Conservation laws and hierarchies of potential symmetries for certain diffusion equations

Conser v ation la ws and hie rarc hies of p oten tial symmetries for certain diffu sion equations N.M. Iv anov a † , R.O. P op o vych ‡ , C. Sophocleous ♦ and O.O. V aneev a § † ‡ § Institute of Mathematics of NAS of Ukr aine, 3 T er eshchenkivska Str., 01601 Kyiv, Ukr aine e-mail: ivanova@imath.kiev.ua, r op@imath .ki ev.ua, vane eva@imath.kiev.ua ‡ F akult¨ at f¨ ur Mathematik, U niversit¨ at Wien, Nor db er gstr aße 15, A-1090 Wien, Austria ♦ Dep artment of Mathema tics and Statistics, University of Cyprus, CY 167 8 Nic osia, Cyprus e-mail: christo d@ucy.ac.cy W e sho w that the so-called hidden po ten tia l symmetries considered in a recent pap e r [1 8] are ordinary p otential symmetries that ca n b e obtained using the method in tr o duced by Bluman and collab or ators [7, 8]. In fa ct, these a re s imples t p otential s ymmetries asso ci- ated with po ten tia l sys tems whic h are co nstructed with single cons erv ation laws ha v ing no constant characteristics. F ur ther more we classify the conserv ation laws for cla sses of p oro us medium equations and then using the co r resp onding conserved (po ten tial) systems we sear ch for p otential symmetries. This is the approach one nee ds to adopt in order to determine the complete list of p otential symmetries. The provenance of p otential symmetr ies is ex- plained for the p or ous medium equa tions by using p otential equiv alence transforma tions. Poin t a nd p otential eq uiv a lence transformatio ns are also applied to deriving new results on po ten tia l symmetries and corr e spo nding inv aria nt solutions from k nown ones . In particula r, in this way the potential systems, potential conserv ation laws a nd potential symmetries o f linearizable equations fr om the classe s of differential equations under co nsideration are ex- haustively describ ed. Infinite series of infinite- dimens io nal alg ebras of p o tent ia l s ymmetries are constructed for such eq uations. Keyw ords : P oten tial symmetries; Conserv ation la ws; Diffusion equations; Pote ntial equiv alence transformations 1 In tro du ction A Lie sym metry group of a system of differential equations is a group of tr an s formations that dep end on conti nuous parameters and map any solution to another solution of the system. While there is n o existing general theory f or s olving nonlinear equations, employmen t of the concept of Lie sy m metry has b een v ery h elpful in d etermining new exact solutions. Details on the theory of Lie symmetry groups and their app licatio n s to differen tial equ ations can b e found in a num b er of textb o oks. See for example, [7, 29, 30]. Bluman et al. [7, 8] introd uced a metho d f or finding a new class of symmetries for a system of partial differen tial equations ∆( x, u ) , in the case that this system has at least one conserv ation la w. If we introdu ce p oten tial v ariables v for th e equations w ritten in conserved form s as fur th er unknown fu nctions, we obtain a system Z ( x, u, v ). Any Lie sym metry for Z ( x, u, v ) indu ces a symmetry for ∆( x, u ). When at least one of th e infinitesimals whic h corresp ond to the v ariables x and u dep ends explicitly on p otent ials, then the lo cal symmetry of Z ( x, u, v ) induces a n onlo cal symmetry of ∆( x, u ). These nonlo cal symmetries are called p otential symmetries . More details ab out p oten tial sym metries and their applications can b e found in [7 , 9, 10]. P otenti al symm e- tries were inv estigate d for quite general classes of d ifferen tial equations. T he problem of finding criteria for the existence of p oten tial s y m metries f or classes of d ifferential equations was p osed in [37 ]. Some u seful criteria were derive d for partial d ifferen tial equations in t wo ind ep endent v ariables. Nonclassical p oten tial s ymmetries of suc h equations we r e discussed in [39]. 1 F or p artial differen tial equations in tw o in dep endent v ariables, t and x , the general form of (lo cal) conserv ation laws is D t F + D x G = 0 , (1) where D t and D x are the total deriv ativ es with r esp ect to t and x . Th e equalit y (1) is assum ed to b e satisfied for any solution of the corr esp onding system of equations. The comp onents F and G of the conserved v ector ( F , G ) are functions of t , x and deriv ativ es of u and are called the density and the flux of the conserv ation la w. B asic defin itions and statemen ts on conserv ation la ws (conserved v ector, equiv alence of conserv ed ve ctors, c haracteristic, No ether’s theorem etc.) are present ed, e.g., in [29]. In the recen t p ap er [18] th e construction of hidd en p otent ial symmetries for some classes of diffusion equations is claimed. Here we sho w that these symmetries are usu al p otenti al symmetries that can b e derived using the con ven tional m etho d b y Bluman and collab orators. In f act, th e wa y prop osed in [18] for obtaining p oten tial systems is a particular case of fin ding c haracteristics of conserv ation laws of order zero and the so-calle d hid den p oten tial symmetries are ord inary simplest p oten tial symmetries corresp onding to p oten tial s y s tems constructed with conserv atio n la ws w h ose charact eristics are nonconstan t. These sy m metries are simp lest since eac h of them is asso ciated w ith a single conserv ation la w and hence inv olv es only a single p oten tial. F urthermore p oten tial symmetries der ived in [18] were also found in [42]. Essen tially more general results on p otentia l conserv ation laws and p oten tial symmetries of w ider classes of p orous medium equations including that with an arbitrary num b er of p oten tials were obtained in [23, 24, 36]. A pr o cedure of classification of related quasilo cal s ymmetries w as prop osed in [46] for the general class of (1+1)- d imensional evo lution equations. In the subsequ ent analysis, we examine eac h diffu sion equ ation considered in [18] in m ore detail. W e also stud y a class of p orous medium equations which w as stated in [18 ]. Sp ecial p oten tial s y m metries for a p artial case of su c h equations were earlier d eriv ed in [17]. The firs t step in the inv estigat ion of p otentia l symm etries is to calculate the conserv ation la ws. The conv en tional symmetry app roac h for this is based on No ether’s theorem bu t it cannot b e directly u sed for ev olution equ ations. Ther e exists no Lagrangian f or which an ev olution equation is an E u ler–Lagrange equation. Hence the app licatio n of No ether’s theorem in this case is p ossible only for particular equations and after sp ecial tec hnical tric ks. A t th e same time, the definition of conserv ation la ws itself give s rise to a metho d of fin ding conserv ation la ws, which is called dir e c t and can b e applied to an y system of differen tial equations w ith no restriction on its structure. T he tec hnique of calculations u sed w ithin the framewo rk of this metho d is similar to the classical Lie metho d yielding symm etries of d ifferential equations. F our ve r sions of it are d istinguished in the literature dep endin g on the wa y of taking in to account systems under consideration and the usage either the definition of conserv ed v ectors or the c haracteristic form of conserv ation la ws. See, e.g., [4, 5, 33] on details of th e calculatio n tec hn iqu e as w ell as [45] for comparison of the v ersions and their realizatio n s in computer algebra programs. The necessary theoretical bac kgroun d is giv en in [29]. In the present wo r k we emp lo y the most direct v ersion [33 ] b ased on immediate solving of determinin g equations for conserved vecto rs of conserv atio n s la ws on the s olution manifolds of in vestig ated systems and additionally com bin ed with tec hniques inv olving symmetry or equiv alence transformations. A complete classification of p oten tial symmetries can b e ac h iev ed b y considering all p oten tial systems that corresp on d to th e conserv ation la ws. It is known [33] that th e equ iv alence group for a class of systems of differentia l equations or the sym metry group for a single system can b e pr olonged to p otentia l v ariables. I t is natural to use these prolonged equiv alence groups for classification of p ossib le p otenti al symmetries. In view of this statemen t w e will classify p oten tial symmetries of diffusion equations up to th e (trivial) prolongatio n of their equiv alence groups to the corresp ondin g p otentia ls. 2 In the n ext four sections we use classes of p orous medium equations as examp les and we find the firs t generation of p oten tial symmetries. In the end of section 3 and in section 6 we present applications of p otent ial equiv alence transformations. I n particular, a connection b e- t we en p oten tial and Lie symmetries is established, n ew exact solutions are found and exh austiv e description of p oten tial symmetries of linearizable d iffusion equatio n s is giv en. In fi nite series of infinite-dimensional algebras of p oten tial sym metries are constructed for suc h equations. 2 F okk er–Planc k equations The class of the F okke r –Planck equations u t = u xx + [ f ( x ) u ] x (2) is con tained in the class of (1+1)-dimensional second-order linear ev olution equations which ha ve the general form u t = A ( t, x ) u xx + B ( t, x ) u x + C ( t, x ) u, (3) where the co efficient s A , B and C are smo oth fun ctions of t and x , A 6 = 0. Due to its imp ortance and r elativ e simplicit y , class (3) is th e most inv estigated in the fr amew ork of group analysis of differen tial equations. Equ ations from this class are often used as examples f or the first presen- tations of inv estigat ions on different n ew kinds of symmetries, illustrativ e examples in textb o oks on the su b ject and b enc hmark examples for computer programs calculating sym metries of d if- feren tial equ ations. In fact, the complete group classification of (1+1)-dimensional linear parab olic equations (i.e., th e complete d escription of their Lie sym metries up to the equ iv alence relation generated b y the corresp onding equiv alence group) wa s p erformed by S oph u s Lie [26] h im s elf as a part of the more general group classification of linear second-order partial differentia l equations in tw o indep end en t v ariables. A mo d ern treatmen t of the sub ject is give n in [30]. T h ere exist also a n u m b er of recen t pap ers partially redisco vering the classical results of Lie and Ovsiann ik o v. Th e lo cal conserv ation la ws of equations fr om class (3) were calculated in [11] using d ifferential forms. This result w as reobtained in [36] by the direct metho d of calculations of conserv ation la ws. First the w hole p oten tial symmetry f rame o ver class (3 ) including lo cal and p oten tial con- serv ation la w s and us u al and generalized p oten tial symmetries w ith an arb itrary num b er of indep end en t p oten tials of any lev el wa s inv estiga ted in [36] with com b in ing a num b er of so- phisticated tec h niques su c h as a r ather intricate int er p la y b et w een differen t repr esen tations of p oten tial systems, the notion of a p otenti al equation asso ciated with a tup le of characte r istics, prolongation of the equ iv alence group to the whole p otent ial frame and application of multiple dual Darb oux transformations. In p articular, all p ossible p oten tial conserv ation la ws of equ a- tions f rom class (3 ) p ro ve d to in fact b e exhausted by lo cal conserv ation la ws . Based on th e to ols dev elop ed, a preliminary analysis of generalized p otenti al symmetries is carried out and then applied to substantia te the choic e of canonical forms for p oten tial sys tems. Effecti ve criteria for the existence of p oten tial sy m metries are pr op osed for the case of an arb itrary num b er of in vol ved p oten tials. Equations f r om class (3), p ossessin g infi nite ser ies of p oten tial symm etry algebras, are studied in detail. Nonclassical (conditional) symmetries of equations from class (3) and all the p ossible reduc- tions of these equations to ordinary differen tial ones are exhaustive ly describ ed in [32]. This problem prov es to b e equiv alen t, in some sense, to solving the in itial equations. The “no-go” result is extend ed to the inv estigat ion of p oin t transformations (adm iss ible transformations, equiv alence transform ations, Lie symmetries) and Lie redu ctions of the determining equations for the nonclassical symmetries. 3 There exist a n u m b er of earlier pap ers on inv estigation of particular p oten tial or nonclassical symmetries of narro w sub classes of equations from class (3) or even sin gle equations f rom this class (see references in [32, 36]). Belo w w e d iscuss only some results whic h are directly related to class (2). Note that any pair ( L , F ), where L is an equation f rom class (3 ) and F is a conserv ation la w of it, is red uced by a p oint transform ation from the equiv alence group of class (3 ) to a pair ( ˜ L , ˜ F ), where ˜ L is a F okk er–Planc k equ ation u t = u xx + ( F ( t, x ) u ) x and ˜ F is its conserv ation la w with the c h aracteristic 1 (i.e., w ith the charact eristic which is identic ally equal to 1) [36, Prop osition 8]. Class (2) admits the generalize d extended equiv alence group formed by the transform ations ˜ t = δ 2 1 t + δ 2 , ˜ x = δ 1 x + δ 3 , ˜ u = ψ u + ψ ζ , ˜ f = f δ 1 − 2 δ 1 ψ x ψ , where ψ = δ 4 R e R f ( x ) dx dx + δ 5 ; δ i , i = 1 , . . . , 5 , are arbitrary constants, δ 1 ( δ 2 4 + δ 2 5 ) 6 = 0; ζ = ζ ( t, x ) is an arbitrary solution of (2 ). Eac h lo cal conserv ation la w of (2) is canonically r epresen ted by a conserv ed v ector ( αu, − αu x + ( α x − αf ) u ), where α = α ( t, x ) is an arbitrary s olution of the adjoin t equation α t + α xx − f α x = 0 [38]. Then, in view of Theorem 5 and Corollaries 20 and 30 of [36] any p otentia l system of (2), which is essenti al for find ing p oten tial sym metries, has the form v i x = α i u, v i t = α i u x − ( α i x − α i f ) u, i = 1 , . . . , k , (4) where k ∈ N and α i = α i ( t, x ), i = 1 , . . . , k , are arbitrary linearly indep enden t solutions of the adjoin t equation. T he same statemen t f or th e particular case of the linear h eat equ ation was earlier obtained in [33]. In [38] the p otenti al sys tems of the f orm (4) w ith a single p otent ial (the case k = 1) w ere considered. T h eir Lie sy m metries we r e p r eliminary in vestig ated. The so-called n atural p oten tial systems asso ciated with the c h aracteristic 1 were studied in more detail. As a resu lt, the corresp onding s p ecial simplest p oten tial s ymmetries were completely classified. (W e call simplest p otential symmetries ones arising under the consideration of p oten tial system with a sin gle p oten tial.) T he p otenti al symmetries of th e same kind we r e also foun d in [37] for the particular equation of form (2) with f = x , i.e., u t = u xx + ( xu ) x . It wa s shown in [20] that these p oten tial symmetries are obtained from the “natural” simp lest p otent ial symmetries of the linear heat equation u t = u xx via a p oint transformation connecting the equ ations u t = u xx and u t = u xx + ( xu ) x . Recen tly these results w ere essentia lly generalized in [36 ]. Any c h aracteristic of the linear heat equations, w hic h giv es rise to nontrivial simplest p oten tial symmetries, pro ve s to b e equiv alen t, with resp ect to the corresp ond ing Lie symmetry group , to either the charac teristic 1 or the c haracteristic x [36, Th eorem 7]. Th e Lie algebra of s im p lest p oten tial symmetries of the linear heat equation, connected with the characte r istic x , w as calculated in [20]. The ab ov e p oin t transformation allo ws u s to easily extend the description of charact eristics whic h are essen tial f or finding simplest p oten tial symmetries to the F okk er-Planc k equation u t = u xx + ( xu ) x . Namely , an y nontrivia l simp lest p oten tial sym metry of the equation u t = u xx + ( xu ) x is asso ciated, up to equiv alence generated by the corresp ondin g Lie symm etry group, with either the c h aracteristic 1 or the charact eristic e t x . The Lie sy m metry algebras of the p otenti al systems constructed w ith the ab ov e c haracteristics are calculated. I n other w ord s, in [36] all simplest p otenti al symm etries of the linear heat equation u t = u xx and th e F okk er-Planc k equation u t = u xx + ( xu ) x w ere foun d. Moreo v er, these equations are shown to p ossess infin ite series of algebras of p otent ial symmetries dep endin g on an arbitrary n u m b er of p otent ials [36, Prop osition 13]. Other equations from class (3) (including those from class (2)) p ossessing suc h series are singled out. System (43) of [18] is a partial case of the p oten tial system (4) with k = 1 and α dep ending only on x . As ab o ve shown, suc h systems were intensiv ely inv estigated earlier and/or in a more general frame than in [18]. 4 Results discu ssed in this section are easily extended by p oin t or n onlo cal equiv alence trans - formations to linearizable second-order ev olution equations [33]. Examples of such extension are present ed in Sections 3 and 6. All th e p ossible p otent ial conserv ation laws of the corresp ond - ing linearizable equations and their p otentia l symmetries dep ending on an arbitrary n u m b er of p oten tials are exh austiv ely d escrib ed in this w a y . 3 Inhomogeneous diffusion equations In this section w e consider v ariable co efficien t n on lin ear d iffusion equations of the general f orm f ( x ) u t = ( g ( x ) u n u x ) x . (5) Using the transformation ˜ t = t , ˜ x = R dx g ( x ) , ˜ u = u , w e can r educe equation (5) to ˜ f ( ˜ x ) ˜ u ˜ t = ( ˜ u n ˜ u ˜ x ) ˜ x , where ˜ f ( ˜ x ) = g ( x ) f ( x ) and ˜ g ( ˜ x ) ≡ 1. That is wh y , with ou t loss of generalit y , we restrict ourselv es to the inv estigatio n of equations ha ving the form f ( x ) u t = ( u n u x ) x . (6) The gauge g = 1 could b e rep laced by other gauges of arbitrary elemen ts. F or example, any equation of form (5) can b e red u ced b y a transformation s im ilar to the ab o ve one to an equation of the same form with f = 1. The gauge f = 1 is con ve ntionally used for arb itrary elements of class (5) b ut the app lication of th e gauge g = 1 instead of f = 1 leads to simplifying all in vesti gations of sym m etry and transformational prop erties of class (5) and hence is optimal. The usage of d ifferen t gauges is discuss ed in [22, 43, 44]. The equiv alence group G ∼ of class (6) h as a simp le s tructure and consists of the transforma- tions ˜ t = δ 1 t + δ 4 , ˜ x = δ 2 x + δ 5 , ˜ u = δ 3 u, ˜ f = δ 1 δ − 2 2 δ n 3 f , ˜ n = n, (7) where δ i , i = 1 , . . . , 5, are arb itrary constants, δ 1 δ 2 δ 3 6 = 0. A t the same time, class (6) p ossesses a generali zed equiv alence group whic h is wider than G ∼ . The notion of generalized equiv alence groups was pr op osed by Meleshk o [27]. See also [22, 35, 44] for discussions and generalizat ions of this notion as wel l as a n umber of examples of classes ha ving non trivial generalized equiv alence groups and w a y s to use them in solving differen t classification problems. In contrast to usual equiv alence groups [30], comp onen ts of transform ations fr om generalized equiv alence groups , as- so ciated with indep enden t v ariables an d u n kno wn fu n ctions, may d ep ends on arb itrary elemen ts of the corresp onding classes. Theorem 1. The gener alize d e quivalenc e gr oup ˆ G ∼ of class (6) under the c ondition n 6 = − 1 c onsists of the tr ansformat ions ˜ t = δ 1 t + δ 2 , ˜ x = δ 3 x + δ 4 δ 5 x + δ 6 , ˜ u = δ 7 | δ 5 x + δ 6 | − 1 n +1 u, ˜ f = δ 1 δ 7 n | δ 5 x + δ 6 | 3 n +4 n +1 f , ˜ n = n . wher e δ j , j = 1 , . . . , 7 , ar e arbitr ary c onstants, δ 1 δ 7 6 = 0 and δ 3 δ 6 − δ 4 δ 5 = ± 1 . In the c ase n = − 1 tr ansform ations fr om the gr oup ˆ G ∼ take the form ˜ t = δ 1 t + δ 2 , ˜ x = δ 3 x + δ 4 , ˜ u = δ 5 e δ 6 x u, ˜ f = δ 1 δ − 2 3 δ − 1 5 e − δ 6 x f , wher e δ j , j = 1 , . . . , 6 , ar e arbitr ary c onstants, δ 1 δ 3 δ 5 6 = 0 . 5 Since the parameter n is an inv arian t of all adm issible (p oint) transformations in class (6), this class can b e p r esen ted as the union of d isjoin t s ub classes where eac h from the su b classes corresp onds to a fixed v alue of n . T h is represen tation allo w s u s to give the inte r pretation of the generalized equiv alence grou p ˆ G ∼ as a family of th e usu al conditional equiv alence group s of the sub classes parameterized with n , and the v alue n = 0 and n = − 1 b eing sin gular. In the case n = 0 a part of the corresp onding conditional equiv alence group h as to b e neglected. The conserv ation la w s for the class (6) are stated in the follo w ing theorem [21, 23]. Theorem 2. The sp ac e of lo c al c onservation laws of any e quation of form (6) with n 6 = 0 is two-dimensional and sp anne d by c onservation laws with the c onserve d ve ctors ( f u, − u n u x ) and ( xf u, − xu n u x + R u n du ) . The sp ac e of lo c al c onservation laws of the line ar e quation f u t = u xx ( n = 0 ) is infinite-dimensional and sp anne d by ( αf u, − αu x + α x u ) . Her e α = α ( t, x ) runs thr ough the solution set of the line ar e quation f α t + α xx = 0 . Up to G ∼ -equiv alence, these conserv ation laws giv e rise to the follo wing inequ iv alent p oten tial systems for equations (6) with n 6 = 0: 1. v 1 x = f u , v 1 t = u n u x ; 2. v 2 x = xf u , v 2 t = xu n u x − R u n du ; 3. v 1 x = f u , v 1 t = u n u x , v 2 x = xf u , v 2 t = xu n u x − R u n du . Systems 1 and 2 are asso ciated with the conserv ation la ws havi n g the c haracteristics 1 and x , resp ectiv ely . The united system 3 corresp onds to the whole space of conserv ation la w s. The generalized equiv alence group ˆ G ∼ prolonged to p oten tials establishes additional equiv alence b et ween p oten tial systems. T h u s , in the case n 6 = − 1 the transformation ˜ t = t, ˜ x = x − 1 , ˜ u = | x | − 1 n +1 u, ˜ v 1 = − (sign x ) v 2 , ˜ v 2 = − (sign x ) v 1 (8) maps systems 1 and 2 to systems 2 and 1 in the tilde v ariables with ˜ f = | ˜ x | − 3 n +4 n +1 f ( ˜ x − 1 ), resp ectiv ely . Systems 1 and 2 are ˆ G ∼ -inequiv alen t for an arbitrary pair of v alues of f iff n = − 1. System (17) of [18] with g = x ± 2 is a p articular case of the ab ov e system 2 , and all symmetries obtained in [18] are n othing but u sual p oten tial symmetries derived p reviously in [42 ]. Thus, the op erators v 4 and v 6 from (20) in [18] coincide with the op erators Γ 2 / 2 and Γ 1 / 4, wh ere the op erators Γ 2 and Γ 1 are present ed in formulas (5.7) and (5.6) of [42], resp ectiv ely . The op erator w 4 giv en in (25) of [18] differs from th e op erator Γ 1 / 3 b y scaling the corresp onding p oten tial, where the op erator Γ 1 is defined in (5.9) of [42 ]. Additionally note that system (26) of [18] do es not define the p oten tial in a prop er w ay . P oten tial sym metries of equation (6), asso ciated with sys tem 1 , w ere first obtained in [40, 41], see also [3, 7, 8] for the constan t co efficien t case f = 1. Ther e exist t w o inequiv alen t equations of form (6) admitting su c h nonlo cal symm etries. Belo w we adduce the v alues of arbitrary elemen ts together with bases of the corr esp onding maximal Lie in v ariance algebras. 1.1. f = 1, n = − 2: h ∂ t , ∂ v 1 , 2 t∂ t + u∂ u + v 1 ∂ v 1 , x∂ x − u∂ u , − v 1 x∂ x + ( xu + v 1 ) u∂ u + 2 t∂ v 1 , 4 t 2 ∂ t − (( v 1 ) 2 + 2 t ) x∂ x + (( v 1 ) 2 + 6 t + 2 xuv 1 ) u∂ u + 4 tv 1 ∂ v 1 , ϕ∂ x − ϕ v 1 u 2 ∂ u i ; 1.2. f = x − 4 / 3 , n = − 2: h ∂ t , ∂ v 1 , 2 t∂ t + u∂ u + v 1 ∂ v 1 , 3 x∂ x − u∂ u − 2 v 1 ∂ v 1 , 3 xv 1 ∂ x − ( v 1 + 3 x − 1 / 3 u ) u∂ u − ( v 1 ) 2 ∂ v 1 i . Here and b elo w ϕ = ϕ ( t, v 1 ) is an arb itrary solution of the linear heat equation ϕ t = ϕ v 1 v 1 . 6 P oten tial symmetries of equation (6) asso ciated with system 2 w ere fi rst inv estigated in [42] (see also [24]). Up to the equiv alence group G ∼ , th er e exist exactl y tw o cases of equations in class (6) admitting suc h p oten tial symmetries: 2.1. f = x − 2 , n = − 2: h ∂ t , ∂ v 2 , x∂ x , 2 t∂ t + u∂ u + v 2 ∂ v 2 , v 2 x∂ x − u 2 ∂ u + 2 t∂ v 2 , 4 t 2 ∂ t + (( v 2 ) 2 + 2 t ) x∂ x + 2(2 t − uv 2 ) u∂ u + 4 tv 2 ∂ v 2 , x 2 ψ ∂ x − xu ( ψ + ψ v 2 u ) ∂ u i . 2.2. f = x − 2 ( c 1 + c 2 x − 1 ) − 4 / 3 , c 2 6 = 0, n = − 2: h ∂ t , ∂ v 2 , 2 t∂ t + u∂ u + v 2 ∂ v 2 , 3( c 1 x + c 2 ) x∂ x − (2 c 2 + 3 c 1 x ) u∂ u + 2 c 2 v 2 ∂ v 2 , 3 v 2 ( c 1 x + c 2 ) x∂ x − (3 x 4 / 3 ( c 1 x + c 2 ) − 1 / 3 u + (2 c 2 + 3 c 1 x ) v 2 ) u∂ u + c 2 ( v 2 ) 2 ∂ v 2 i ; Here and b elow ψ = ψ ( t, v 2 ) is an arbitrary solution of the linear heat equation ψ t = ψ v 2 v 2 . By transformation (8), cases 2.1 and 2.2 are reduced to cases 1.1 and 1.2, resp ectiv ely . F or the precise red u ction 2 . 2 → 1 . 2 the transformation ˆ t = ˜ t , ˆ x = c 1 + c 2 ˜ x , ˆ u = c − 1 2 ˜ u from G ∼ has to b e additionally carried out. The united system 3 is equiv alen t to the second-lev el p oten tial system v 1 x = f u, w x = v 1 , w t = R u n du (9) constructed fr om system 1 using its conserved vecto r ( v , − R u n du ), and w = xv 1 − v 2 . Nont r ivial G ∼ -inequiv alen t cases of p oten tial symmetries asso ciated with system (9) are exhausted b y the follo wing ones: 3.1. f = 1, n = − 2: h ∂ t , ∂ w , ∂ v 1 + x∂ w , 2 t∂ t + u∂ u + v 1 ∂ v 1 + w ∂ w , x∂ x − u∂ u + w ∂ w , ( w − 2 v 1 x ) ∂ x + (2 xu + v 1 ) u∂ u + 2 t∂ v 1 + (2 t − ( v 1 ) 2 ) x∂ w , 4 t 2 ∂ t + (2 v 1 w − 3 x ( v 1 ) 2 − 6 tx ) ∂ x + (6 xuv 1 − 2 uw + ( v 1 ) 2 + 10 t ) u∂ u + 4 tv 1 ∂ v 1 + (( v 1 ) 2 w − 2 tw − 2 x ( v 1 ) 3 ) ∂ w , ϕ v 1 ∂ x − ϕ t u 2 ∂ u + ( v 1 ϕ v 1 − ϕ ) ∂ w i ; 3.2. f = 1, n = − 2 / 3: h ∂ t , ∂ x , ∂ w , ∂ v 1 + x∂ w , 2 t∂ t + 3 u∂ u + 3 v 1 ∂ v 1 + 3 w ∂ w , x∂ x − 3 u∂ u − 2 v 1 ∂ v 1 − w ∂ w , w∂ x − 3 uv 1 ∂ u − ( v 1 ) 2 ∂ v 1 i ; 3.3. f = x − 2 , n = − 2: h ∂ t , ∂ w , ∂ v 1 + x∂ w , x∂ x − v 1 ∂ v 1 , 2 t∂ t + u∂ u + v 1 ∂ v 1 + w ∂ w , x (2 xv 1 − w ) ∂ x − u ( xv 1 + 2 u ) ∂ u + v 1 ( w − xv 1 ) ∂ v 1 + ( x 2 ( v 1 ) 2 − 2 t ) ∂ w , 4 t 2 ∂ t + x (6 t + 3 x 2 ( v 1 ) 2 − 4 xv 1 w + w 2 ) ∂ x + 2 u (2 t − 3 xuv 1 + 2 uw − x 2 ( v 1 ) 2 + xv 1 w ) ∂ u +(2 xv 1 w − 2 t − x 2 ( v 1 ) 2 − w 2 ) v 1 ∂ v 1 + 2(2 tw + x 3 ( v 1 ) 3 − x 2 ( v 1 ) 2 w ) ∂ w , x 2 ψ v 2 ∂ x + ( ψ v 2 + uψ t ) xu∂ u − ψ ∂ v 1 + x ( xv 1 ψ v 2 − ψ ) ∂ w i ; 3.4. f = x − 6 , n = − 2 / 3: h ∂ t , ∂ w , ∂ v 1 + x∂ w , 2 t∂ t + 3 u∂ u + 3 v 1 ∂ v 1 + 3 w ∂ w , x∂ x + 6 u∂ u + v 1 ∂ v 1 + 2 w ∂ w , x 2 ∂ x + 3 xu∂ u + ( w − xv 1 ) ∂ v 1 + xw ∂ w , xw ∂ x − 3( xv 1 − 2 w ) u∂ u − ( xv 1 − w ) v 1 ∂ v 1 + w 2 ∂ w i . Here v 2 = xv 1 − w . By transformation (8) wh ic h is rewritten for v 1 and w as ˜ v 1 = w sign x − | x | v 1 and ˜ w = | x | − 1 w , cases 3.3 and 3.4 are reduced to cases 3.1 and 3.2, resp ectiv ely . 7 The linear equation f u t = u xx ( n = 0) p ossesses an infinite series of p oten tial systems of the form 4. v i x = α i f u , v i t = α i u x − α i x u , i = 1 , . . . , k , where k ≥ 1 and α i = α i ( t, x ) are arbitrary linearly indep endent solutions of the lin ear equa- tion f α t + α xx = 0 . F or th e classification of p oten tial sym metries of lin ear p arab olic equations (in particular, the inv estiga tion of Lie sym m etries of system 4 ) we refer the reader to [36]. See also the previous section for the discussion of th is result. The app earance of non trivial p oten tial symmetries for equations from class (6) can b e easily explained usin g tran s formations inv olving p oten tials and called p oten tial equiv alence transfor- mations (PETs), cf. [34]. It is sufficien t to consider only the p otent ial ho dograph transformation ˜ t = t, ˜ x = v 1 , ˜ u = u − 1 , ˜ v 1 = x, (10) and ˜ v 2 = v 2 (resp. ˜ w = w ). Un d er transformation (10), system 1 is m app ed to the system f ( v 1 ) ˜ v 1 ˜ x = ˜ u, ˜ v 1 ˜ t = ˜ u − n − 2 ˜ u ˜ x . (11) Since transformation (10) is a p oint transformation in the sp ace of v ariables supplemented with the p oten tials, it establishes an isomorphism b et w een maximal Lie in v ariance algebras of systems 1 and (11) . Note that transformation (10) is an in volution, i.e., it coincides with its in verse. If f = 1 and n = − 2, system (11) coincides with the p oten tial s y s tem of the linear heat equation ˜ u ˜ t = ˜ u ˜ x ˜ x , asso ciated with the c haracteristic 1. The p oten tial systems of the linear heat equation are exhausted by the p otentia l systems of form 4 with f = 1 [33, 36]. Th erefore, the whole set of p otenti al systems (of all lev els) of the u − 2 -diffusion equation u t = ( u − 2 u x ) x consists of the images of the p otenti al systems of the linear heat equation, constru cted with the tuples of c haracteristics including the c haracteristic 1 as the fi rst elemen t, with resp ect to transformation (10) iden tically p rolonged to th e other p oten tials. The maximal lev el of p oten tials and p oten tial systems equals t wo. All prop erties of p oten tial sym metries of the u − 2 -diffusion equation follo w from the corresp ond ing p rop erties of the linear heat equation. In p articular, Prop osition 12 of [36] imp lies that the u − 2 -diffusion equ ation admits an infin ite series { g p , p ∈ N } of p oten tial sy m metry algebras. F or any p ∈ N the algebra g p is of strictly p th p oten tial order and is asso ciated with p -tup les of the linearly ind ep endent lo w est degree p olynomial solutions of th e backw ard heat equation. Moreo ver, eac h algebra g p is isomorp hic to the m aximal Lie inv ariance algebra of the linear heat equ ation. The case 3.1 is a sp ecificatio n of the ab ov e general frame, corresp onding to the pair (1 , x ) of the simplest solutions of the linear heat equation. If f = x − 4 / 3 and n = − 2, sys tem (11) is equiv alent to the ˜ v − 4 / 3 -diffusion equation ˜ v ˜ t = ( ˜ v − 4 / 3 ˜ v ˜ x ) ˜ x . T he p o w er n onlinearit y of degree − 4 / 3 is a well -kn o wn case of Lie symm etry extension fr om the O v s iannik o v’s group classification of nonlinear diffusion equ ations [30]. Eac h Lie sy m metry of system (11) is the first-order prolongation, w ith resp ect to x , of a Lie symm etry of the ˜ v − 4 / 3 -diffusion equation. Eac h Lie s y m metry of the ˜ v − 4 / 3 -diffusion equation is prolonged to a Lie symmetry of system (11) and then mapp ed, b y transf orm ation (10 ), to a Lie symmetry of the system 1 with f = x − 4 / 3 and n = − 2. Th erefore, the Lie inv ariance algebra of case 1.2 is isomorphic to the Lie in v ariance algebra of the ˜ v − 4 / 3 -diffusion equation. If f = 1 and n = − 2 / 3, system (11) coincides w ith th e p oten tial system of the ˜ u − 4 / 3 -diffusion equation ˜ u ˜ t = ( ˜ u − 4 / 3 ˜ u ˜ x ) ˜ x , asso ciated with the charact eristic 1. Th e n on trivial Lie symmetry op erator x 2 ∂ x − 3 xu∂ u of this equation cannot b e pr olonged to a single p oten tial. Th is is why the corresp onding case of p otentia l sym metry of equations from class (6) arises only under consideration of the united p oten tial system 3 havi n g t wo p oten tial v ariables. 8 4 A class of p orous medium equations The second class examined in [18] consists of p orous med ium equations ha ving the form u t = ( u n ) xx + h ( x )( u m ) x . (12) Here f ( x ) /m from [18] is redenoted b y h ( x ) for con v en ience. Only the case m = n was considered in [18] b ut th is inv estiga tion in fact is n eedless. T he general principle of group analysis of differen tial equ ations is that ob jects (differential equ ations, classes of differen tial equations, exact s olutions, subalgebras of Lie in v ariance algebras etc.) are assum ed similar if they are related via p oin t transformations [29, 30, 35, 44]. Suc h ob j ects ha ve similar transformational or other prop erties relev an t to the fr amew ork of group analysis and h ence a single representati ve from a set of similar ob jects is enough to b e in vestig ated. An y consid eration in another st yle m us t b e additionally justified. Th e sub class of class (12 ), singled out b y th e condition m = n , is mapp ed to class (6) via a family of p oin t transform ations p arameterized by the arb itrary elemen t h . I ndeed, an y equation of form (12) is reduced b y the transform ation ˜ t = t, ˜ x = Z e − R h dx dx, ˜ u = u, (13) to the equation of form (6) with f ( ˜ x ) = e 2 R h dx /n an d ˜ n = n − 1. T ransformation (13) was found in [21, 22] as an elemen t of the generaliz ed extended equ iv alence group of the wider class of nonlinear v ariable-co efficien t diffu sion–con v ection equations of the general form f ( x ) u t = ( A ( u ) u x ) x + h ( x ) B ( u ) u x , where f A 6 = 0. (The additional attribute “extended” means that transformations from the group ma y dep end on arbitrary elemen ts of the class in a nonlo cal wa y .) The prop erties of p ossessing conserv ation la ws and p oten tial symmetries completely agree with the similarit y relation of differen tial equatio n s [33, 36]. Hence an y resu lt on p oten tial symmetries of equations from class (12) can b e easily deriv ed via th e application of th e in verse to transform ation (13) as a r eform ulation of the corresp ondin g r esu lt for class (6 ). See the discussion on p oten tial symmetries of equations from class (6) in Section 3. T h u s , p otent ial sym metries of (12 ) is found in [18] only in th e case h = − 1 / (2 x ) (formula (36) of [18]). All of them are the preimages of symmetries presented in case 2.2 of Section 3, where c 1 has to b e set to equal 0. In other w ords, more general results on p oten tial symmetries of equations fr om class (12) than pr esen ted in [18] are in fact kno wn. 5 A second class of p orous medium equations In [18] the p orous medium equations of th e form u t = (( u n ) x + f ( x ) u m ) x , (14) with n 6 = 0 w as give n without considering its p oten tial symmetries. It w as stated that the complete classification of p oten tial symmetries was carried out in [17]. Th er e are three remarks on this statemen t. 1) In [17] the case m = 0 was omitted fr om th e consideration since another representa tion for w hic h th e v alue m = 0 is singular w as used for equ ations from class (14). A t the same time, the corresp ond ing sub class of class (14) con tains we ll-known equations, e.g., the linearizable equation u t = ( u − 2 u x ) x + 1. Moreo ver, s ome equations from the cases m = 0 and m 6 = 0 are connected within b oth the p oin t and p otenti al frame. 9 2) Th e description of p oten tial s y m metries in [17] was not a classification since no equiv alence relations of equations or symmetries w ere used . 3) On ly simplest p oten tial sym metries arising und er the stud y of the corresp onding “n atur al” p oten tial systems we r e foun d. Th e problem on the construction of the other simplest and, moreo v er, general p oten tial symmetries of equatio n s fr om class (14) wa s still op en. W e emplo y class (14) in order to giv e th e basic steps for the exhaustiv e classification of the simplest p oten tial symmetries. Moreo v er, in the next section we completely d escrib e p oten tial symmetries of some equations from class (14). The equiv alence group G ∼ of class (14) is formed b y the transformations ˜ t = δ 1 t + δ 2 , ˜ x = δ 3 x + δ 4 , ˜ u = δ 1 − 1 n − 1 δ 3 2 n − 1 u, ˜ f = δ 1 m − n n − 1 δ 3 n − 2 m +1 n − 1 f , ˜ n = n, ˜ m = m, where δ i , i = 1 , . . . , 4 , are arb itrary constant s, δ 1 δ 3 6 = 0 . Additionally , the sub class singled out from (14) b y the condition m = n can b e mapp ed to the sub class consisting of the equations e − n +1 n R f ( x ) dx ˜ u ˜ t = ( ˜ u n ) ˜ x ˜ x b y the transform ation ˜ t = t, ˜ x = Z e R f ( x ) dx dx, ˜ u = e 1 n R f ( x ) dx u. (15) W e presen t the conserv ation la ws for (14) and the subs equen t p oten tial systems. Theorem 3. Any e quation fr om class (14) has the c onservation law of form (1 ) whose density and flux ar e, r esp e ctively, 1 . F = u, G = − nu n − 1 u x − f u m . (16) A c omplete list of G ∼ -ine quivalent e quations (14) having additional (i.e. line ar indep endent with (16 ) ) c onservation laws is exhauste d by the fol lowing ones 2 . m = n 6 = 1 : F = u R e R f dx dx, G = − R e R f dx dx ( nu n − 1 u x + f u n ) + e R f dx u n , 3 . n 6 = 1 , m = 0 : F = xu, G = − x ( nu n − 1 u x + f ) + u n + R f dx, 4 . n 6 = 1 , m = 1 , f = 1 : F = ( t + x ) u, G = − ( t + x )( nu n − 1 u x + u ) + u n , 5 . n 6 = 1 , m = 1 , f = εx : F = e εt xu, G = − e εt x ( nu n − 1 u x + xu ) + e εt u n , 6 . n = 1 , m = 0 : F = αu, G = − α ( u x + f ) + α x u + R α x f dx, 7 . n = 1 , m = 1 : F = β u, G = − β ( u x + f u ) + β x u, wher e ε = ± 1 mo d G ∼ , α = α ( t, x ) and β = β ( t, x ) ar e arbitr ary solutions of the line ar e quations α t + α xx = 0 and β t + β xx − f β x = 0 , r esp e ctively. (T o gether with r estrictions on values f , n and m we also adduc e densities and fluxes of additiona l c onservation laws.) These conserv ation laws can b e us ed for the construction of p oten tial systems that lead to p oten tial symmetries f or the equation (14). The asso ciated c h aracteristics are equal to the co efficien ts of u in the pr esented expr essions for F . Here w e consider only simplest p oten tial systems (i.e., p oten tial systems with one p oten- tial v ariable, constructed with usage of sin gle conserved v ectors of b asis conserv ation la ws) of equations from class (14), ha ving the form v x = F , v t = − G. 10 Cases 6 and 7 of Th eorem 3 can b e exclud ed from the inv estigatio n since they concern lin ear equations studied in [36]. Th en, the equations of case 2 are reducible to diffusion equations of form (6 ) by tran s formation (15). The equations of cases 4 an d 5 are r educible to th e constant co efficien t diffu sion equation ˜ u ˜ t = ( ˜ u n ) ˜ x ˜ x b y means of the Galilei transformation ˜ t = t, ˜ x = x + t, ˜ u = u and the transformation ˜ t = ( 1 ε ( n +1) e ε ( n +1) t , n 6 = − 1 , t, n = − 1 , ˜ x = e εt x, ˜ u = e − εt u, resp ectiv ely . Therefore, w e hav e to in vestig ate only t wo p oten tial sy s tems v x = u, v t = nu n − 1 u x + f u m , (17) v ∗ x = xu, v ∗ t = x ( nu n − 1 u x + f ) − u n − R f dx (18) corresp onding to cases 1 and 3 of Th eorem 3. (T o distinguish the p oten tial introd uced, we denote the second p oten tial by v ∗ .) Lie p oint symmetries of the p oten tial system (17) giv e nontrivial p oten tial symmetries of equation (14) in the follo wing cases: 1. f = x − 2 , n = − 1, m = − 2: h ∂ t , ∂ v , x∂ x − u∂ u , 12 t∂ t + (3 ln x − v ) x∂ x + (3 − 3 ln x + xu + v ) u∂ u + 2(3 v − t ) ∂ v i ; 2. f = εx ln | x | , ε = ± 1, n = − 1, m = 1: h ∂ t , ∂ v , e − εt ( x∂ x − u∂ u ) , e − εt ( − εxv ∂ x + ε ( xu 2 + uv ) ∂ u + 2 ∂ v ) i ; 3. f = x , n = − 1, m = 0: h ∂ t , ∂ v , ∂ x + t∂ v , 2 t∂ t − x∂ x + 2 u∂ u + v ∂ v , t 2 ∂ t + ( v − tx ) ∂ x + (2 t − u ) u∂ u + tv ∂ v i ; 4. f = 0, n = − 1, m = 0: h ∂ t , ∂ v , x∂ x − u∂ u , 2 t∂ t + u∂ u + v ∂ v , v x∂ x − ( v + xu ) u∂ u + 2 t∂ v , 4 t 2 ∂ t + ( v 2 − 2 t ) x∂ x − ( v 2 + 2 v xu − 6 t ) u∂ u + 4 tv ∂ v , β ∂ x − β v u 2 ∂ u i ; 5. f = 1, n = − 1, m = 1: h ∂ t , ∂ v , ( x + t ) ∂ x − u∂ u , 2 t∂ t + 2 x∂ x − u∂ u + v ∂ v , v ( x + t ) ∂ x − [( x + t ) u + v ] u∂ u + 2 t∂ v , 4 t 2 ∂ t + [( x + t ) v 2 − 2 tx − 6 t 2 ] ∂ x − [ v 2 − 6 t + 2( x + t ) v u ] u∂ u + 4 tv ∂ v , β ∂ x − β v u 2 ∂ u i ; 6. f = εx , ε = ± 1, n = − 1, m = 1: h ∂ t , ∂ v , x∂ x − u∂ u , 2 t∂ t − 2 εtx∂ x + (1 + 2 εt ) u∂ u + v ∂ v , v x∂ x − ( xu + v ) u∂ u + 2 t∂ v , 4 t 2 ∂ t +  v 2 − 2 t − 4 εt 2  x∂ x −  v 2 − 6 t + 2 xv u − 4 εt 2  u∂ u + 4 tv ∂ v , e − εt ( β ∂ x − β v u 2 ∂ u ) i ; 7. n = − 1, m = − 1: h ∂ t , ∂ v , ψ ∂ x − (1 − f ψ ) u∂ u , 2 t∂ t + u∂ u + v ∂ v , ψ v ∂ x − ((1 − f ψ ) v + ψ u ) u∂ u + 2 t∂ v , 4 t 2 ∂ t + ( v 2 − 2 t ) ψ ∂ x − [2 ψ v u + ( v 2 − 2 t )(1 − f ψ ) − 4 t ] u∂ u + 4 tv ∂ v , ϕ ( β ∂ x − ( β v u 2 − f β u ) ∂ u ) i ;. 8. f = 1, n = 1, m = 2: h ∂ t , ∂ x , ∂ v , 2 t∂ t + x∂ x − u∂ u , 2 t∂ x − ∂ u − x∂ v , 4 t 2 ∂ t + 4 tx∂ x − 2(2 tu + x ) ∂ u − (2 t + x 2 ) ∂ v , e − v [( αu − α x ) ∂ u − α∂ v ] i . Here α = α ( t, x ) and β = β ( t, v ) r u n th r ough the solution sets of the linear heat equation α t − α xx = 0 and bac kward linear heat equation β t + β vv = 0, resp ectiv ely , ϕ ( x ) = e − R f dx , ψ ( x ) = e − R f dx R e R f dx dx. 11 Note 1. Equ ations from class (14) with n = − 1, m = 0 and f ∈ { 0 , 1 } are just different represent ations of the same equation. P oten tial systems corresp onding to these tw o cases are connected via the tr ansformation ˜ v = v + t of p oten tial v ariable v . This transformation maps case 4 to th e case f = 1, n = − 1, m = 0: h ∂ t , ∂ ˜ v , x∂ x − u∂ u , 2 t∂ t + u∂ u + ( t + ˜ v ) ∂ ˜ v , ( ˜ v − t ) x∂ x − ( ˜ v − t + xu ) u∂ u + 2 t∂ ˜ v , 4 t 2 ∂ t + [( ˜ v − t ) 2 − 2 t ] x∂ x − [( ˜ v − t ) 2 + 2( ˜ v − t ) xu − 6 t ] u∂ u + 4 t ˜ v ∂ ˜ v , e t 4 − ˜ v 2 [ ˜ β ∂ x − ( ˜ β ˜ v − 1 2 ˜ β ) u 2 ∂ u ] i , where the fun ction ˜ β = ˜ β ( t, ˜ v ) r uns through the solution set of the bac kwa r d linear h eat equation ˜ β t + ˜ β ˜ v ˜ v = 0. Note 2. C ases 5, 6 and 7 are r educed to case 4 by the p oin t transf ormations { ˜ x = x + t, ˜ u = u } , { ˜ x = e εt x, ˜ u = e − εt u } and (15) , resp ectiv ely . The v ariables t and v are iden tically transformed. As men tioned, Lie symmetries of p oten tial system (17) were inv estigated in [17] only for m 6 = 0 and hence cases 3 and 4 were omitted there. It is explained b y c hoice of another representat ion of equation (14). F or all v alues of m , p oten tial s ymmetries of equation (14) asso ciated with p oten tial s y s tem (17 ) are first classified abov e. There exists only one inequ iv alen t case of p oten tial system (18) that give s non trivial p otenti al symmetries for equation (14), namely , f = x , n = − 1 and m = 0. Lie algebra of p otent ial symmetries in this case has the form 9. f = x , n = − 1, m = 0: h ∂ t , ∂ v , 2 t∂ t − x∂ x + 2 u∂ u , e − v ∗ / 2 (2 x∂ x + ( x 2 u − 2) u∂ u − 4 ∂ v ∗ ) i . Note 3. In this section w e h a v e first pr esen ted the classification of lo cal conserv atio n laws and the simplest p oten tial sym metries of equations fr om class (14). T o complete inv estigation of p oten tial symmetries in this class, it is necessary to study p oten tial systems dep ending on sev eral p otent ials and systems constructed with p oten tial conserv ation la ws of (14). Using the p oten tial equiv alence metho ds, in the n ext section w e describ e general p oten tial symmetries of linearized equations from this class. Here we giv e only one example on p otent ial symmetries of equations from class (14) with f 6 = 0, whic h inv olv e t wo p oten tials. They are easily constru cted via the p oin t transformation (15) from p oten tial symmetries presente d in S ection 3. Namely , the equation x − 6 u t = ( u − 2 / 3 u x ) x from class (6) (case 3.4, f = x − 6 , n = − 2 / 3) is mapp ed b y the transf ormation ˜ t = 3 4 t , ˜ x = | x | − 1 / 2 , ˜ u = | x | − 9 / 2 u to the equation ˜ u ˜ t = (( ˜ u 1 / 3 ) ˜ x − 3 ˜ x − 1 ˜ u 1 / 3 ) ˜ x (19) from class (14), where ˜ n = ˜ m = 1 / 3 and ˜ f = − 3 ˜ x − 1 . F or co efficient s to b e simpler, w e h a v e additionally com bined the corresp ond ing transformation of form (15) with a scaling. The second order p oten tial system for equation (19), wh ich is constructed from (9) via the transformation prolonged to the p oten tials as ˜ v = − v 1 / 2 and ˜ w = w / 4, is ˜ v ˜ x = ˜ u, ˜ w ˜ x = ˜ x − 3 ˜ v , ˜ w ˜ t = ˜ x − 3 ˜ u 1 / 3 . The asso ciated p oten tial symmetry algebra of (19) is (we omit tildes f or con v enience) h ∂ t , ∂ w , 2 ∂ v − x 2 ∂ w , 2 t∂ t + 3 u∂ u + 3 v ∂ v + 3 w ∂ w , x∂ x − 3 u∂ u − 2 v ∂ v − 4 w ∂ w , x − 1 ∂ x + 3 x − 2 u∂ u + 2(2 w + x − 2 v ) ∂ v − 2 x − 2 w∂ w , xw ∂ x − 3( x − 2 v + w ) u∂ u − ( x − 2 v + 2 w ) v ∂ v − 2 w 2 ∂ w i . 12 6 Applications of p oten tial equiv alence transformations In a w ay analogous to class (6) (see the end of Section 3), p oten tial equiv alence transformations (PETs) can b e effectiv ely applied to explain the pro venance of p otent ial symmetries of equations from class (14). The main tool again is the p oten tial ho dograph transformation ˜ t = − t, ˜ x = v , ˜ u = u − 1 , ˜ v = x. (20) T ransformation (20) differs f rom (10) in the sign of t du e to a difference in the repr esentati ons of classes (6) and (14) and will b e additionally mo difi ed via comp osing w ith p oin t transformations in order to p resen t the imaged equ ations in canonical form s. W e need to interpret only cases 1–3, 8 and 9 of non trivial p oten tial s y m metries from Section 5 since cases 5–7 are reduced to case 4 b y p oint transform ations and case 4 coincides, up to alternating th e sign of t , with case 1.1 from Section 3 whose in terpr etatio n is pr esen ted in the end of th at section. W e also consider s ub- classes including cases to b e in terpr eted and sho w that the prov enance of p oten tial sym m etries is connected with extensions of Lie symmetry groups of th e corresp onding p oten tial equations. f = 1, n = 1, m = 2 (case 8). Th e equation of form (14) with these v alues of the arbitrary elemen ts is the Burgers equation u t = u xx + 2 uu x . T he asso ciated p otentia l system v x = u , v t = u x + u 2 is mapp ed to th e p otentia l system ˜ v ˜ x = ˜ u , ˜ v ˜ t = ˜ u ˜ x of the linear heat equation ˜ u ˜ t = ˜ u ˜ x ˜ x , constructed w ith the conserv ation la w h a ving the c haracteristic 1, by the transformation T : ˜ t = t , ˜ x = x , ˜ u = ue v , ˜ v = e v . Analogously to the u − 2 -diffusion equation, the whole s et of p oten tial systems (of all lev els) of the Burgers equation consists of the preimages of the p oten tial systems of the linear heat equation, corresp onding to the tuples of c haracteristics including the c haracteristic 1 as the fi rst elemen t, with resp ect to trans f ormation T iden tically prolonged to th e other p oten tials. Let us recall that th e p oten tial sys tems of the linear heat equation are exhausted by the firs t-lev el p otenti al systems of form 4 with f = 1 (see section 3) [33, 36 ]. T herefore, the maximal level of p oten tials and p oten tial systems of the Burgers equation equals t wo. All prop erties of p otent ial symmetries of the Bu rgers equation follo w fr om the similar prop erties of th e linear heat equation, including the existence of an infinite series of p otentia l symmetry algebras of all p ossible p oten tial orders, which are isomorphic to the maximal Lie in v ariance algebra of the linear heat equation. n = − 1, m = 0. W e ha v e the mapping v x = u, v t = − u − 2 u x + f ∼ u t = ( u − 1 ) xx + f x    y (20) ˜ v ˜ x = ˜ u, ˜ v ˜ t = ˜ u ˜ x + f ˜ u ∼ ˜ v ˜ t = ˜ v ˜ x ˜ x + f ( ˜ v ) ˜ v ˜ x . The imaged ‘p oten tial’ equations form the class of semilinear conv ection–diffusion equations whose group classification is known [34]. Only t wo in equiv alen t cases of this classification (the linear h eat equation with f = 0 and the Burgers equation with f = ˜ v ) lead to inequiv alen t cases of p oten tial symmetries in class (14). W e neglect the case f = 0 as w as already discussed in Sections 2 and 3. In the case f = x = ˜ v b oth p oten tial systems (17) and (18) giv e non trivial p oten tial sym- metries. The corresp ond in g equation from class (14) is a w ell kno wn in tegrable equation [28, p. 129] (see also [29, p . 328 ]). The maximal Lie symmetry algebra of th e p oten tial sys tem (17 ) (case 3) is the preimage, with resp ect to the transformation v x = u, v t = − u − 2 u x + x ∼ u t = ( u − 1 ) xx + 1    y (20) ˜ v ˜ x = ˜ u, ˜ v ˜ t = ˜ u ˜ x + ˜ v ˜ v ˜ x ∼ ˜ v ˜ t = ˜ v ˜ x ˜ x + ˜ v ˜ v ˜ x , 13 of the maximal Lie symmetry alge b ra h ∂ ˜ t , ∂ ˜ x , ˜ t∂ ˜ x − ∂ ˜ v , 2 ˜ t∂ ˜ t + ˜ x∂ ˜ x − ˜ v ∂ ˜ v , ˜ t 2 ∂ ˜ t + ˜ t ˜ x∂ ˜ x − ( ˜ t ˜ v + ˜ x ) ∂ ˜ v i of the Burgers equation ˜ v ˜ t = ˜ v ˜ x ˜ x + ˜ v ˜ v ˜ x , p rolonged to ˜ u according to the equ ality ˜ u = ˜ v ˜ x . Therefore, these algebras are isomorphic. F or th e p oten tial system (18) with f = x , n = − 1 and m = 0 (case 9) w e need to mo dify transformation (20): v ∗ x = xu, v ∗ t = − xu − 2 u x − u − 1 + 1 2 x 2 ∼ u t = ( u − 1 ) xx + x     y ˜ t = − t 4 , ˜ x = v ∗ 2 , ˜ u = 2 u , ˜ v = 1 x ˜ v ˜ x = − ˜ v 3 ˜ u, ˜ v ˜ t = − ( ˜ v ˜ u ) ˜ x + ˜ v − 2 ˜ v ˜ x ∼ ˜ v ˜ t = ( ˜ v − 2 ˜ v ˜ x ) ˜ x + ˜ v − 2 ˜ v ˜ x The equation ˜ v ˜ t = ( ˜ v − 2 ˜ v ˜ x ) ˜ x + ˜ v − 2 ˜ v ˜ x arises u nder the group classification of conv ection–diffusion equations [34]. It is red u ced to the remark able diffusion equation ˆ v ˆ t = ( ˆ v − 2 ˆ v ˆ x ) ˆ x b y the p oin t transformation ˆ t = ˜ t , ˆ x = e ˜ x , ˆ v = e − ˜ x ˜ v . Its maximal Lie in v ariance algebra is A = h ∂ ˜ t , ∂ ˜ x , 2 ˜ t∂ ˜ t + ˜ v ∂ ˜ v , e − ˜ x ∂ ˜ x + e − ˜ x ˜ v ∂ ˜ v i . The prolongation of A to ˜ u according to the equalit y ˜ u = − ˜ v − 3 ˜ v ˜ x is an image of the algebra of case 9. Therefore, th e algebra of case 9 is isomorph ic to A and the maximal Lie inv ariance algebra of the diffusion equation ˆ v ˆ t = ( ˆ v − 2 ˆ v ˆ x ) ˆ x . The united p oten tial s ystem of (17) and (18) w ith f = x , n = − 1 and m = 0 also admits Lie symmetries indu cing p oten tial sym metries of the equation u t = ( u − 1 ) xx + 1. I t is mapp ed to a system equiv alen t to the linear heat equation b y transf orm ation (20) su pplemen ted with ˜ v ∗ = e v ∗ / 2 . Namely , w e h a v e the transformation v x = u, v t = − u − 2 u x + x, v ∗ x = xu, v ∗ t = − xu − 2 u x − u − 1 + 1 2 x 2 ∼ u t = ( u − 1 ) xx + 1    y (20) , ˜ v ∗ = e v ∗ / 2 ˜ v ˜ x = ˜ u, ˜ v ˜ t = ˜ u ˜ x + ˜ v ˜ v ˜ x , ˜ v ∗ ˜ x = 1 2 ˜ v ∗ ˜ v , ˜ v ∗ ˜ t = 1 4 ˜ v ∗ ˜ v 2 + 1 2 ˜ v ∗ ˜ u ∼ ˜ v ∗ ˜ t = ˜ v ∗ ˜ x ˜ x . As a result, th e w hole set of inequiv alen t p oten tial systems (of all lev els) of th e equation u t = ( u − 1 ) xx + 1 consists of s y s tems (17) and (18), the un ited system of (17) and (18) and the systems obtained b y the follo wing pr o cedure: W e take an y p oten tial system of the linear heat equation ˜ v ∗ ˜ t = ˜ v ∗ ˜ x ˜ x and s upplement it with the equations ˜ v = 2 ˜ v ∗ ˜ x / ˜ v ∗ and ˜ u = ˜ v ˜ x defining ˜ v and ˜ u . Th e equations ˜ v ˜ t = ˜ u ˜ x + ˜ v ˜ v ˜ x and ˜ v ∗ ˜ t = 1 4 ˜ v ∗ ˜ v 2 + 1 2 ˜ v ∗ ˜ u are differentia l consequ ences of the ab o ve equations in ˜ v ∗ , ˜ v and ˜ u . Th en the inv erse of transformation (20) extended to v ∗ b y ˜ v ∗ = e v ∗ / 2 and iden tically extended to the other p oten tials giv es the p oten tial system of the equation u t = ( u − 1 ) xx + 1. The studied structur e of the set of p oten tial symmetries allo ws us to conclude that the p oten tial symmetries of this equation are exhausted by cases 3 and 9 and the p otenti al symm etry algebras constructed fr om Lie and p oten tial symmetry algebras of the linear heat equation u sing the describ ed pro cedur e of extension and mapping. Finally , the ab ov e consideration sho ws that th e well known linear and lineariza b le equations u t = u xx , u t = ( u − 2 u x ) x , u t = ( u − 2 u x ) x + u − 2 u x , u t = u xx + uu x and u t = ( u − 2 u x ) x + 1 are singled out f rom classes of quasilinear second-order ev olution equations with the pr op erties of p ossessing infinite series of infin ite-dimensional algebras of p otent ial symm etries, isomorphic to the maximal Lie inv ariance algebra of the linear heat equation. Moreo v er, these equations are connected to eac h other by p oten tial equiv alence transformations. 14 n = − 1, m = 1. The corresp onding sub class of class (14) is mapp ed b y (20) in the follo win g w ay v x = u, v t = − u − 2 u x + f u ∼ u t = (( u − 1 ) x + f u ) x    y (20) ˜ v ˜ x = ˜ u, ˜ v ˜ t = ˜ u ˜ x + f ∼ ˜ v ˜ t = ˜ v ˜ x ˜ x + f ( ˜ v ) The imaged ‘p oten tial’ equations ˜ v ˜ t = ˜ v ˜ x ˜ x + f ( ˜ v ) (21) form the class of semilinear heat equations with sources. Th e problem of group classification in class (21) was solv ed in [13] (see also [19 ]). Th e linear equations from th e list presented in [13] induce cases 4–6 already discus s ed. Excludin g them, there is only one represent ative in the list, whose preimage w ith resp ect to (20) p ossesses p oten tial sym metries. Namely , the equation of form (21) with f = ε ˜ v ln ˜ v has the maximal Lie in v ariance algebra h ∂ ˜ t , ∂ ˜ x , e ε ˜ t ˜ v ∂ ˜ v , 2 e ε ˜ t ∂ ˜ x − εe ε ˜ t ˜ x ˜ v ∂ ˜ v i . Preimaging this case of Lie symmetry extension in the class (21), w e obtain case 2 of the classification of p oten tial s y m metries in class (14 ). The same tric k with preimaging with resp ect to th e p oten tial h o dograph transformation (20) can b e applied to constr u ct p oten tial nonclassical symmetries in the initial sub class singled out b y the conditions n = − 1 and m = 1. This is easy to do b ecause nonclassical symmetries of the equations from the class (21) ha ve b een already inv estigate d [6, 12, 16, 44]. T he corresp onding exact solutions w ere constru cted by the red u ction metho d in [6, 12], see also [44]. Nonlinear equations of form (21) p ossess purely nonclassical s y m metries with nonv anishing co efficien ts of ∂ ˜ t if and only if f is a cu bic p olynomial in ˜ v , i.e., f = a ˜ v 3 + b ˜ v 2 + c ˜ v + d , where a , b , c , and d are arbitrary constant s , a 6 = 0 and the co efficien t b can b e pu t equ al to 0 by translations with resp ect to v . Up to the equiv alence generated b y trans lations with resp ect to v and x , suc h symmetries are exhausted by the follo wing op erators (hereafter b = 0 and µ = p | c | / 2): a < 0 : ∂ ˜ t ± 3 2 √ − 2 a ˜ v∂ ˜ x + 3 2 ( a ˜ v 3 + c ˜ v + d ) ∂ ˜ v , c = 0 , d = 0 : ∂ ˜ t − 3 x − 1 ∂ ˜ x − 3 x − 2 ˜ v ∂ ˜ v , c < 0 , d = 0 : ∂ ˜ t + 3 µ tan ( µ ˜ x ) ∂ ˜ x − 3 µ 2 sec 2 ( µ ˜ x ) ˜ v ∂ ˜ v , c > 0 , d = 0 : ∂ ˜ t − 3 µ tanh ( µ ˜ x ) ∂ ˜ x + 3 µ 2 sec h 2 ( µ ˜ x ) ˜ v ∂ ˜ v , ∂ ˜ t − 3 µ coth ( µ ˜ x ) ∂ ˜ x − 3 µ 2 cosec h 2 ( µ ˜ x ) ˜ v ∂ ˜ v . Using PET (20), we are able to obtain p oten tial n onclassical symmetries of equation (14) with n = − 1, m = 1 and f = ax 3 + bx 2 + cx + d , where a 6 = 0 and we assume b = 0, whic h are asso ciated with p oten tial s ystem (17): a < 0 : ∂ t − 3 2 ( ax 3 + cx + d ) ∂ x ± 3 2 √ − 2 a x∂ v + 3 2 [3 ax 2 u + cu ± √ − 2 a ] ∂ u , c = 0 , d = 0 : ∂ t + 3 v − 2 x∂ x + 3 v − 1 ∂ v + 6 uv − 3 ( xu − v ) ∂ u , c < 0 , d = 0 : ∂ t + 3 µ 2 sec 2 ( µv ) x∂ x − 3 µ tan ( µv ) ∂ v − 6 µ 2 sec 2 ( µv )[ µxu tan( µv ) + 1] u∂ u , c > 0 , d = 0 : ∂ t − 3 µ 2 sec h 2 ( µv ) x∂ x + 3 µ tanh ( µv ) ∂ v − 6 µ 2 sec h 2 ( µv )[ µxu tanh ( µv ) − 1] u∂ u , ∂ t + 3 µ 2 cosec h 2 ( µv ) x∂ x + 3 µ coth ( µv ) ∂ v + 6 µ 2 cosec h 2 ( µv )[ µxu coth( µv ) − 1] u∂ u . 15 Note that the op erator pr esen ted for the case a < 0 is in fact usu al nonclassical symmetry since it is pro jectable to the sp ace ( t, x, u ). There exist tw o w ays to use mappings b et w een equations in the inv estigat ion of nonclassical symmetries. Sup p ose that nonclassical symmetries of the imaged equation are known. The first wa y is to tak e the preimages of the constru cted op erators. Then we can reduce the initial equation with resp ect to the preimaged op erators to find its non-Lie exact solutions. T h e ab o ve wa y seems to b e non-optimal since the ultimate goal of the inv estigat ion of nonclassical symmetries is the constru ction of exact solutions. This observ ation is confirmed b y the fact that the imaged equation and the asso ciated nonclassical symm etry op erators often h a v e a simpler form and therefore, are more suitable than their p r eimages. This is why the second wa y based on the implementat ion of redu ctions in the imaged equation and pr eimaging the obtained exact solutions instead of preimaging the corresp ond ing reduction op erators is pr eferable. The same observ ation is tru e for Lie symmetries. F or example, the equ ation u t = [( u − 1 ) x − x 3 u ] x is mapp ed by (20) to the equation ˜ v ˜ t = ˜ v ˜ x ˜ x − ˜ v 3 . Therefore, w e hav e the mapping b et ween solutions ˜ v = 2 √ 2 ˜ x ˜ x 2 + 6 ˜ t → u = − √ 2 √ 1 + 3 tx 2 ± 1 x 2 √ 1 + 3 tx 2 . Analogously , the equ ation u t = [( u − 1 ) x − x ( x 2 − 1) u ] x is transformed via (20) to th e equation ˜ v ˜ t = ˜ v ˜ x ˜ x − ˜ v 3 + ˜ v . Under the in verse tr ansformation, the kno wn non-Lie exact solution [6, 12] ˜ v = C 1 exp  √ 2 2 ˜ x  − C ′ 1 exp  − √ 2 2 ˜ x  C 2 exp  − 3 2 ˜ t  + C 1 exp  √ 2 2 ˜ x  + C ′ 1 exp  − √ 2 2 ˜ x  of the imaged equation is mapp ed to the exact solution u = √ 2 ln      C 2 xe 3 2 t ± p C 2 2 x 2 e 3 t − C 3 ( x 2 − 1) C 1 ( x − 1)      of the initial equation. f = x − 2 , n = − 1 , m = − 2 (case 1). Th e red u cing trans formation in this case is the most complicated: v x = u, v t = − u − 2 u x + x − 2 u − 2 ∼ u t = (( u − 1 ) x + x − 2 u − 2 ) x     y ˜ t = − t, ˜ x = v + t 3 , ˜ u = 1 xu + 1 3 , ˜ v = ln x + v 3 + 2 27 t ˜ v ˜ x = ˜ u, ˜ v ˜ t = ˜ u ˜ x + ˜ u 3 ∼ ˜ v ˜ t = ˜ v ˜ x ˜ x + ˜ v 3 ˜ x The imaged ‘p otent ial’ equation ˜ v ˜ t = ˜ v ˜ x ˜ x + ˜ v 3 ˜ x p ossesses only trivial shift and scale Lie symme- tries. Namely , its maximal Lie inv ariance algebra is generated by the op erators ∂ t , ∂ x , ∂ v and 4 t∂ t + 2 x∂ x + v ∂ v . Its fi r st prolongation w ith resp ect to x give s the maximal Lie inv ariance algebra h ∂ t , ∂ x , ∂ v , 4 t∂ t + 2 x∂ x + v ∂ v − u∂ u i of the imaged p oten tial system. Usually such simple op era- tors do not indu ce purely p otent ial symmetries via p oten tial equiv alence transformations similar to the p oten tial h o dograph transformation. This is n ot the case here sin ce du e to the complexit y of the applied transformation the pr eimage of the scale op erator 4 t∂ t + 2 x∂ x + v ∂ v − u∂ u is a purely p otent ial symm etry . 16 7 Conclusion The complete classification of p oten tial symmetries for a giv en partial differentia l equation can b e ac h iev ed b y considerin g all the p oten tial systems corresp onding to the fin ite-dimensional subspaces of th e sp ace of conserv ation la ws of the equation. F urther study can b e introduced in determinin g conserv ation laws for th e p otent ial systems whic h , in turn , lead to th e second generation of p oten tial systems. Lie symmetries can b e deriv ed for the second generation of p oten tial systems with the optimal goal to obtain p oten tial sym metries f or the original equation. This pro cedur e can b e iterated. The so-called hidden p otenti al sy m metries that app ear recently in [18] are ordinary p otent ial symmetries in th e primary s en se of [7, 8]. They can b e obtained using the standard appr oac h describ ed in the presen t pap er and hav e b een earlier found f or a num b er of different equations. Moreo v er, these symmetries are s implest ones s ince eac h of them in vol ves a sin gle p oten tial. The attribute “hidd en” serves in [18] for emph asizing a difference b et w een the case s of constan t and nonconstan t c haracteristics (“natural” and “hidden ” p oten tial systems, resp ectiv ely) but this difference is not essent ial in any w ay . W e p oin t out th at, to our knowledge, hidden p o- ten tial symm etries did not app ear in th e literature b efore. As a ru le, the attribute “hidden” is used for symmetries of a s ystem of differenti al equations, whic h in fact are symmetries of a related system in certain “usu al” sense. Th u s, hidden p oint sy m metries of a system of partial differen tial equations arise as p oint symmetries of systems obtained b y Lie r eductions of the initial one [29, p. 197]. F or details the reader can refer to [1]. T h e first n on trivial example of suc h hidden symmetries w as foun d by Kapitansky [25] for the Euler equations. It is also present ed in [29]. Wide families of su c h h idden symmetries of th e Na vier–Stok es and Eu ler equations w ere constructed in [15, 31]. Differen t n otions of hidden symmetries w ere inv en ted for ordinary differential equations [2]. Th ey are connected with low ering or in creasing the order of the corresp onding equations via differentia l sub s titutions. In [14] another n otion of hidd en symmetries was p rop osed for linear partial differentia l equations with extendin g the class of admissible symmetry op erators b y ps eu do-differen tial ones. Hidden s y m metries of all the ab o ve kind s are stable under p oint transformations in the sense that the image of a hidden symmetry of an initial ob ject (a differen tial equation or a system of suc h equations) under a p oin t tran s formation is a hidden symmetry of the corresp ond ing trans - formed ob ject. This p rop ert y is imp ortant since in volving differen t p oint equiv alence relations in statemen ts and solutions of problems is one of the m ain prin ciples of group analysis of dif- feren tial equations. Sin ce the “hidden” p oten tial symmetries that app ear in [18] do not p ossess the prop ert y of the stability with resp ect to p oin t transf ormations, the usage of th e attribute “hidden” is not justified therein. Another p oint w hic h sh ou ld b e emphasized is that p oin t and p oten tial equiv alence trans - formations in classes of differenti al equations pla y an imp ortan t role in the inv estigatio n and application of p oten tial fr ames ov er them. Often new conserv atio n la ws, p oten tial symm etries and corresp onding inv ariant solutions can b e easier constru cted fr om known ones via su c h trans- formations th an by dir ect calculations. In the present pap er this conclusion has b een illustr ated b y a num b er of examples. Ac knowledge ments The r esearc h of NMI and OOV w as p artially sup p orted by the Grant of th e Pr esiden t of Ukraine for yo u ng scien tists (pro ject num b er GP/F26/000 5). Th e researc h of R OP wa s supp orted by the Austrian Science F und (FWF), ST AR T-pro ject Y237 and pro ject P20632. NMI, ROP and OO V are grateful for the hosp italit y shown during their visits to the Univ ersity of C yprus. T he authors thank all four referees for a lot of h elpful remarks and suggestio n s. 17 References [1] Ab rah am- Shrauner B., Govinder K.S., Pro venance of t y p e I I hidden symmetries from nonlinear p artial differential equations, J. Nonline ar Math. Phys. , 2006, V.13, 612–622. [2] Ab rah am- Shrauner B., Leach P .G.L., Go vin d er K.S. and R atcliff G., Hid d en and contact symmetries of ordinary differential equations, J. Phys. A , 1995, V.28, 6707–6716. [3] Ak hato v I.Sh ., Gazizo v R .K. and Ibragimov N .Kh., Nonlo cal symm etries. A heuristic app roac h, Ito gi Nauki i T ekhniki, Curr ent pr oblems in mathematics. Newest r esults , 1989, V .34, 3–83 (Russian, translated in J. Soviet Math. , 1991, V .55, 1401–1450). [4] An co S .C. and Bluman G., Direct constru ction metho d for conserv ation la ws of partial differential equa- tions. I. Examples of conserv ation la w classifications, Eur. J. A ppl . Math. , 2002, V.13, Pa rt 5, 545–566; arXiv:math-ph/0108023. [5] An co S .C. and Bluman G., Direct constru ction metho d for conserv ation la ws of partial differential equa- tions. I I. General treatment, Eur. J. Appl. Math. , 2002, V.13, Part 5, 567–585; arXiv :math-ph/0108024. [6] Arrigo D.J., Hill J.M., Broadbridge P ., N onclassical symmetry reductions of the linear diffusion equ ation with a nonlinear source, IMA J. Appl. Math. , 1994, V.52, 1–24. [7] Bluman G.W., Kumei S., S ymmetries and Differential Eq uations, Sp ringer, New Y ork, 1989. [8] Bluman G.W., Reid G.J. and Kumei S., New classes of symmetries for partial differential eq uations, J. Math. Phys. , 1988, V.29, 806–811. [9] Bluman G.W., Use and construction of p otential sym met ries, Math. Comput. Mo del ling , 1993, V.18, 1–14. [10] Bluman G.W., Poten tial symmetries and equiv alent conserv ation laws, in: N.H. I bragimo v, M. T orrisi, A. V alenti (Eds.), Mod ern Group Analysis: A dv anced A nalytical and Computational Method s in Mathematical Physics (Acireale, 1992), Kluw er, Dordrech t, 1993, pp. 71-84. [11] Bryan t R.L. and Griffiths P .A., Characteristic cohomology of differential sy stems I I: Conserv ation law s for a class of p arab olic equations, Duke Math. J. , 1995, V.78, 531–676. [12] Clarkson P .A., Mansfield E.L., S ymmetry reductions and exact solutions of a class of nonlinear heat eq ua- tions, Physic a D , 1994, V.70, 250–288. [13] Doro dnitsyn V.A., Group prop erties and inv arian t solutions of a nonlinear h eat equation with a source or a sink, Preprin t N 57, Mosco w, Keldysh Institute of A pplied Mathematics of Academy of Sciences U SSR, 1979. [14] F u shchic h W.I. and Nik itin A .G., Symmetries of Maxwel l’s e quations , D.Reidel, Dord recht, 1987. [15] F u shch ych W.I. an d Popovyc h R.O., Symmetry red uction and exact solution of t he Navier-Stok es equations, J. Nonline ar Math. Phys. , 1994, V .1, 75–113, 156–188; arXiv:math-ph /020701 6 . [16] F u shchic h W.I., Serov N .I., Conditional inv ariance and reduction of nonlinear heat equa- tion, Dokl. Akad. Nauk Ukr ain. SSR Ser. A , 1990, no.7 , 24–27 (in Russian). Ava iliable at http://w ww.imath.kiev.ua/ ∼ fushch ych. [17] Gandarias M.L., P otential sy mmetries of a p orous medium equ ation, J. Phys. A: Math. Gen. , 1996, V.29, 5919–59 34. [18] Gandarias M.L., New p otential symmetries for so me evolution equations, Physic a A , 2008, V.387, 2234–2242. [19] Ibragimov N.H. (Editor), Lie group analysis of differential eq uations — symmetries, exact solutions and conserv ation law s, V.1, CRC Press, Boca R aton, FL, 1994. [20] Iv anov a N.M. and P op ovych R.O., Equ iv alence of conserv ation la ws and equival ence of p otential sys- tems, Internat. J. The or. Phys , 2007, V .46, 2658–2668 ; preprint N1885 of ESI for Mathematical Physics, arXiv:math-ph/0611032. [21] Iv anov a N .M., Popovych R.O. and Soph ocleous C., Conserv ation laws of v ariable coefficient diffusion– conv ection equ ations, Pr o c e e dings of T enth International Confer enc e i n Mo dern Gr oup A nalysis , (Larnaca, Cyprus, 2004), 107–113. [22] Iv anov a N.M. , Popovych R.O. and Sopho cleous C., Group analysis of v ariable coefficient diffusion–conv ection equations. I. En h anced group classification, 2007, [23] Iv anov a N.M. , Popovych R.O. and Sopho cleous C., Group analysis of v ariable coefficient diffusion–conv ection equations. I I I. Conserv ation la ws, 2007, arXiv:0710.3053 . [24] Iv anov a N.M. , Popovych R.O. and Sopho cleous C., Group analysis of v ariable coefficient diffusion–conv ection equations. IV. Poten tial symmetries, 2007, arXiv :0710. 4251 . 18 [25] Kapitanskiy L.V., Group analysis of the Navier–Stok es and Euler equ ations in the presence of rotational symmetry and new exact solutions of t hese equations, Dokl. A c ad. Sci. USSR , 1978, V.243, 901–904. [26] Lie S., ¨ Ub er die In t egration durc h b estimmte I ntegra le von einer Klasse li n ear partieller Differentialg leich un g, Ar ch. for Math. , 1881, V.6, N 3, 328–368. ( T ranslation by N.H. Ibragimov: Lie S. O n integration of a class of linear partial differential equations by means of definite integral s, CRC Handb o ok of Lie Gr oup A nalysis of Differ ential Equations , V ol. 2, 1994, 473–508). [27] Meleshko S.V., Group classificati on of equations of tw o-dimensional gas motions, Prikl. Mat. Mekh. , 1994, V.58, 56–62 (in Russian); translation in J. Appl. Math. Me ch. , 1994, V.58, 629–635. [28] Mikhailov A.V., S habat A.B. and Sokolo v V.V., The symmetry approac h to classification of integrable equations (p p. 115–184), in What is inte gr abil ity? , Edited by V.E. Z ak haro v . Springer S eries in Nonlinear Dynamics. Springer-V erlag, Berlin, 1991. [29] Olver P .J., Applications of Lie groups to d ifferen t ial equations. Second edition. Graduate T exts in Mathe- matics, 107. S p ringer-V erlag, New Y ork, 1993. [30] Ovsianniko v L.V., Group analysis of differential equ ations, Academic Press, New Y ork, 1982. [31] Popovyc h H.V., Lie, partially in vari ant, and nonclassical submod els of Euler equations, Pr o c e e dings of Institute of Mathematics of NAS of Ukr aine , 2002, V.43, Part 1, 178–183. [32] Popovyc h R.O., R eduction op erators of linear second-order p arabolic equ ations, J. Phys. A , 2008, V .41, 185202; [33] Popovyc h R.O. and Iv anov a N.M., H ierarc hy of conserv ation la ws of diffusion–conv ection equations, J. Math. Phys. , 2005, V .46, 043502, arXiv:math-p h/0407008. [34] Popovyc h R.O. and Iv anov a N .M., Poten t ial equiv alence transformations for n onlinear diffu sion–con vection equations, J. Phys. A , 2005, V.38, 3145–3155, arXiv:math-ph/0402066. [35] Popovyc h R.O., Kunzinger M., Eshraghi H., Admissible transformations and normalized classes of nonlinear Schr¨ odinger equ ations, A cta Appl. Math. , d oi:10.1007/s10 440-008-9321-4, 45 p., arXiv:math-ph/0611061. [36] Popovyc h R .O., Kunzinger M. and Iva n ov a N.M., Conserv ation laws and p otentia l symmetries of linear parab olic equations, A cta Appl. Math. , 2008, V.100, no.2, 113–185 , arXiv:0706.04 43 . [37] Pucci E. and Saccomandi G., P otential symmetries and sol u tions by reduction of partial differential equ ations J. Phys. A , 1993, V.26 681–690. [38] Pucci E. and Saccomandi G., Poten tial symmetries of F okker Plank equations, Mo dern Group Analysis: Adv anced Analytical and Computational Metho ds in Mathematical Physics (Acireale, 1992), Kluw er Acad. Publ., Dordrech t, 1993, 291–298 . [39] Saccomandi G., Poten tial symmetries and direct red uction meth ods of order tw o, J. Phys. A , 1997, V.30, 2211–22 17. [40] Sopho cleous C., Poten tial symmetries of inhomogeneous nonlinear d iffusion eq uations, Bul l. Au str al. Math. So c. , 2000, V.61, 507–521 . [41] Sopho cleous C., Classification of p otential symmetries of generalised inhomogeneous nonlinear diffusion equations, Physic a A , 2003, V.320, 169–183. [42] Sopho cleous C., F urth er transformatio n prop erties of generalised inhomogeneous nonlinear diffusion equa- tions with va riable coefficients, Physic a A , 2005, V.345, 457–471. [43] V aneev a O .O., Johnpillai A .G., Popovych R.O. and Sopho cleous C., Enhanced group analysis and conserv a- tion laws of vari able co efficien t reaction–diffusion equations with p ow er nonlinearities, J. Math. Anal. Appl. , 2007, V.330, 1363–1386; arXiv:math-ph /0605081. [44] V aneev a O .O., Po p ovych R.O. and Soph ocleous C., Enhanced group analysis and exact solutions of v ariable coefficient semilinear d iffusion equations with a p ow er source, A cta Appl. Math. , doi:10.1007 /s10440-008- 9280-9, 46 p ., arXiv:0708.345 7 . [45] W olf T., A comparison of four app roac hes to th e calculation of conserva tion laws, Eur. J. Appl. Math. , 2002, V.13, Part 5, 129–152 . [46] Zhdanov R. and Lahn o V., Group Classification of th e General Evolution Equation: Lo cal and Quasilocal Symmetries SIGMA , 2005, V.1 Paper 009, 7 pages; arXiv :n lin.SI/051000 3 . 19