Conservation laws and normal forms of evolution equations

Conser v ation La ws and Normal F orms of Ev olu tion Equations Roman O. POPOVYCH † and Artur SERGYEYEV ‡ † F akult¨ at f¨ ur Mathematik, Universit¨ at Wien, Nor d b er gstr aße 15, A-1090 Wien, A ustria Institute of Mathema tic s of NAS of Ukr aine, 3 T er eshchenkivska Str., Kyiv-4, Ukr aine ‡ Mathematic a l Institute, Silesian University in O p ava, Na Rybn ´ ıˇ cku 1, 746 01 Op ava, Cze ch R epublic E-mail: † r op@ima th.kiev .ua, ‡ Artur.Ser g yeyev@math.slu.cz W e study lo cal conserv ation laws for evolution equations in tw o indep endent v aria bles. In particular, we present normal forms for the equations admitting one or tw o low-order con- serv ation laws. Examples include Har ry Dym equation, Ko rteweg–de-V ries-type equations, and Sc hw a rzian KdV equatio n. It is also shown that for linear evolution equations all their conserv ation laws are (mo dulo tr ivial conser ved vectors) at most qua dratic in the dependent v ariable and its deriv a tives. 1 In tro d uction The role pla y ed in th e sciences by linear and nonlinear evo lu tion equations and, in particular, by conserv ation la ws th er eof, is hard to ov er estimate (recall e.g. linear and nonlinear S chr¨ odinger equations and the Kortew eg–de V ries (KdV) equation in physics, reaction-diffusion systems in c hemistry and biology , and the Blac k–Schole s equation in th e finance, to name just a few). F or in stance, the disco very of higher conserv ation l aws for the Kd V equations provided an imp ortant milestone on th e wa y that has eve ntually lead to th e disco very of the inv erse s cattering transform and the mo d ern theory of integrable systems, see e.g. [21, 22]. Ho wev er, the theory of conserv ation laws for ev olution equations is still far from b eing complete ev en for the simplest case of t wo ind ep endent v ariables, and in the present pap er w e addr ess some issues of the th eory in question for this very case. W e shall d eal with an evol u tion equation in t w o indep endent v ariables, u t = F ( t, x, u 0 , u 1 , . . . , u n ) , n ≥ 2 , F u n 6 = 0 , (1) where u j ≡ ∂ j u/∂ x j , u 0 ≡ u , and F u j = ∂ F /∂ u j . W e shall also emplo y , d ep ending on con ve- nience or necessit y , th e follo win g notation f or lo w-order deriv ativ es: u x = u 1 , u xx = u 2 , and u xxx = u 3 . There is a considerable b o dy of r esults on conserv ation la ws of evo lu tion equations of the form (1). F or instance, in the seminal pap er [8] the authors stud ied, inter alia , conserv ation la ws o f Eq. (1 ) with ∂ F /∂ t = 0 for n = 2. They prov ed that the possib le d im en sions of spaces of inequiv alen t conserv ation la ws for suc h equ ations are 0, 1, 2 and ∞ , and describ ed the equations p ossessing spaces of conserv ation la ws of these dimensions (the precise d efinitions of equiv alence and order of conserv ation la ws are giv en in the n ext section). These results were further generalized in [28] for the case when F explicitly dep ends on t . Imp ortant results on conserv ation la ws of (1), t yp ically und er the assumptions of p olyno- mialit y and t , x -indep endence of F and of the conserv ation la ws themselve s , w ere obtained in [1, 2, 3, 4, 10, 11, 12, 15]. Ho wev er, for general Eq. (1) there is no simple picture analogous to that of the second-order case d iscussed ab o ve. F or instance, unlik e the second-order case, there exist od d-order ev olution equations that p ossess infin itely man y inequ iv alent conserv ation la ws of increasing orders without b eing linearizable. Rather, suc h equations are integrable via 1 the in ve r se scattering transform, the famous Kd V equation providing a prime example of su c h b ehavio r , see e.g. [13, 15, 21] and r eferences therein; for the fifth-order equations see [9]. Note that man y r esults on s ymmetries and conserv ation la ws w ere ob tained using the formal symmetry approac h and mo difications thereof, see e.g. the r ecen t surve y [20] and references therein, in particular [19, 34]. F or instance, it was sho wn that an equation (1) of even order ( n = 2 m ) has no conserv ation la ws (mo du lo trivial ones) of order greater than m , see [1, 10, 13, 14] for details. There also exists a closely related app roac h to the stud y of symmetries and conserv ation la ws of ev olution equations, th e so-called sym b olic metho d, see [18, 29, 30, 31 ] and references therein for details. Ho w ever, many imp ortant questions concerning the conserv ation la w s of ev olution equations w ere not answe r ed so far. F or example, w e are not a wa r e of an y s ignifi can t adv ances in the study of normal forms of ev olution equations admitting lo w -order conserv ation la ws considered in [8, 11, 28]. In the presen t pap er w e provide su c h normal forms with resp ect to conta ct or p oint transformations for equations admitting one or tw o lo w-order conserv ation la w s, resp ectiv ely , see Theorems 1 and 2 b elo w. Let us stress that in what follo ws we restrict ou r selv es to considering only lo c al conserv ation la ws wh ose densities and flu xes dep end only on the in dep end en t and dep end en t v ariables and a finite num b er of the deriv ativ es of the latter. The complete d escription of conserv ation la ws for line ar evo lution equations with t, x - dep end en t co efficien ts was also missin g so far. Belo w we sh o w that linear ev en-order equations of the f orm (1) can only p ossess conserv ation la ws linear in u j for all j = 0 , 1 , 2 , . . . wh ile the o dd-order equations can f urther admit the conserv ation la ws (at most) quadratic in u j , see The- orems 3 and 4, Corollary 6 and Theorem 5 b elo w. This n aturally generalizes some earlier results from [3, 12]; cf. also [5]. The generation of linear and qu adratic conserv ation la ws for linear differen tial equations is also discuss ed in some depth in [24, Section 5.3]. Belo w we denote by CL( E ) the space of local conserv ation la w s of E (cf. Section 3), w here E denotes a fixed equation from the class (1). In what f ollo ws D t and D x stand for the total deriv ativ es (see e.g. [24] for details) with resp ect to the v ariables t and x , D t = ∂ t + u t ∂ u + u tt ∂ u t + u tx ∂ u x + · · · , D x = ∂ x + u x ∂ u + u tx ∂ u t + u xx ∂ u x + · · · . As usual, the subscripts lik e t , x , u , u x , etc. stand for the partial deriv ativ es in the resp ective v ariables. 2 Admissible transformations of ev olution equations The conta ct tr ansformations mapping an equation from class (1) in to an other equation from th e same class are well kno wn [17] to ha v e the form ˜ t = T ( t ) , ˜ x = X ( t, x, u, u x ) , ˜ u = U ( t, x, u, u x ) . (2) The functions T , X and U m u st satisfy the n ondegeneracy assumptions, namely , T t 6 = 0 and rank  X x X u X u x U x U u U u x  = 2 , (3) and the con tact condition ( U x + U u u x ) X u x = ( X x + X u u x ) U u x . (4) The transformation (2) is u niquely extended to the deriv ativ e u x and to the higher d eriv ative s b y the form ulas ˜ u ˜ x = V ( t, x, u, u x ) and ˜ u k ≡ ∂ k ˜ u/∂ ˜ x k = ((1 /D x X ) D x ) k V , wh ere V = U x + U u u x X x + X u u x or V = U u x X u x 2 if X x + X u u x 6 = 0 or X u x 6 = 0, resp ectiv ely; th e p ossib ilit y of s imultaneous v anishin g of these t wo quan tities is ruled out b y (3 ). The transformed equation (1) r eads ˜ u ˜ t = ˜ F where ˜ F = U u − X u V T t F + U t − X t V T t , (5) and ( X u , U u ) 6 = (0 , 0) b ecause of (3) and (4). An y transformation of the form (2 ) lea ves the class (1 ) in v arian t, and therefore its extension to an arbitrary element F b elongs to the con tact equiv alence group G ∼ c of class (1), so there are no other elemen ts in G ∼ c . In other w ords, the equiv alence group G ∼ c generates th e wh ole set of admissible con tact transformations in the class (1), i.e., this class is normalized with resp ect to con tact transformations, see [26] for details. The ab ov e results can b e su m marized as follo ws. Prop osition 1. The class of e quations (1) is c ontact-normalize d. The c ontact e quivalenc e gr oup G ∼ c of the class (1) is forme d by the tr ansformatio ns (2) , satisfying c onditions (3) and (4) and pr olonge d to the arbitr ary element F by (5) . F urther m ore, the class (1) is also p oin t-normalized. The p oin t equiv alence group G ∼ p of th is class consists of th e transformations of the form ˜ t = T ( t ) , ˜ x = X ( t, x, u ) , ˜ u = U ( t, x, u ) , ˜ F = ∆ T t D x X F + U t D x X − X t D x U T t D x X , (6) where T , X and U are arbitrary smo oth f u nctions that satisfy the n on d egeneracy conditions T t 6 = 0 and ∆ = X x U u − X u U x 6 = 0. Notice that there exist sub classes of the class (1) whose s ets of admissible con tact transfor- mations are exhauste d by p oint transformations. In the p resen t pap er w e do not consider more general transformations, e.g., d ifferen tial sub- stitutions such as the C ole–Ho p f transformation. 3 Some basic results on conserv ation la ws It is well known that for any evolutio n equation (1) we can assume without loss of generalit y that the asso ciated quan tities lik e symmetries, cosymmetries, densities, etc., can b e tak en to b e indep end en t of the t -d er iv ativ es or mixed deriv ativ es of u . W e shall stic k to this assum p tion throughout the r est of the p ap er. F ollo wing [24] w e shall r efer to a (smooth) function of t , x and a fi nite n u m b er of u j as to a differ ential function . Giv en a differen tial function f , its or der (denoted b y ord f ) is the greatest in teger k suc h that f u k 6 = 0 but f u j = 0 for all j > k . F or f = f ( t, x ) w e assum e that ord f = 0. Th u s, for a (fixed) ev olution equatio n (1), whic h we denote b y E as before, we lose no generalit y [24] in considering only the c onserve d ve ctors of the form ( ρ, σ ), w h ere ρ and σ are differen tial functions whic h satisfy the condition D t ρ + D x σ = 0 m o d ˇ E , (7) and ˇ E means th e equation E together with all its differentia l consequences. Here ρ is the density and σ is the flux for th e conserv ed v ector ( ρ, σ ). Let δ δ u = ∞ X i =0 ( − D x ) i ∂ u i , f ∗ = ∞ X i =0 f u i D i x , f † ∗ = ∞ X i =0 ( − D x ) i ◦ f u i , 3 denote the op erator of v ariational d er iv ativ e, the F r´ ec het deriv ativ e of a differen tial fu nction f , and its formal adjoin t, resp ectiv ely . With this notation in mind we readily infer that the condi- tion (7) can b e rewritten as ρ t + ρ ∗ F + D x σ = 0. As ρ ∗ F = F δ ρ/δ u + D x ζ for some differentia l function ζ , see e.g. [13, Section 22.5], there exists a differentia l fun ction Ψ (in fact, Ψ = − ζ − σ ) suc h th at ρ t + F δ ρ δ u = D x Ψ . (8) A conserved v ector ( ρ, σ ) is called trivial if it satisfies the condition D t ρ + D x σ = 0 on the entire jet sp ace. It is easily seen that the conserved vect or ( ρ, σ ) is trivial if and only if ρ ∈ Im D x , i.e., ther e exists a different ial function ζ su c h that ρ = D x ζ . Two conserved v ectors are e quivalent if they differ by a trivial conserv ed v ector. W e shall call a c onservation law of E an equiv alence class of conserv ed v ectors of E . The set CL( E ) of conserv ation laws of E is a v ector space, and the zero element of this sp ace is the conserv ation la w b eing the equiv alence class of trivial conserv ed v ectors. Th is is why nonzero conserv ation laws are u sually called nontrivial . F or an y c ons er v ation la w L of E there exists a u nique d ifferen tial function γ called the char acteristic of L suc h that for an y conserved ve ctor ( ρ, σ ) asso ciated w ith L (we shall write this as ( ρ, σ ) ∈ L ) there exists a trivial conserv ed vecto r ( ˜ ρ, ˜ σ ) satisfying the condition D t ( ρ + ˜ ρ ) + D x ( σ + ˜ σ ) = γ ( u t − F ) . (9) It is imp ortan t to stress that, unlik e (7), the ab o ve equation holds on the en tire jet space r ather than merely mo d ulo ˇ E . The c haracteristic γ of an y conserv ation la w satisfies the equation (see e.g. [24]) D t γ + F † ∗ γ = 0 mo d ˇ E , or equiv alently , γ t + γ ∗ F + F † ∗ γ = 0 . (10) Ho w ever, in general a solution of (10) is not necessarily a c haracteristic of some conserv ation la w f or (1 ). Solutions of (10) are called c osymmetries , see e.g. [6]. It can b e sho wn that the c haracteristic of th e conserv ation la w asso ciated with a conserv ed v ector of the form ( ρ, σ ) equals δ ρ/δ u . This yields a necessary and su fficien t condition for a cosymmetry γ to b e a charac teristic o f a conserv ation la w (see e.g. [24]): γ ∗ = γ † ∗ . This condition means simply that the F r´ echet deriv ativ e of γ is f ormally self-adjoin t. The follo wing results are we ll kno wn, see e.g. [10] for Lemma 2. Lemma 1. Supp ose that an e quation fr om the class (1) admits a nontrivial c onserve d ve ctor ( ρ, σ ) , wher e ord ρ = k > 0 , and ρ u k u k = 0 . Then the c onserve d ve ctor ( ρ, σ ) is e q u ivalent to a c onserve d ve ctor ( ˜ ρ, ˜ σ ) with ord ˜ ρ 6 k − 1 . Pr o of. By assum ption, ρ = ρ 1 u k + ρ 0 , and h ence σ = − ρ 1 D k − 1 x ( F ) + σ 0 , wh ere ρ 1 = ρ 1 ( t, x, u, u 1 , . . . , u k − 1 ), ρ 0 = ρ 0 ( t, x, u, u 1 , . . . , u k − 1 ), and σ 0 = σ 0 ( t, x, u, u 1 , . . . , u k + n ). Pu t ˜ ρ = ρ − D x Φ and ˜ σ = σ + D t Φ, wh er e Φ = R ρ 1 du k − 1 . Then ˜ ρ u k = 0, ˜ σ u k + n = 0 and ( ˜ ρ, ˜ σ ) is a conserv ed v ector equiv alen t to ( ρ, σ ), and ord ˜ ρ 6 k − 1. In what follo ws, for any giv en conserv ation law L w e sh all, un less otherw ise explicitly stated, c ho ose a representat ive (that is, the conserv ed ve ctor) w ith th e low est p ossible order k of the asso ciated densit y ρ . The order in question (i.e., th e greatest in teger k such th at ρ u k u k 6 = 0 but ρ u j = 0 for all j > k ) will b e called the density or der of L and den oted b y ord d L . It equals one half of the order of the asso ciated c haracteristic. Lemma 2. F or any c onservation law L of a (1 + 1) -dimensional even-or der ( n = 2 q ) evolution e q uation of the form (1) we have ord d L 6 q . 4 4 Ev olution equations ha ving lo w-ord er conserv ation l a ws Con tact and p oin t equ iv alence transformations can b e used for bringing equ ations from th e class (1) that admit (at least) one or tw o nontrivia l lo w-order conserv ation la ws in to certain sp ecial forms. Th is is ac hiev ed through bringing the conserv ation la ws in q u estion to normal forms. Theorem 1. Any p air ( E , L ) , wher e E is an e quation of the form (1) and L is a nontrivial c onservation law of E with ord d L 6 1 is G ∼ c -e quivalent to a p air ( ˜ E , ˜ L ) , wher e ˜ E is an e quation of the same form and ˜ L is a c onservation law of ˜ E with the char acteristic e qual to 1 . Pr o of. Let T ∈ G ∼ c map an equation E in to (another) equation ˜ E from the same class (1), see Section 2. Q uite ob viously , th e inv ers e T − 1 of T in duces (through p ullbac k) a mapping from the s p ace CL( E ) of conserv ation la ws of E to CL ( ˜ E ). The conserved ve ctors of E are transf orm ed in to those of ˜ E acco r ding to the form ula [25, 27] ˜ ρ = ρ D x X , ˜ σ = σ T t + D t X D x X ρ T t . No w let an equatio n E from the class (1 ) ha ve a non trivial conserv ation la w L with ord d L 6 1. Fix a conserve d vec tor ( ρ, σ ) asso ciated with L , and set T = t . The densit y ˜ ρ of the transformed conserv ed ve ctor ( ˜ ρ, ˜ σ ) is easily seen to dep end at most on ˜ t , ˜ x , ˜ u , ˜ u ˜ x and ˜ u ˜ x ˜ x . Moreo ver, it is immediate that ˜ ρ is linear in ˜ u ˜ x ˜ x , so w e can pass to an equiv alen t conserved vec tor ( ¯ ρ , ¯ σ ) such that ∂ ¯ ρ/∂ ˜ u ˜ x ˜ x = 0, and hence f or the transformed coun terpart ˜ L of L we ha v e ord d ˜ L 6 1. Next, the conserv ation la w ˜ L asso ciated w ith ( ˜ ρ, ˜ σ ) has c h aracteristic 1 if and only if there exists a fu nction ˜ Φ = ˜ Φ( ˜ t, ˜ x, ˜ u, ˜ u ˜ x ) such that ˜ ρ = ˜ u + D ˜ x ˜ Φ. Up on going b ack to the old co ordinates x, t, u, u x and b earing in mind th at ˜ u = U ( t, x, u, u x ) and ˜ x = X ( t, x, u, u x ) this b oils d o wn to D x Φ + U D x X = ρ , w here Φ( t, x, u, u x ) = ˜ Φ( ˜ t, ˜ x, ˜ u , ˜ u ˜ x ). Splitting the equation D x Φ + U D x X = ρ with resp ect to u xx yields the s ystem Φ x + U X x + (Φ u + U X u ) u x = ρ, Φ u x + U X u x = 0 . (11) This s ystem in conjun ction with the con tact condition (4) has, i nter alia , the follo win g different ial consequence: Φ u + U X u = ρ u x . It is obtained as follo ws. W e subtract th e result of action of the op erator ∂ x + u x ∂ u on the second equation of (11 ) from the partial u x -deriv ativ e of the fir s t equation of (11) while taking in to accoun t the conta ct cond ition (4). Moreo v er , th e system (11) also implies the equation Φ x + U X x = ρ − u x ρ u x . Th u s, w e arrive at the system Φ x + U X x = ρ − u x ρ u x , Φ u + U X u = ρ u x , Φ u x + U X u x = 0 . (12) Rev ersing these steps sho ws that the s y s tem (12) implies (4) and (11). He n ce the com bined system of (4 ) and (11) is equ iv alen t to (12). T o complete th e pro of, it suffices to c hec k that for an y function ρ = ρ ( t, x, u, u x ) w ith ( ρ u , ρ u x ) 6 = (0 , 0) the system (12) has a solution ( X, U, Φ) wh ich satisfies the n ondegeneracy condition (3). Consider firs t the case ρ u x u x 6 = 0 and seek for solutions with X u x 6 = 0. The equation Φ u x + U X u x = 0 implies th at Φ u x 6 = 0 and U = − Φ u x /X u x . Then the r emaining equations in (12) tak e the form Φ x − X x X u x Φ u x = ρ − u x ρ u x , Φ u − X u X u x Φ u x = ρ u x . (13) 5 Eq. (13) can b e considered as an o v erd etermined system with resp ect to Φ. The compatibilit y condition for this system is ρ u x u x X x + u x ρ u x u x X u + ( ρ u − u x ρ uu x − ρ xu x ) X u x = 0; it sh ould b e treated as an equation f or X . As ρ u x u x 6 = 0 b y assumption, the equation in q u estion has a lo cal solution X 0 with X 0 u x 6 = 0. S ubstituting X 0 in to (13) yields a compatible p artial differen tial sys tem for Φ. T ake a lo cal solution Φ 0 of this system and set U 0 = − Φ 0 u x /X 0 u x . Th e c hosen triple ( X 0 , U 0 , Φ 0 ) satisfies (12). The nondegeneracy condition (3) is also satisfied. Ind eed, if w e assume the conv er s e, then U = Ψ( t, X ) for some function Ψ of t wo arguments, and (11) implies the equalit y ρ = Φ x + Ψ X x + (Φ u + Ψ X u ) u x + (Φ u x + Ψ X u x ) u xx = D x (Φ + R Ψ dX ) , i.e., ( ρ, σ ) is a trivial conserv ed vect or, whic h con tradicts the initial assumption on ( ρ, σ ). No w tur n to the case when ρ u x u x = 0. Then up to the equiv alence of conserved v ectors we can assu me th at ρ u x = 0 and ρ u 6 = 0, where the latter condition ensures non trivialit y of the asso ciated conserve d v ector. T he triple ( X, U, Φ) = ( x, ρ, 0) obviously satisfies (12) and (3), and the result follo ws. Corollary 1. A ny p air ( E , L ) , wher e E is an e quation of the f orm (1 ) and L is a nontrivial c onservation law of E with the density or der 0 is G ∼ p -e quivalent to a p air ( ˜ E , ˜ L ) , wher e ˜ E also is an e quation of form (1) and ˜ L is a c onservation law of ˜ E with the char acteristic e qual to 1 . Corollary 2. An e quation E fr om c lass (1) admits a nontrivial c onservation law L with ord d L 6 1 (r esp. ord d L = 0 ) if and only i f it c an b e lo c al ly r e duc e d by a c ontact (r esp. p oint) tr ansformation to the form ˜ u ˜ t = D ˜ x G ( ˜ t, ˜ x, ˜ u 0 , . . . , ˜ u n − 1 ) , G ˜ u n − 1 6 = 0 . (14) Note that up on setting n = 3 and ord d L = 0 in th is corollary w e reco v er Theorem 1.1 from [11]. Pr o of. Fix a nontrivia l conserv ation la w L of E w ith ord d L 6 1 (resp. ord d L = 0). By Theorem 1 (resp. Corollary 1), the pair ( E , L ) is reduced b y a con tact (resp. p oin t) transform ation to a p air ( ˜ E , ˜ L ), where the equ ation ˜ E has the form ˜ u ˜ t = ˜ F ( ˜ t, ˜ x, ˜ u 0 , . . . , ˜ u n ) and ˜ L is its conserv ation la w with the unit c haracteristic. Therefore, the equalit y D ˜ t ˜ ρ + D ˜ x ˜ σ = ˜ u ˜ t − ˜ F is satisfied for a conserv ed vec tor ( ˜ ρ, ˜ σ ) from ˜ L , i.e., up to a summand b eing a n ull divergence we ha ve ˜ ρ = ˜ u and ˜ F = − D ˜ x ˜ σ . T o complete the pro of, it suffices to pu t G = − ˜ σ . Con versely , let the equation E b e lo cally reducible by a con tact (resp. p oint) transformation T to the equation ˜ u ˜ t = D ˜ x G ( ˜ t, ˜ x, ˜ u 0 , . . . , ˜ u n − 1 ), wh ere G ˜ u n − 1 6 = 0. The transf ormed equation ˜ u ˜ t = D ˜ x G admits at least the conserv ation law ˜ L with th e unit c haracteristic. The preimage L of ˜ L with resp ect to T is a n on trivial conserv ation la w of E with ord d L 6 1 (resp. ord d L = 0). Corollary 3. If an e q u ation E of the form (1) with n > 4 (r esp. n > 5 ) has two line arly indep endent c onservation laws L I and L II , wher e ord d L I 6 1 and ord d L II 6 n / 2 − 1 (r esp. ord d L II < n/ 2 − 1 ) then it c an b e lo c al ly r e duc e d by a c ontact tr ansforma tion to the form (14) wher e G is line ar fr actional (r esp. line ar) with r esp e ct to ˜ u n − 1 , i.e., G = G 1 ˜ u n − 1 + G 0 G 3 ˜ u n − 1 + G 2 ( r esp. G = G 1 ˜ u n − 1 + G 0 ) , wher e G 0 , . . . , G 3 (r esp. G 0 and G 1 ) ar e differ ential fu nctions of or der less than n − 1 . If ord d L I = 0 then the c ontact tr ansformation i n question is a pr olongation of a p oint tr ansfor- mation. 6 Pr o of. Without loss of generalit y we can assume that ord d L I 6 ord d L II . By T heorem 1 and Corollary 2, the pair ( E , L I ) is reduced b y a con tact transformation to a pair ( ˜ E , ˜ L I ), where the equation ˜ E is of form (14 ) and the conserv ation la w ˜ L I has the density ˜ u . The transformed conserv ation la w ˜ L II satisfies the same inequalit y as th e original one, L II , i.e., ord d ˜ L II 6 n/ 2 − 1 (resp. ord d ˜ L II < n/ 2 − 1) if n > 4 (resp . n > 5). Belo w we omit tildes o ver the transformed v ariables for con v enience and assume that the conserv ation la w L I p ossesses the density u and, therefore, the equation E has the form (14). Let ( ρ II , σ II ) b e a conserv ed vect or asso ciated with L II and ord ρ II = ord d L II . By (8), it satisfies the cond ition ρ II t + ( D x G ) δρ II /δ u = D x Ψ for some differen tial function Ψ. Th e last equalit y can b e rewritten as ρ II t − GD x ( δ ρ II /δ u ) = D x Φ , (15) where Φ = Ψ − Gδ ρ II /δ u . Note that D x ( δ ρ II /δ u ) 6 = 0 b ecause otherwise the conserv ation la w s L I and L II are linearly dep endent. As ord ρ II t < n − 1, ord G = n − 1 and ord D x ( δ ρ II /δ u ) 6 n − 1 (resp. ord D x ( δ ρ II /δ u ) < n − 1), we ha ve ord Φ < n − 1. Finally , as D x ( δ ρ II /δ u ) must b e linear in the h ighest-order x -deriv ative of u it con tains, expressing G fr om (15) and taking in to accoun t the ab ov e inequalities for ord D x ( δ ρ II /δ u ) immed iately yields the d esired result. Theorem 2. L et E b e an e quation of the form (1) and L I and L II b e line arly indep endent c onservation laws of E of density or der 0. Any su c h triple ( E , L I , L II ) is G ∼ p -e quivalent to a triple ( ˜ E , ˜ L I , ˜ L II ) , wher e ˜ E is an e quation fr om the same class (1) that admits c onservation laws ˜ L I and ˜ L II with the char acteristics e qual to 1 and ˜ x , r e sp e ctively. Pr o of. Let ( ρ i , σ i ) ∈ L i and ord ρ i = 0, i = I , I I. Then γ i = ρ i u is the charac teristic of L i , i = I , I I. Mo r eo v er, γ I and γ II are lin early indep endent differential fu n ctions in view of the linear indep end ence of conserv ation la ws L I and L II . Th erefore, w e ha v e ( λ x , λ u ) 6 = (0 , 0), wh er e λ = γ II /γ I . (Indeed, otherwise the su bstitution of these charac teristics in to (10) would imply that λ t = 0 as well, i.e., the c h aracteristics γ I and γ II w ould b e linearly dep endent.) W e will pr ov e the existence of (and, in fact, construct) a p oin t equiv alence transformation of the form (6) with T ( t ) = t suc h that th e transformed conserv ed vecto r s ( ˜ ρ I , ˜ σ I ) and ( ˜ ρ II , ˜ σ II ) are equiv alent to the conserved vect ors with the densities ˜ u and ˜ x ˜ u , resp ectiv ely . In other words, w e w ant to ha ve ˜ ρ I = ˜ u + D ˜ x Φ and ˜ ρ II = ˜ x ˜ u + D ˜ x Ψ for some functions Φ = Φ( t, x, u ) and Ψ = Ψ( t, x, u ). In the old co ordinates these conditions take the form D x Φ + U D x X = ρ I and D x Ψ + X U D x X = ρ II . Splitting them with resp ect to u x yields Φ x + U X x = ρ I , Φ u + U X u = 0 and Ψ x + X U X x = ρ II , Ψ u + X U X u = 0 . After the elimination of Φ and Ψ f rom these systems through cross-differentia tion, w e arriv e at the conditions X x U u − X u U x = ρ I u and ρ I u X = ρ II u . If w e set X = λ = ρ II u /ρ I u then ( X x , X u ) 6 = (0 , 0). This ensures existence of a function U = U ( t, x, u ) whic h lo cally satisfies the equation X x U u − X u U x = ρ I u . I t is obvious th at the so c h osen functions X and U are functionally indep end en t and th at the ab o ve systems are then compatible with r esp ect to Φ and Ψ, and hence the p oin t transformation we sough t for do es exist. Corollary 4. An e quation E of the form (1) has (at le ast) two line arly indep endent c onservation laws of density or der 0 i f and only if it c an b e lo c al ly r e duc e d by a p oint tr ansformation to the form ˜ u ˜ t = D 2 ˜ x H ( ˜ t, ˜ x, ˜ u 0 , . . . , ˜ u n − 2 ) , H ˜ u n − 2 6 = 0 . (16) 7 Pr o of. If E of the form (1) admits (at least) tw o linearly ind ep endent conserv ation la ws of densit y order 0, then b y Th eorem 2 w e can assume (mo d ulo a suitably c h osen p oint transformation, if necessary) that E has the conserv ation la ws L I and L II with the c haracteristics 1 and x , resp ectiv ely . Then th ere exist conserved vec tors ( ρ I , σ I ) ∈ L I and ( ρ II , σ II ) ∈ L II suc h that D t ρ I + D x σ I = u t − F , D t ρ II + D x σ II = x ( u t − F ) . Up to the equiv alence of conserved vect ors mo dulo trivial ones we hav e ρ I = u and ρ II = xu . Hence D x σ I = − F and D x σ II = − xF . Com b ining these equalitie s , we fi nd that σ I = − D x ( σ II − xσ I ), i.e., F = D 2 x ( σ II − xσ I ). As a result, w e can represent the equ ation E in the form u t = D 2 x H , where H = σ II − xσ I , ord H = n − 2. Con versely , assume that E is red u ced to the equation ˜ u ˜ t = D 2 ˜ x H ( ˜ t, ˜ x, ˜ u, . . . , ˜ u n − 2 ), where H ˜ u n − 2 6 = 0, thr ough a p oin t transformation T . The transf ormed equation ˜ u ˜ t = D 2 ˜ x H admits at least tw o linearly indep en d en t conserv ation la ws, in particular, those with the c haracteristics 1 and ˜ x . T heir preimages under T are linearly in d ep end ent conserv ation la ws of E whose densit y orders are zero, and the r esu lt follo ws. Corollary 5. If an e quation E of the form (1) with n > 5 (r esp. 2 6 n 6 4 ) has two line arly indep endent c onservation laws L I and L II with ord d L I 6 1 and ord d L II < n/ 2 − 1 (r esp. ord d L I = ord d L II = 0 ), then the right-hand side F of E has the form F = F 3 u n + F 2 u 2 n − 1 + F 1 u n − 1 + F 0 , wher e F 0 , . . . , F 3 ar e diffe r ential functions of or der less than n − 1 . Remark 1. If in the pro of of Corollary 4 w e replace the conserv ation laws L I and L II b y linear com binations thereof, ˆ L I = a 11 L I + a 12 L II and ˆ L II = a 21 L I + a 22 L II , wh ere a ij , i, j = 1 , 2, are arbitrary constants su ch that a 11 a 22 − a 12 a 21 6 = 0, then the asso ciated equations of the form (16) are related through th e transformation ˆ t = ˜ t, ˆ x = a 22 ˜ x + a 21 a 12 ˜ x + a 11 , ˆ u = ( a 12 ˜ x + a 11 ) 3 a 11 a 22 − a 12 a 21 ˜ u, ˆ H = a 11 a 22 − a 12 a 21 a 12 ˜ x + a 11 H , where ˆ t , ˆ x , ˆ u and the d ifferen tial function ˆ H corresp ond to the conserv ation la ws ˆ L I , II . Suc h transformations, considered for all admissible v alues of a ij , i, j = 1 , 2, form a sub group G of the p oint equ iv alence group for the class of equations of the form (16). Th u s, u p to the G -equiv alence w e can assume that the form (16) of the equation E is asso ciated with the t wo- d imensional subsp ac e spanned b y its conserv ation la w s L I and L II rather than w ith L I and L II p er se . 5 Examples: third-order ev olution equatio n s Example 1. W e start with the so-c alled Harry Dym (HD) equation, see e.g. [13, Section 20.2] and references therein for m ore details: u t = u 3 u xxx . The subspace of its conserv ation la ws of densit y order not great er than one is five -dim en sional and generated b y the zero-order conserv ation la ws L i , i = I , . . . , IV , with the densities ρ I = u − 2 , ρ II = xu − 2 , ρ II I = x 2 u − 2 , and ρ IV = u − 1 , and the fir st-order conserv ation la w L V with the densit y ρ V = u 2 x u − 1 . The first three densities agree in the sense that ρ II /ρ I = ρ II I /ρ II . Hence up on in tro d ucing new v ariables ˜ t = − 2 t sign u , ˜ x = x , ˜ u = u − 2 obtained by app lyin g Theorem 2 to L I and L II , the HD equation can b e rewritten in an ev en more sp ecific than (16 ), and also well -kn o wn, conserv ativ e 8 form ˜ u ˜ t = D 3 ˜ x ( ˜ u − 1 / 2 ) . (W e transformed t ab o ve in ord er to simp lify the transformed equation.) The transf ormed equation obviously admits conserv ation la ws with the charact eristics equ al to 1, x and x 2 . F or the pair of conserv ation la ws L IV and L I Theorem 2 yields the trans formation ˜ t = t , ˜ x = − 2 /u , ˜ u = x/ 2 whic h maps the HD equation in to the equation ˜ u ˜ t = D 2 ˜ x  1 2 ˜ x 3 ˜ u 2 ˜ x  . The conserv ation la w L V is mapp ed into a conserv ation la w with the charac teristic 1 by the con tact transf ormation ˜ t = t , ˜ x = u 2 x /u , ˜ u = u − 2 u/u x , ˜ u = u 2 /u 3 x constructed using the metho d from the pro of of Theorem 1. Th e corresp onding transformed equation r eads ˜ u ˜ t = D ˜ x  − ˜ x 8 ˜ u 6 ˜ x 4(2 ˜ x ˜ u ˜ x ˜ x + 3 ˜ u ˜ x ) 2  . Example 2. Consider no w th e class of K d V-t yp e equations u t = u xxx + f ( u ) u x . (17) An y equation from this class admits at least th ree conserv ation laws L i , i = I , . . . , I I I, with the d ensities ρ I = u , ρ II = u 2 / 2, ρ II I = − u 2 x / 2 + ˇ f ( u ), wher e ∂ ˆ f /∂ u = f , ∂ ˇ f /∂ u = ˆ f . It is straigh tforwa r d to v er if y that if ∂ 3 f /∂ u 3 6 = 0 these conserv ation la ws f orm a b asis in the space of the conserv ation la ws of densit y order not greater than one. The reduction (17) to the form (14) u sing L I (resp. L II ) according to Theorem 1 is immediate. The conserv ation la w L I giv es rise to the ident ity transform ation and the representa tion u t = D x ( u xx + ˆ f ( u )) for (17). The transformation asso ciated with L II is ˜ t = t , ˜ x = x and ˜ u = ρ II = u 2 / 2. It maps equation (17) in to ˜ u ˜ t = D ˜ x  ˜ u ˜ x ˜ x − 3 4 ˜ u 2 ˜ x ˜ u + ε √ 2 ˜ u ˆ f ( ε √ 2 ˜ u ) − ˇ f ( ε √ 2 ˜ u )  , where ε = sign u . No w consider the conserv ation la w s L I and L II and apply Theorem 2. W e can directly follo w the pro cedure from the pro of of this theorem and set ˜ t = t , ˜ x = ρ II u = u and ˜ u = x . This is nothing but the ho dograph transformation inte r c hanging x and u . It reduces equation (17) to the equation (cf. [11]) ˜ u ˜ t = D 2 ˜ x  1 2 ˜ u 2 ˜ x − ˇ f ( ˜ x )  . Example 3. T he K dV equation, i.e., equation (17) w ith f ( u ) = u , p ossesses one more linearly indep end en t zero-order conserv ation la w L IV with the densit y ρ IV = xu + tu 2 / 2, cf. [21]. This giv es more p ossibilities for redu ction to the forms (14) an d (16). In analog y with the previous example, w e find that th e transformation associated with L IV is ˜ t = t , ˜ x = x and ˜ u = ρ IV = xu + tu 2 / 2. It maps the Kd V equation in to ˜ u ˜ t = D ˜ x ˜ u ˜ x ˜ x − 3 2 ˜ t Z ˜ u 2 ˜ x − 3 ˜ x Z ˜ u ˜ x ± Z 3 / 2 3 ˜ t 2 + 3 ˜ u Z − ˜ x ˜ t ˜ u − ˜ x 3 3 ˜ t 2 ! , where Z = ˜ x 2 + 2 ˜ t ˜ u . 9 F or the p air of the conserv ation laws L I and L IV w e h av e the transform ation of the f orm ˜ t = t , ˜ x = ρ IV u = x + tu an d ˜ u = u , and the transformed equ ation reads ˜ u ˜ t = D ˜ x  ˜ u ˜ x ˜ x (1 − t ˜ u ˜ x ) 3  = D 2 ˜ x  (1 − t ˜ u ˜ x ) − 2 2 t  . Another pair of the conserv ation la ws, L II and L IV , giv es rise to a more complicated trans- formation ˜ t = t , ˜ x = x/u + t , ˜ u = u 3 / 3, and a more cu m b ersome transformed equation, ˜ u ˜ t = D 2 ˜ x  (( ˜ x − ˜ t ) ˜ u ˜ x + 6 ˜ u ) ˜ u ˜ x 2(( ˜ x − ˜ t ) ˜ u ˜ x + 3 ˜ u ) 2  . Note that exhaustiv e lists of one- and t wo-dimensional su bspaces of zero-order conserv ation la ws of the KdV equation that are not equiv alen t with resp ect to the Lie p oin t sym metry group of the latter are {hL I i , hL II i , hL IV i} and {hL I , L II i , hL I , L IV i , hL II , L IV i} , resp ectiv ely . Therefore, the ab o ve d escription of normal forms (14 ) and (16) related to zero-order conserv ation la ws of the K dV equation is complete mo dulo th e action of the Lie p oin t s ymmetry group of the KdV equation, cf. Remark 1. Example 4. The Sc hw arzian K d V equation u t = u xxx − 3 2 u 2 xx u x p ossesses no zero-order conserv ation la ws. The subspace of its firs t-order conserv ation la ws is spanned b y the conserv ation la w s L i , i = I , . . . , I I I, w ith the densities ρ I = 1 /u x , ρ II = u/u x , ρ II I = u 2 /u x . F or transformin g L I in to a conserv ation la w with the density u , we constru ct, follo wing the p ro of of Th eorem 1, the con tact transformation ˜ t = t, ˜ x = u x , ˜ x = 2 x u 2 x − 2 u u 3 x , ˜ u = − 4 x u 3 x + 6 u u 4 x , whic h m aps the Sch w arzian KdV equation in to ˜ u ˜ t = D ˜ x  − 4 ˜ x − 5 ( ˜ x 2 ˜ u ˜ x ˜ x + 6 ˜ x ˜ u ˜ x + 6 ˜ u ) 2  . F urther e xamp les of n ormal forms for lo w -order n onlinear ev olution equations, including physic ally rele v ant examples like the nonlinear diffusion-con vec tion equations, can b e found in [28]. 6 Conserv ation la ws of linear ev olution equatio n s An y linear partial differenti al equation admits conserv ation la ws whose characte r istics dep en d on indep endent v ariables only and run thr ough the set of solutions of the adjoin t equation. The corresp ond ing conserved v ectors are linear w ith resp ect to the unkno w n function and its deriv ativ es. It is natural to call the conserv ation la ws of th is k in d line ar [24, Section 5.3]. Let u s stress that, f ollo wing th e literature, here and b elo w w e allo w f or a sligh t abuse of terminology b y calling a conserv ation la w linear (resp. qu adratic) when it con tains a conserve d v ector whic h is linear (resp. quadr atic) in the totalit y of v ariables u 0 , u 1 , u 2 , . . . . The problem of describin g other kind s of conserv ation la ws for general linear partial d ifferen - tial equations is quite difficult. Ho we ver, it can b e solv ed for certain sp ecial classes of equations including linear (1 + 1)-dimensional ev olution equations. 10 Consider an equation E of form (1), where the fu nction F is linear in u 0 , . . . , u n , i.e., F = F u = n X i =0 A i ( t, x ) u i , wh ere F = n X i =0 A i ( t, x ) D i x , A n 6 = 0 . Th u s, the equation E reads u t = F u. (18) Then the condition (10) for cosymmetries tak es th e form D t γ + F † γ = 0 mo d ˇ E , wh ere F † = n X i =0 ( − D x ) i ◦ A i ( t, x ) . The op erator F † is the formal adj oin t of F . W riting out the condition (10) yields γ t + X k γ u k n X i =0 k X j =0  k j  A i k − j u i + j + n X i =0 ( − 1) i i X s =0  i s  A i i − s D s x γ = 0 , (19) where A i j = ∂ j A i /∂ x j . A function v = v ( t, x ) is a cosymmetry of the equation E if and only if it is a solution of th e adjoint equation E ∗ : v t + F † v = 0 . (20) An y cosymmetry of E that do es not dep end on u and the deriv ativ es thereof is a c haracteristic of a linear conserv ation la w of E , and an y linear conserv ation la w of E has a c haracteristic of this form. Namely , a solution v = v ( t, x ) of the adjoin t equation E ∗ corresp onds to the conserv ed v ector ( ρ, σ ) of E with ρ = v ( t, x ) u and σ = P n − 1 i =0 σ i ( t, x ) u i . The co efficien ts σ i are f ound recursiv ely from the equations σ n − 1 = − v A n , σ i = − v A i +1 − σ i +1 x , i = n − 2 , . . . , 0 . (21) It turn s ou t that al l cosymmetries of even-or der equations (18) are of this f orm. Theorem 3. F or any line ar (1 + 1) -dimensional evolution e quation of even or der, al l its c osym- metries dep end only on x and t , and the sp ac e of al l c osymmetries i s isomorphic to the solution sp ac e of the asso ciate d adjoint e q uation. Pr o of. Supp ose that there exists a ν ∈ N ∪ { 0 } suc h that γ u ν 6 = 0 and d en ote r = max { ν ∈ N ∪ { 0 } | γ u ν 6 = 0 } . F or even n v anishing of the co efficient at u n + r in (19) yields the equation 2 A n γ u r = 0, whence γ u r = 0. This contradict s the original assu mption, and hence γ dep end s only on t and x . Corollary 6. F or any line ar (1 + 1) -dimensional evolution e quation of even or der its sp ac e of c onservation laws is exhauste d by line ar ones and is isomorphic to the solution sp ac e of the c orr esp onding adjoint e quation. F or o dd n things b ecome somewhat more inv olv ed. Theorem 4. F or any line ar (1 + 1) -dimensional evolution e quation of o dd or der, al l its c osym- metries ar e affine in the totality of variables u 0 , u 1 , u 2 , . . . . 11 Pr o of. In con trast with the case of ev en n , no w the co efficien t at u n + r in (19) v anish es identic ally . Requiring the co efficient at u n + r − 1 in (19) to v anish yields nA n 0 D x γ u r = (2 A n − 1 0 + ( r − 1 − n ) A n 1 ) γ u r , so γ u r dep end s only on t and x . Using this resu lt wh ile ev aluating the co efficient at u n + r − 2 yields nA n 0 D x γ u r − 1 = (2 A n − 1 0 + ( r − 1 − n ) A n 1 ) γ u r − 1 + ψ r − 1 , where ψ r − 1 is a function of t and x , whic h is expressed via γ u r and A i ; the explicit form of ψ r − 1 is n ot imp ortant here. T h u s, γ u r − 1 also dep ends only on t and x . Iterating th e ab o ve pro cedu re allo ws us to conclud e that the function γ is affin e in u 0 , . . . , u r , that is, γ = Γ u + v ( t, x ) , Γ = r X k =0 g k ( t, x ) D k x , (22) and the result f ollo ws. Note that if w e restrict ourselv es to the case of p olynomial cosymmetries for equations with constan t co efficient s , then up on com b ining Theorems 3 and 4 we reco ver (part of ) Prop osition 1 of [12]. F or the equations (18) with F † = − F the determining equations for cosymmetries and for c haracteristics of generalized sym m etries coincide. Th is observ ation in conjunction with Theo- rem 4 implies the f ollo wing assertio n (cf. [33]). Corollary 7. F or any line ar (1 + 1) -dimensional evolution e quation (18) of o dd or der such that F † = − F , al l its gener alize d symmetries ar e affine in u j for al l j . No w let u s get bac k to the general case of (18) with o dd n . S u bstituting the represen tation (22) for γ int o (19) r ev eals that v = v ( t, x ) satisfies the adjoin t equation (20) wh ic h is decoupled from the equations for g i . T h u s, v is a cosymmetry p er se . Just as b efore, to any such cosym- metry there corresp onds a linear conserv ation la w w ith the densit y ρ = v ( t, x ) u . Ho wev er , the issue of existence of conserv ation laws asso ciated with cosymmetries linear in u j is non trivial. Indeed, let γ = Γ u . As we wan t γ to b e a c h aracteristic of a conserv ation la w, we sh ould require that γ ∈ Im δ /δ u (cf. Section 3). Hence, the op er ator Γ should b e form ally self-adjoin t and, in particular, its order should b e ev en (note, ho wev er, th at if Γ is not formally s elf-adjoint, w e can tak e its formally self-adjoin t part ˜ Γ = (Γ + Γ † ) / 2; ˜ γ = ˜ Γ u is easily v er ifi ed to b e a cosymmetry if so is γ ). The density of the conserv ation law asso ciated with the c haracteristic γ reads, up to the usual add ition of a total x -deriv ativ e of something, ρ = 1 2 u Γ u . Without loss of generalit y we can also assu me the corresp ondin g fl ux to b e quadratic in u 0 , u 1 , . . . , se e Theorem 5.104 of [24 ], so the conserv ation la w in question is quadr atic , and we obtain the follo wing result. Theorem 5. F or any line ar (1 + 1) -dimensional evolution e quation of o dd or der, the sp ac e of its c onservation laws is sp anne d by line ar and quadr atic ones. F or linear conserv ation la ws with the densities of the form ρ = v ( t, x ) u w here v solv es the adjoin t equatio n w e still ha v e (21). No w tu r n to the qu adratic conserv ation laws. Th e d ifferen tial function Γ u is a charact eristic of a conserv ation la w f or E if and only if the op erator Γ satisfies the f ollo wing equiv alen t conditions: 1) it maps the solutions of the equ ation E int o solutions of the adjoint equation E ∗ ; 2) ∂ Γ /∂ t + Γ F + F † Γ = 0; 3) ( ∂ t + F † )Γ = Γ( ∂ t − F ), i.e., th e op erator Γ( ∂ t − F ) is form ally sk ew-adjoint. 12 Note that if the op er ator F is formally skew-adjoin t ( F † = − F ) th en the op erators ∂ t − F and Γ comm ute: [ ∂ t − F , Γ] = 0, i.e., Γ is a symm etry op erator for the equation E . An y linear equation admits a symmetry u∂ u , and the asso ciated op erator Γ is the iden tity op erator which is obvio u sly f ormally self-adjoin t. Com bin ing this result with the ab o ve we obtain the follo wing assertion. Prop osition 2. Any line ar (1 + 1) -dimensional evolution e quation (18) of o dd or der such that F † = − F p ossesses a c onservation law with the density ρ = u 2 . Moreo v er, linear (1 + 1)-dimensional ev olution equations of o dd order can p ossess infi nite series of quadratic conserv ation la ws of arbitrarily high orders, as illustrated by the follo wing example. Example 5. Consider the equation u t = u xxx . (23) It is straigh tforwa r d to ve r ify that in this case the determinin g equations for cosymmetries and c haracteristics of (generaliz ed ) sym metries coincide b ecause Eq. (23) is iden tical with its adjoin t. Denote by S the space of a ll ge n eralized symm etries of (23) a n d let Q b e the space of symmetries of the form f ( t, x ) ∂ u , where f solv es (23): f t = f xxx . By Corollary 7 the quotient space S / Q is exhausted by linear generalized sy m metries. S uccessiv ely solving the determining equations (cf. e.g. [23]) we find that the s p ace S / Q is spanned by the symmetries of the form ( D k x Υ l u ) ∂ u , where k , l = 0 , 1 , 2 , . . . and Υ = x + 3 tD 2 x . As the determining equations for symmetries and cosymmetries of (23) coincide, the sp ace of cosymmetries for (23) is sp anned b y the follo wing ob jects: 1) the cosymmetries of the f orm f ( t, x ) where u = f ( t, x ) is any solution of (23); 2) the cosymmetries of the f orm D k x Υ l u , where k , l = 0 , 1 , 2 , . . . . An y cosymmetry of the first kind is asso ciated with a conserv ation la w with the density ρ = f ( t, x ) u . As f or cosymmetries of the second kind, only those w ith eve n k = 2 m are c haracteristics of the conserv ation la w s. The conserv ation la ws in question can (mo d ulo trivial ones) b e c hosen to b e quadratic, with the densities ρ lm = 1 2 u D m x Υ l D m x u and the den sit y orders l + m , l, m = 0 , 1 , 2 . . . . Ho w ever, there also exist linear (1 + 1)-dimensional evol u tion equations of o dd order which ha ve no quadratic conserv ation la ws. Example 6. The op erator F = D 3 x + x asso ciated with the equations u t = u xxx + xu (24) is not formally sk ew-adjoint. Equation (24) p ossesses n on trivial symmetries w hic h are linear com binations of the op erators (( D 3 x + x ) k ( D x + t ) l u ) ∂ u but they cannot b e emplo y ed for con- struction of quadr atic conserv ation laws of (24) in th e ab ov e fashion. In fact, all cosymmetries of (24) dep end only on x and t , and therefore this equation has no quadratic conserv ation la ws. Indeed, using the pro of b y con tradiction, supp ose that (24) has a cosymmetry γ = Γ u , and ord γ = r , i.e., g r 6 = 0. T he cond ition ∂ Γ /∂ t + Γ F + F † Γ = 0 implies the follo wing system of determining equations for the co efficients of Γ: 3 g i x = ( i + 3) g i +3 + g i +2 t − g i +2 xxx + 2 xg i +2 − 3 g i +1 xx , i = 1 , . . . , r, (25) 2 g 2 + g 1 t − g 1 xxx + 2 xg 1 − 3 g 0 xx = 0 , (26) g 1 + g 0 t − g 0 xxx + 2 xg 0 = 0 , (27) 13 where the fun ctions g r +3 , g r +2 and g r +1 v anish by definition. W e successiv ely inte grate (25) starting from the equations with the greatest v alue of i and going do w n. The equations for i = r and i = r − 1 imply that the co efficien ts g r and g r − 1 dep end on t but not on x . Pro ceeding b y induction, we find that f or any j = 0 , . . . , r the fu nction g r − j is a p olynomial in x of degree 2[ j / 2]. The ratio of the co efficient at the highest p o wer of x in g r − j to g r (resp. g r − 1 ) is a constan t if j is eve n (resp . o dd). Then (26) an d (27) imply g r = 0 and g r − 1 = 0. This con tradicts our assumption that g r 6 = 0, and the result follo ws. 7 Conclusions In this pap er w e ha ve p resen ted normal form s f or the ev olution equations in t wo indep end en t v ariables p ossessing lo w-order conserv ation la ws, s ee Theorems 1 and 2, and Corollaries 2–5 for details. Using these n ormal forms considerably simplifies the construction of nonlo cal v ariables asso ciated with the conserv ation la ws in question an d hence the s tudy of the Ab elian co verings and nonlo cal symmetries, includ ing p oten tial symmetries, for the equ ations in question in spirit of [7, 25, 27, 16, 32], and references th erein. As these norm al forms are asso ciated, up to a certain natur al equiv alence (see Remark 1), with the subsp ac es spanned by conserv ation la ws rather than conserv ation la ws p er se , w e are naturally led to p ose the problem of cla s sification of in equiv alent sub spaces of (low-o r der) conserv ation laws for the classes or sp ecial cases of ev olution equations of in terest. As f or the line ar ev olution equations in t w o ind ep endent v ariables, we ha ve shown that their conserv ation la ws are (mo d ulo trivial conserved vec tors, of course) at most quadr atic in the dep end en t v ariable and the deriv ativ es thereof, see Theorem 3. Moreo v er, for the linear evo lu tion equations of even order their conserv ation la ws are at most linear in these quan tities, and th e asso ciated d ensities can b e c hosen to h a ve the form of a pro duct of the dep endent v ariable with a solution of the adjoint equation (Theorem 4). It is natural to ask whether similar resu lts can b e obtained for more general lin ear PDEs (cf. [33] f or the case of symm etries), an d w e intend to address this issu e in ou r future w ork. Ac knowledgem ents The r esearc h of R.O.P . was supp orted by the p ro ject P20632 of the Austrian Science F und . The researc h of A.S. w as sup p orted in part by the Ministry of Education, Y outh and Sp orts of the Czec h Republic (M ˇ SMT ˇ CR) under gran t MSM 4781 305904, and b y S ilesian Universit y in Opa v a under grant IGS 2/2009. T h e au th ors are pleased to th ank M. Kunzinger for stim ulating discussions. A.S. gratefully ac k n o wledges th e warm hospitalit y extended to h im b y the Depart- men t of Mathematics of the Universit y of Vienna d uring his visits in the course of preparation of the present pap er. It is our great p leasure to thank the referees for useful suggestions that ha ve considerably impro ved the pap er. 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