Some Recent Developments in Symmetries and Conservation Laws for Partial Differential Equations

Transcription

1 Some Recet Developmets i Symmetries ad Coservatio Laws for Partial Differetial Equatios This course will be cocered with recet developmets o how to fid ad use symmetries for PDEs; how to fid ad use coservatio laws for PDEs; coectios betwee symmetries ad coservatio laws for PDEs. Much of the material i this course will appear i a forthcomig book to be published i the Spriger Applied Mathematical Scieces series. Backgroud material appears i Symmetry ad Itegratio Methods for Differetial Equatios (Spriger 2002) by George W. Bluma ad Stephe C. Aco, which focused o Lie groups of trasformatios ad their applicatios to solvig ordiary differetial equatios (ODEs), the costructio of coservatio laws (itegratig factors) for ODEs, ad fidig ivariat solutios of PDEs. Topics to be selected from: 1. Local symmetries poit, cotact, higher-order. 2. How to fid local symmetries admitted by a give PDE system. 3. Noether s theorem ad its limitatios. 4. How to directly fid local coservatio laws of a give PDE system. 5. Coectios betwee symmetries ad coservatio laws. 6. Liearizatio of PDEs by ivertible mappigs through admitted symmetries; through admitted coservatio law multipliers. 7. How to systematically fid trees of equivalet but olocally related PDE systems for a give PDE system. 8. How to systematically fid olocal symmetries ad olocal coservatio laws for a give PDE system. Overview of the subject matter of this course. I the latter part of the 19 th cetury Sophus Lie iitiated his studies o cotiuous groups (Lie groups) i order to put order to, ad thereby exted systematically, the hodgepodge of heuristic techiques for solvig ODEs. He showed that the problem of fidig the Lie group of poit trasformatios (poit symmetries) admitted by a DE (ordiary or partial) reduced to solvig related liear systems of determiig equatios for its ifiitesimal geerators. Lie also showed that a admitted poit symmetry led to reducig the order of a ODE (irrespective of ay imposed iitial coditios) ad, i the case of a PDE, to fidig special solutios called ivariat (similarity) solutios. Moreover, he showed that a admitted poit symmetry geerates a oe-parameter family of solutios from a kow solutio of a DE. Most importatly, the applicability of Lie s work exteds to oliear DEs. This work is discussed i the above-metioed book as well as may other excellet refereces therei. The direct applicability of Lie s work to PDEs (especially oliear PDEs) is rather limited, eve whe a give PDE admits a poit symmetry, sice the resultig ivariat solutios yield oly a small subset of the solutio set of the PDE ad hece few posed boudary value problems ca be solved. The extesios of Lie s work to PDEs have focused o how to fid ad implemet further applicatios for admitted symmetries, how to exted the applicatios arisig from admitted symmetries to applicatios arisig from geeralizatios of symmetries (e.g., coservatio laws arisig from multipliers, solutios arisig from the 1

2 oclassical method ), how to exted the spaces of admitted symmetries, ad how to efficietly solve the (overdetermied) liear system of symmetry determiig equatios through the developmet of symbolic computatio software as well as related calculatios to fid multipliers for coservatio laws or to solve the oliear system of determiig equatios for the oclassical method. A symmetry of a PDE is ay trasformatio of its solutio maifold ito itself, i.e., a symmetry trasforms (maps) ay solutio of a PDE to aother solutio of the same PDE. Cosequetly, cotiuous symmetry trasformatios are defied topologically ad are ot restricted to admitted poit symmetries. Thus, i priciple, ay otrivial PDE admits symmetries. The problem is to fid ad use admitted symmetries. Practically, to fid a admitted symmetry oe must cosider trasformatios, actig o spaces of a fiite umber of variables, which leave ivariat the solutio maifold of the give PDE ad its differetial cosequeces. Oe such extesio is to cosider higher-order symmetries (local symmetries) where the solutios of the determiig equatios for symmetries are allowed to deped o a fiite umber of derivatives of the give depedet variables of the PDE (poit symmetries deped at most liearly o the first derivatives of the depedet variables ad cotact symmetries allow depedece at most o first derivatives). I makig this extesio, it is essetial to realize that the liear determiig equatios for local symmetries are the liearized system of the give PDE that holds for all of its solutios. Globally, poit ad cotact symmetries act o fiite-dimesioal spaces whereas higherorder symmetries act o ifiite-dimesioal spaces cosistig of the depedet ad idepedet variables as well as all of their derivatives. Well-kow itegrable equatios of mathematical physics such as the Korteweg-de-Vries equatio admit a ifiite umber of higher-order symmetries. Aother extesio is to cosider solutios of the determiig equatios that allow a ad-hoc depedece o olocal variables such as itegrals of the depedet variables. Usually such symmetries are foud formally through recursio operators that deped o iverse differetiatio. Itegrable equatios such as the sie-gordo ad cubic Schrodiger equatios admit a ifiite umber of such symmetries. I her celebrated 1918 paper, Emmy Noether showed that if a system of DEs admits a variatioal priciple, the ay local trasformatio group leavig ivariat the actio itegral for its Lagragia desity, i.e., a admitted variatioal symmetry, yields a coservatio law. Coversely, ay coservatio law admitted by a variatioal system of DEs arises from a variatioal symmetry, ad hece there is a direct correspodece betwee coservatio laws ad variatioal symmetries (Noether s theorem). There are several limitatios to Noether s theorem for fidig coservatio laws for a give system of DEs. First of all, it is restricted to variatioal systems. Cosequetly, for this theorem to be applicable to a give system as writte, the system must be of eve order, have the same umber of depedet variables as the umber of equatios i the give system, ad have o dissipatio. I particular, a give system of DEs is variatioal if ad oly if its liearized system is self-adjoit. There is also the difficulty of fidig symmetries admitted by the actio itegral. I geeral, ot all admitted local symmetries of a variatioal system of DEs are variatioal symmetries. Moreover, the use of Noether s theorem to fid coservatio laws is coordiate depedet. 2

3 A coservatio law of a give system of PDEs is a divergece expressio that vaishes o all solutios of the PDEs. I geeral all such divergeces arise from a scalar product formed by multipliers, depedig o ay idepedet ad depedet variables as well as at most a fiite umber of derivatives of the depedet variables of the system, with each PDE i the system. It the follows that a give system of PDEs has a coservatio law if ad oly if there exist multipliers whose scalar product with the PDEs i the system is idetically aihilated by the Euler operators associated with each of its depedet variables without restrictig these variables i the scalar product to solutios of the system of PDEs. If a give PDE system is variatioal the its multipliers are variatioal symmetries. I this case, it turs out that all multipliers satisfy the liearized PDE system augmeted by additioal determiig equatios that correspod to the actio itegral beig ivariat uder the associated variatioal symmetry. I geeral, all multipliers are the solutios of a liear determiig system that icludes the adjoit system of the liearized PDE system. For ay multiplier yieldig a coservatio law, there is a itegral formula that yields the fluxes ad desities of ay admitted coservatio law without eed of a specific Lagragia eve i the case whe the give system of DEs is variatioal. Aother importat applicatio of symmetries to PDEs is to determie whether or ot a give PDE ca be mapped ito a equivalet target PDE of iterest. This is especially sigificat if a target class of PDEs ca be completely characterized i terms of admitted symmetries. Target classes which such characterizatios iclude liear systems ad liear PDEs with costat coefficiets. Cosequetly, from the admitted poit or cotact symmetries of a give system of PDEs, oe ca determie whether or ot it ca be mapped ito a liear PDE by a poit or cotact trasformatio ad fid such a explicit mappig whe oe exists. Moreover, oe ca also see whether or ot such a liearizatio is possible from its admitted multipliers for coservatio laws. From the admitted poit symmetries of a liear PDE with variable coefficiets, oe ca determie whether or ot it ca be mapped by a poit trasformatio ito a liear PDE with costat coefficiets ad fid such a explicit mappig whe oe exists. I order to effectively apply symmetry methods to PDEs, oe eeds to work i some coordiate frame i order to perform calculatios. A systematic procedure to fid symmetries that are olocal ad yet are local i some related coordiate frame ivolves embeddig a give PDE system i aother PDE system obtaied by adjoiig olocal variables i such a way that the related PDE system is equivalet to the give PDE system ad the give PDE system arises through projectio. Cosequetly, ay local symmetry of the related system yields a symmetry of the give system. If the local symmetry of the related PDE system has a essetial depedece o the olocal variables after projectio, the it yields a olocal symmetry of the give PDE system. A systematic way to fid such a embeddig is through coservatio laws of a give PDE system of. For each coservatio law, oe ca itroduce potetials. By adjoiig the resultig potetial equatios to the give system, oe ca costruct a augmeted PDE system (potetial system). By costructio, such a potetial system is olocally equivalet to the give PDE system sice, through built i itegrability coditios, ay solutio of the give PDE system yields a solutio of the potetial system ad, coversely, through projectio ay solutio of the potetial system yields a solutio of the give PDE system. But this relatioship is olocal sice there is ot a oe-to-oe 3

4 correspodece betwee solutios of the give ad potetial systems. If a local symmetry of the potetial system depeds essetially o the potetial variables whe projected to the give PDE system, the it yields a olocal symmetry (potetial symmetry) of the give system. It turs out that may PDE systems admit such potetial symmetries. Moreover, oe ca fid further olocal symmetries of the give PDE system through seekig local symmetries of equivalet subsystems of the give system or potetial system provided such subsystems are olocally equivalet to the give PDE system. Ivariat solutios of such potetial systems ad subsystems ca yield further solutios of the give PDE system. Sice a potetial symmetry is a local symmetry of a potetial system, it geerates a oe-parameter family of solutios from ay kow solutio of the potetial system that i tur yields a oe-parameter family of solutios from a kow solutio of the give PDE system. Furthermore, coservatio laws of potetial systems (ad subsystems) ca yield olocal coservatio laws of give systems. Liearizatios of such potetial systems through local symmetry or coservatio law aalysis ca yield explicit olocal liearizatios of give systems of PDEs. Moreover, through a potetial system oe ca exted the mappigs of liear systems with variable coefficiets to liear systems with costat coefficiets to iclude olocal mappigs betwee such systems. Oe ca further exted embeddigs through usig coservatio laws to systematically costruct trees of olocally related but equivalet systems of PDEs. If a give PDE system has local coservatio laws, the each coservatio law yields potetials ad a correspodig potetial system. Most importatly, from the coservatio laws, oe ca directly costruct 2 1 idepedet olocally related systems of PDEs by cosiderig the correspodig potetial systems idividually ( siglets), i pairs (( 1)/2 couplets),, take all together (oe -plet). I tur, ay oe of these 2 1 systems could lead to the discovery of ew olocal symmetries ad/or olocal coservatio laws of the give PDE system or ay of the other 2 olocally related systems. Moreover, such olocal coservatio laws could yield further olocally related systems, etc. Furthermore, subsystems of such olocally related systems could yield further olocally related systems. Correspodigly, a tree of olocally related systems is costructed. Through such costructios, oe ca systematically relate Euleria ad Lagragia coordiate descriptios as well as fid other descriptios of gas dyamics ad oliear elasticity equatios. I both cases, subsystems of potetial systems arisig from their systems writte i Euleria coordiates yield the correspodig systems i Lagragia coordiates. For a give class of PDEs with costitutive fuctios, it is of iterest to classify its trees of olocally related systems ad correspodig symmetries ad coservatio laws with respect to various forms of its costitutive fuctios. Whe a system is variatioal, i.e., its liearized system is self-adjoit, the of course the coservatio laws arise from a subset of its symmetries ad, i particular, the umber of liearly idepedet coservatio laws caot exceed the umber of higher-order symmetries. But from the above, we see that this will ot be the case whe a system is ot variatioal. Here a costitutive fuctio could yield more coservatio laws tha symmetries as well as vice versa. For ay give system of PDEs, a admitted symmetry (cotiuous or discrete) yields a formula that maps a coservatio law to a coservatio law of the same system, 4

5 whether or ot the give system is variatioal. If the symmetry is cotiuous, the i terms of a parameter expasio a give coservatio law could map ito more tha oe ew coservatio law for the give system. Aother importat extesio relates to Lie s work o fidig ivariat solutios for PDEs. As metioed previously, a poit symmetry admitted by a PDE maps each of its solutios ito a oe-parameter family of solutios. But some solutios map ito themselves, i.e., they are themselves ivariat. Such solutios satisfy the characteristic PDE that is the ivariat surface coditio yieldig the ivariats for the poit symmetry. The ivariat solutios arisig from the poit symmetry are the solutios of the give PDE that satisfy the augmeted system cosistig of its characteristic PDE with kow coefficiets (obtaied from the poit symmetry) ad the give PDE itself. The ivariat solutios arise as solutios of a reduced system with oe less idepedet variable. This method ( classical method ) of Lie to fid ivariat solutios of a give PDE geeralizes to the oclassical method itroduced i 1967 where oe seeks solutios of a augmeted system cosistig of the give PDE ad the characteristic PDE with ukow coefficiets as well as differetial cosequeces of the characteristic PDE. Here the ukow coefficiets are determied by substitutig the characteristic equatio ito the determiig system for symmetries of the augmeted system. The resultig overdetermied system is oliear (eve if the give PDE is liear) i these ukow coefficiets, but less over-determied tha is the case whe fidig poit symmetries of the give PDE. Each solutio of the determiig system for poit symmetries is a solutio of the determiig system for the ukow coefficiets of the characteristic PDE. Solvig for the ukow coefficiets, oe the proceeds to fid the correspodig oclassical solutios of the augmeted system that, by costructio, iclude the classical ivariat solutios. The solutios of a PDE that ca be obtaied by the oclassical method iclude all of its solutios that satisfy a particular fuctioal form (asatz) of some geerality that allows a arbitrary depedece o a similarity variable (depedig o the idepedet ad depedet variables of the PDE) ad a arbitrary depedece o a fuctio of a similarity variable ad the idepedet variables of the PDE. The solutios obtaied by the oclassical method iclude all solutios obtaied directly from such a asatz by the direct method itroduced by Clarkso ad Kruskal i