1,C2,C3,C4,and C5are arbitrary constants.Next,a case by case analysis for free constants can resu lt in linearly independent conservation law s.

Case 1:If C1=1,C2=C3=C4=C5=0,then w e can get

and the correspond ing density and fl ux

The fi rst conservation law of BSSK system is

Case 2:If C2=1,C1=C3=C4=C5=0,then w e can get

and the correspond ing density and fl ux

The second conservation law of BSSK system is

Case 3:If C3=1,C1=C2=C4=C5=0,then w e can get

and the correspond ing density and fl ux

The third conservation law of BSSK system is

Case 4:If C4=1,C1=C2=C3=C5=0,then w e can get

and the correspond ing density and fl ux

The fou rth conservation law isw ritten as

Case 5:If C5=1,C1=C2=C3=C4=0,then w e can get

and the correspond ing density and fl ux

Therefore,the fi fth conservation law isw ritten as

5 Conclusion

In conclusion,som e conservation law shavebeen found for the CKdV and BSSK system s bym eansof the fi rsthom otopym ethod and sym bolic com pu tation.Furtherm ore,theobtained conservation law sm ay be usefu l in considering the p roblem of doub le reduction [10,11]and others.

Acknow ledgm en ts

Thisw ork is supported by the NSFC(11462019)and the Scien tific Research Foundation of Inner M ongolia University for Nationalities(NMD 1306).The au thor w ou ld like to thank the referees for help fu l comm ents and suggestions.

Referen ces

[1]N¨oether E.,Nachr.Konig.Gesell.W iss.G ¨ottingen M ath.Phys.Kl.Heft,2(1918),235-257. Eng lish Translation in Transport Theory Statist.Phys.,1(3)(1971),186-207.

[2]Bisw as A.,Kara A.H.,M oraru L.,Bokhari A.H.and Zam an F.D.,Conservation law s of coup led K lein-Gordon equationsw ith cubic and pow er law non linearities.Proceedings ofthe Romanian academy,Series A,15(2)(2014),123-129.

[3]Herem an W.,Sym bolic com putation of conservation law s of non linear partial d ifferential equations in m u lti-d im ensions.International Journal ofQuantum Chemistry,106(1)(2006), 278-299.

[4]Cheviakov A.F.,Com putation of fl uxes of conservation law s.Journal ofEngineering M athematics,66(2010),153-173.

[5]Zhang D.J.,N ing T.K.,Conservation law sof integrab le system s.JournalofShanghaiUniversity(Natu ral Science Ed ition),12(1)(2006),19-30(in Chinese).

[6]Tong B.,Jia M.and Lou S.Y.,A new coup led KdV equation:Pain lev ′e test.Communications in Theoretical Physics,45(6)(2006),965-968.

[7]Wang D.S.,Com p lete in teg rability and the M iu ra transform ation of a coup led KdV equation.Applied M athem atics Letters,23(2010),665-669.

[8]Mao J.J.,Yang J.R.,Sim ilarity reductionsand exactsolu tionso fa coup led Kortew eg de Vries system.Communications in Theoretical Physics,53(4)(2010),605-608.

[9]Liu P.,Zeng B.Q.and Liu L.M.,Bosonized supersymm etric Saw ada-Koteraequations:symm etriesand exactsolu tions.Communications in Theoretical Physics,63(4)(2015),413-422.

[10]Han Z.,Zhang Y.F.and Zhao Z.L.,Doub le reduction and exact solu tions of Zakharov-Kuznetsov m od ified equal w id th equation w ith pow er law non linearity via conservation law s.Communications in Theoretical Physics,60(6)(2013),699-706.

[11]Eerdun Buhe,Fakhar K.,Wang G.W.and Tem uer Chaolu,Doub le reduction of the generalized Zakharov equations v ia conservation law s.Rom anian Reports in Physics,67(2)(2015), 329-339.

? Correspond ing au thor.Emailaddress:zhzhkang@126.com(Z.Z.Kang)

AM SSub ject Classifi cations:35A 25,35G20

Chinese Lib rary C lassifi cations:O 175.4