On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations

On nonlo cal symmetries, nonlo cal conserv ation la ws and no nlo cal transfor mations of ev olution equations: Tw o linearisable hierarc hi es Norb ert Euler and Marianna E uler Dep artment of Mathematics, Lu le ˚ a Univ e rsity of T e chnolo g y SE-971 87 Lule ˚ a, Swe den Norb e rt.Euler@sm.luth.se; Marianna.Euler@sm.luth.se Abstract : W e discuss n onlo cal symmetries and nonlocal conserv ation la ws that f ollo w from the systematic p otentia lisation of ev olution equations. Those are the Lie p oin t sym- metries of the auxiliary systems, also kno wn as p oten tial symmetries. W e defi ne h igher- degree p oten tial symmetries whic h th en lead to nonlo cal conserv ation la ws and nonlo cal transformations for the equations. W e demonstrate our app r oac h and derive second degree p oten tial symmetries for the Bu rgers’ h ierarc hy and the Calogero-Dega sp eris-Ibragimo v- Shabat h ierarc hy . 1 In tro d uction The concept of nonlo cal symmetries of partial differentia l e qu ations and its relations to lo cal Lie p oin t sym metries of its asso ciated auxiliary s y s tems was in tro duced b y Bluman , Kumei and Re id [3 ] and a r e kno wn as p oten tial symmetries. W e p oin t out that more ge n - eral t yp e of p oten tial symm etries were introdu ced earlier b y Krasil’shchik and Vinogrado v (see [13] and [20]). In the present p ap er our starting p oint is based on p otent ial symmetries as int r o duced in [3] and [2]. W e defin e higher-degree nonlo cal symmetries f or ev olution equations by in tro d ucing further auxiliary system b y higher-degree p otentia lisations. This leads to nonlo cal conserv ation la ws for the giv en ev olution equations and to nonlo cal transforma- tions b et w een the ev olution equatio n s a n d its potent ialised equations. W e demonstrate our approac h by considering the w ell-kno wn Bu r gers’ hierarch y and the so-called Calogero- Degasp er is-Ib ragimo v-Shabat hierarc hy ([5], [12 ], ([19], ([4]). Both of these hierarc h ies are kno wn to b e linearisable (see e.g. [4], [7] and [16]. See also [18] for a discussion on n on- lo cal s ymmetries of the Calogero-Deg asp eris-Ibragimo v-Shabat equation). W e s h o w that the linearisations of the t wo hierarc h ies follo w directly from their second p ote ntialisa tions. An inte r esting and unexp ected result of our inv estigation is th at second-degree p oten tial symmetries (as defin ed by Definition 2.1) exist only f or the fi r st mem b ers of b oth the Burgers’ and the Calogero-Deg asp eris-Ibragimo v-Shabat hierarchies. On the notatio n: Throughout this pap er D a [ p ] denotes the total deriv ativ e-op erator of the dep endent v ariable p ( a, b ) with resp ec t to the indep enden t v ariable a , wher e sub scripts 2 N Euler and M Euler of p den ote partial deriv ativ es: D a [ p ] := ∂ ∂ a + p a ∂ ∂ p + p aa ∂ ∂ p a + p ab ∂ ∂ p b + p 3 a ∂ ∂ p aa + · · · . (1.1) The f ormal inv erse-op erator of D a [ p ] is denoted b y D − 1 a , such that D − 1 a ◦ D a [ p ] ϕ = D a [ p ] ◦ D − 1 a ϕ = ϕ. (1.2) Moreo v er D n a [ p ] ϕ = D n − 1 a [ p ] ◦ D a [ p ] ϕ, n ∈ N . (1.3) If the dep endence of the op erator D a [ p ] on p is obvious, w e write jus t D a instead of D a [ p ]. 2 Preliminaries and higher-degree p oten tial symmetries Consider an n th -order ev olution equ ation of the general form u t = F ( x, u, u x , u xx , u 3 x , . . . , u nx ) . (2.1) Assume that (2. 1 ) is a symmetry-in tegrable ev olution equ ation [10], i.e . (2.1 ) admits a hereditary r ecursion op erator R [ u ] suc h that [ L F [ u ] , R [ u ]] = D t [ u ] R [ u ] , (2.2) where L F [ u ] is th e linear op erator L F [ u ] := ∂ F ∂ u + ∂ F ∂ u x D x + ∂ F ∂ u xx D 2 x + · · · . (2.3) Assume further that the hierarc hy of symmetry-in tegrable ev olution equ ations can b e present ed in the form u t = R n [ u ] u x , n ∈ N , (2.4) suc h that (2.1) corresp onds to the first mem b er of the hierarch y (2.4) w ith n = 1. The conserv ed curr en t, Φ t , for (2.1) m ust satisfy the relation ([11], [1]) Λ = ˆ E [ u ] Φ t , (2.5) where Λ denotes an integrati n g factor for (2.1), i.e. ˆ E [ u ] (Λ u t − Λ F ( x, u, u x , u xx , . . . u nx )) = 0 . (2.6) Here ˆ E [ u ] is the Euler op erator ˆ E [ u ] := ∂ ∂ u − D t ◦ ∂ ∂ u t − D x ◦ ∂ ∂ u x + D 2 x ◦ ∂ ∂ u xx − D 3 x ◦ ∂ ∂ u 3 x + · · · . (2.7) Nonlo cal symmetries, conserv ation la ws and transform ations 3 F or the flu x, Φ x , we state Prop osition 2.1: L et Λ b e an inte gr ating factor for the evolution e qu ations (2.1) and assume that the c orr esp onding c onserve d curr ent, Φ t , admits the dep endenc e Φ t = Φ t ( x, u, u x , u xx , u 3 x ) . (2.8) Then the flux, Φ x , f or (2.1) is giv en by Φ x = − D − 1 x (Λ F ) − ∂ Φ t ∂ u x F − ∂ Φ t ∂ u xx D x F − ∂ Φ t ∂ u 3 x D 2 x F + F D x  ∂ Φ t ∂ u xx  − F D 2 x  ∂ Φ t ∂ u 3 x  + ( D x F ) D x  ∂ Φ t ∂ u 3 x  . (2.9) The hier ar chy (2.4) admits the same inte gr ating factor, Λ , as the first memb er of the hier ar chy (for n = 1 ) and henc e the same c orr e sp onding cu rr ent, Φ t . The flux, Φ x , for the hier ar chy, (2.4), for al l n ∈ N then takes the fom Φ x ( x, u, u x , . . . ; n ) = − D − 1 x (Λ R n [ u ] F ) − ∂ Φ t ∂ u x R n [ u ] F − ∂ Φ t ∂ u xx D x ( R n [ u ] F ) − ∂ Φ t ∂ u 3 x D 2 x ( R n [ u ] F ) + ( R n [ u ] F ) D x  ∂ Φ t ∂ u xx  − ( R n [ u ] F ) D 2 x  ∂ Φ t ∂ u 3 x  + D x ( R n [ u ] F ) D x  ∂ Φ t ∂ u 3 x  , (2.1 0) wher e we assume the dep endenc e of Φ t as state d in (2.8). Remark : T he pro of of Proposition 2.1 is straig htforw ard, n amely b y in tegrating the conserv atio n la w  D t Φ t + D x Φ x    u t = R n [ u ] u x = 0 (2.11) of the hierarch y (2.4) with resp ect to x , that is Φ x = − D − 1 x  D t Φ t    u t = R n [ u ] u x . Assume no w that the ev olution equation (2.1) admits a conserv ed cu rrent , Φ t 1 , and flux, Φ x 1 . F ollo wing [3] a first p otential v ariable v is then defined by the auxiliary system: v x = Φ t 1 ( x, u, u x , . . . ) (2.12a ) v t = − Φ x 1 ( x, u, u x , . . . ) . (2.12b) W e name sys tem (2.12a)–(2.12b) the first auxiliary system of (2.1). Assume further that (2.12b) can b e expressed in terms of the first p oten tial v ariable v , i.e. (2.12b) b ecomes by (2.12a) th e first p otential e quation of the general form v t = G ( x, v x , v xx , . . . , v nx ) (2.13) 4 N Euler and M Euler whic h ma y again admit a c onserved curr en t, Φ t 2 , and flux, Φ x 2 . A furth er p ote ntial w is then in tro duced for (2.13), and named the se c ond p otential for (2.1), by the se c ond auxiliary system w x = Φ t 2 ( x, v , v x , . . . ) (2.14a ) w t = − Φ x 2 ( x, v , v x , . . . ) . (2.14b) The corresp ond in g p oten tial equ ation for (2.13) is then obtained from (2.14a) and (2.14b ), whic h we assume to hav e the general form w t = H ( x, w x , w xx , . . . , w nx ) . (2.15) W e name (2.15) the se c ond p otential e quation for (2.1). W e no w in tro duce the follo wing Definition 2.1: The Lie p oint symmetry gener ators Z = ξ 1 ( x, t, u, v ) ∂ ∂ x + ξ 2 ( x, t, u, v ) ∂ ∂ t + η 1 ( x, t, u, v ) ∂ ∂ u + η 2 ( x, t, u, v ) ∂ ∂ v (2.16) of the first auxiliary system (2.12a)–(2.12b) for (2.1), i.e . v x = Φ t 1 ( x, u, u x , . . . ) v t = − Φ x 1 ( x, u, u x , . . . ) , ar e define d as the first-de gr e e p otential symmetries of (2.1) if the infinitesimals ξ 1 , ξ 2 and η 1 dep end essential ly on the first p otential variable v , that is  ∂ ξ 1 ∂ v  2 +  ∂ ξ 2 ∂ v  2 +  ∂ η 1 ∂ v  2 6 = 0 . (2.17) The se c ond-de gr e e p otential symmetries of (2.1) ar e define d by the Lie p oint symmetry gen- er ators of the c ombine d first- and se c ond-auxiliary systems (2.12a)–(2.12b) and (2.14a)– (2.14b), that is the Lie p oint symmetry gener ators of the form Z = ξ 1 ( x, t, u, v , w ) ∂ ∂ x + ξ 2 ( x, t, u, v , w ) ∂ ∂ t + η 1 ( x, t, u, v , w ) ∂ ∂ u + η 2 ( x, t, u, v , w ) ∂ ∂ v + η 3 ( x, t, u, v , w ) ∂ ∂ w (2.18) for the system v x = Φ t 1 ( x, u, u x , . . . ) v t = − Φ x 1 ( x, u, u x , . . . ) w x = Φ t 2 ( x, v , v x , . . . ) w t = − Φ x 2 ( x, v , v x , . . . ) , Nonlo cal symmetries, conserv ation la ws and transform ations 5 wher e the infinitesimals ξ 1 , ξ 2 , η 1 and η 2 dep end essential ly on the se c ond p otential variable w , that is  ∂ ξ 1 ∂ w  2 +  ∂ ξ 2 ∂ w  2 +  ∂ η 1 ∂ w  2 +  ∂ η 2 ∂ w  2 6 = 0 (2.19) It sh ould b e clear that Definition 2.1 can easily b e extended to m th-degree p oten tial symmetries. 3 The Burgers’ hierarc h y Consider the Burgers’ equation in the form u t = u xx + 2 uu x . (3.1) It is w ell kno w n that (3.1) admits only one lo cal integ r ating factor and one lo cal conser- v atio n la w (see e.g. [15]), where Λ = 1 , Φ t 1 = u, Φ x 1 = −  u x + u 2  . (3.2) Equation (3.1) admits the recursion op erator [14 ] R [ u ] = D x + u + u x D − 1 x ◦ 1 (3.3) and the Burgers’ hierarch y then tak es the form u t = R n [ u ] u x , n = 1 , 2 , . . . . (3.4) W e remark that a g eneral class of linearisable sec ond -order ev olution equati ons and its recursion op erato r s , for w hic h the Burgers’ h ierarc hy is a sp ecial case, wa s rep orted in [8] and [7]. 3.1 Nonlo cal conserv ation la ws and linearisation W e pro ve the follo wing Prop osition 3.1: The Bur gers’ hier ar chy (3.4), u t = R n [ u ] u x , n = 1 , 2 , . . . , with R giv e n by (3.3) admits the first p otentialisat ion of the f orm v t = P n [ v x ] v x , n = 1 , 2 , . . . , (3.5) wher e P [ v x ] = D x [ v x ] + v x , (3.6) 6 N Euler and M Euler and the se c ond p otentialisatio n w t = w ( n +1) x , n = 1 , 2 , . . . , (3.7) wher e v x = u (3.8a) v t = P n [ u ] u, n = 1 , 2 , . . . , (3.8b) w x = e v (3.8c) w t = D n x [ v ] e v , n = 1 , 2 , . . . , (3 .8d) and P [ u ] = D x + u. (3.9) The c orr e sp onding nonlo c al c onserve d cu rr ent, Φ t , and flux, Φ x , for hier ar chy (3.4) ar e Φ t = e R u dx (3.10a ) Φ x = − D n x [ u ]  e R u dx  , n = 1 , 2 , . . . (3.10b) and the line arising tr ansformatio n that tr ansforms (3.4) in (3.7) is w x = e R u dx . (3.11) Pro of: By Pr op osition 2.1 the hierarc hy (3.4) admits the follo wing integ rating factor, Λ, conserv ed curr en t, Φ t 1 , and fl ux, Φ x 1 : Λ = 1 (3.12a ) Φ t 1 ( u ) = u ( 3.12b) Φ x 1 ,n = − D − 1 x ( R n [ u ] u x ) , n = 1 , 2 , . . . , (3.12c ) where R is the recursion op erator, (3.3). It is easy to v erify that D − 1 x ( R n [ u ] u x ) = P n [ u ] u, n = 1 , 2 , . . . , (3.13) where P is defined b y (3.9). Th e fir s t auxiliary system for the Burgers’ hierarc hy (3.4) is then d efined in terms of a p oten tial v ariable v in the form (3.8a)–(3.8b ), i.e. v x = u v t = P n [ u ] u, n = 1 , 2 , . . . , and the first p otenti al hierarc hy of the Burgers’ hierarch y , (3.4), b ecomes (3.5), i.e. v t = P n [ v x ] v x , n = 1 , 2 , . . . . Nonlo cal symmetries, conserv ation la ws and transform ations 7 The first p otenti al hierarc hy , (3.5), admits the follo win g integ r ating factor, Λ, conserv ed current, Φ t , and flux, Φ x : Λ = e v (3.15a ) Φ t 2 ( v ) = e v (3.15b) Φ x 2 ,n = − D − 1 x ( e v P n [ v x ] v x ) , n = 1 , 2 , . . . . (3.15c ) By the relation D − 1 x ( e v P n [ v x ] v x ) = D n x [ v ] e v , n = 1 , 2 , . . . , (3.16) the seco n d auxiliary s ystem of the Burgers’ hierarc h y (3.4) is th en defined in the form (3.8c)–(3.8d), i.e. w x = e v w t = D n x [ v ] e v , n = 1 , 2 , . . . , so that the second p otent ial hierarch y b ecomes w t = D n x [ w ] w x ≡ w ( n +1) x , n = 1 , 2 , . . . . The n onlo cal transf ormation (3.11) follo ws directly from (3.8a) and (3.8c), n amely the w ell kno wn C ole-Hopf transformation (see e.g. [15]). The nonlo cal conserved cur ren t (3.10 a ) and flux (3.10b) follo ws d irectly by expressin g (3.8c)–(3.8d) in terms of the original v ariable u . ✷ 3.2 P otential symmetries of the Burgers’ hierarch y W e no w turn our atten tion to the symmetry prop erties of the auxiliary sys tems (3.8a)– (3.8b) and the com bined auxiliary system (3.8a)–(3.8d). Firstly w e discuss in detail the cases n = 1 and n = 2. Case n = 1: Th e fir st auxiliary system (3.8a)–(3.8b) of h ierarch y (3.4) w ith n = 1 is v x = u (3.18a ) v t = u x + u 2 (3.18b) and the first p otenti al equation h as the form v t = v xx + v 2 x . (3.19) By Definition 2.1 the first-degree p oten tial symmetries of (3.1) are the Lie p oin t symm etries 8 N Euler and M Euler of (3.18 a )–(3.18b). W e obtain Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (3.20a ) Z 4 = x ∂ ∂ x + 2 t ∂ ∂ t − u ∂ ∂ u , Z 5 = 2 t ∂ ∂ x − ∂ ∂ u − x ∂ ∂ v (3.20b) Z 6 = 4 xt ∂ ∂ x + 4 t 2 ∂ ∂ t − 2( x + 2 tu ) ∂ ∂ u − (2 t + x 2 ) ∂ ∂ v (3.20c ) Z ∞ = e − v  ∂ f ∂ x − uf ( x, t )  ∂ ∂ u + f ( x, t ) e − v ∂ ∂ v , where f t − f xx = 0 . (3.20d) The fi rst-degree p ote ntial symmetries (3.20a )–(3.20 d ) were fir stly obtained b y Vino’grado v and Krasil’shc h ik [20]. The second auxiliary system (3.8c)–(3.8d) for hierarc hy (3.4) with n = 1 is w x = e v (3.21a ) w t = v x e v (3.21b) and the second p ot ential equation for hierarc hy (3.4) with n = 1 has the form w t = w xx . (3.22) F ollo win g Definition 2.1 the s econd-degree p oten tial symmetries of the Burgers hierarc hy (3.4) for n = 1 are the Lie p oint symmetries of the com b ined auxiliary sys tems (3.18a)– (3.18b) and (3.21a )–(3.21b), i.e. the Lie p oin t s ymmetries of the system v x = u (3.23a ) v t = u x + u 2 (3.23b) w x = e v (3.23c ) w t = v x e v . (3.23d) W e obtain the follo wing s econd-degree p oten tial symmetries of (3.4) for n = 1: Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x (3.24a ) Z 3 = x ∂ ∂ x + 2 t ∂ ∂ t − u ∂ ∂ u + w ∂ ∂ w , Z 4 = w ∂ ∂ w + ∂ ∂ v (3.24b) Z 5 = 2 t ∂ ∂ x −  2 − uw e − v  ∂ ∂ u −  x + w e − v  ∂ ∂ v − xw ∂ ∂ w (3.24c ) Nonlo cal symmetries, conserv ation la ws and transform ations 9 Z 6 = 2 xt ∂ ∂ x + 2 t 2 ∂ ∂ t −  2 x + 2 tu + w e − v − xuw e − v  ∂ ∂ u (3.25a ) −  3 t + 1 2 x 2 + xw e − v  ∂ ∂ v −  tw + 1 2 x 2 w  ∂ ∂ w (3.25b) Z ∞ = e − v  u ∂ f ∂ x − ∂ 2 f ∂ x 2  ∂ ∂ u − e − v ∂ f ∂ x ∂ ∂ v − f ( x, t ) ∂ ∂ w , (3.25c ) where f t − f xx = 0 . Case n = 2 : Th e fir st auxiliary system (3.8a)–(3.8b) of hierarch y (3.4) w ith n = 2 is v x = u (3.26a ) v t = u xx + 3 uu x + u 3 (3.26b) and the first p otenti al equation h as the form v t = v 3 x + 3 v x v xx + v 3 x . (3.27) The fi rst-degree p oten tial symmetries of (3.1 ) are then Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (3.28a ) Z 4 = x ∂ ∂ x + 3 t ∂ ∂ t − u ∂ ∂ u (3.28b) Z ∞ = e − v  ∂ f ∂ x − uf ( x, t )  ∂ ∂ u + f ( x, t ) e − v ∂ ∂ v , (3.28c ) where f t − f 3 x = 0 . The s econd auxiliary system (3.8c)–(3.8d) for h ierarc hy (3.4) with n = 2 is w x = e v (3.29a ) w t = v xx e v + v 2 x e v (3.29b) and the second p ot ential equation for hierarc hy (3.4) with n = 2 has the form w t = w 3 x . (3.30) The second-degree p oten tial sym metries of the Burgers’ hierarc hy (3.4) for n = 2 wo u ld follo w from the Lie p oint symmetries of the combined auxiliary systems (3.18a)–(3.18b) and (3.21 a )–(3.2 1b ), i.e. the Lie p oint sy m metries of the system v x = u (3.31a ) v t = u xx + 3 uu x + u 3 (3.31b) w x = e v (3.31c ) w t = v xx e v + v 2 x e v . (3.31d) 10 N Euler and M Euler W e obtain the follo wing L ie p oin t s ymmetries of system (3.31a)–(3.31d): Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x (3.32a ) Z 3 = x ∂ ∂ x + 3 t ∂ ∂ t − u ∂ ∂ u + w ∂ ∂ w , Z 4 = w ∂ ∂ w + ∂ ∂ v (3.32b) Z ∞ = e − v  u ∂ f ∂ x − ∂ 2 f ∂ x 2  ∂ ∂ u − e − v ∂ f ∂ x ∂ ∂ v − f ( x, t ) ∂ ∂ w , (3.32c ) where f t − f 3 x = 0 . It is clear that the ab o v e L ie p oint sym metry generators are not p ote ntial symmetries of second d egree for the th ird-order Bur gers’ equation (3.27). The same hap p ens for the case n = 3, i.e., second-degree p oten tial symmetries for the Burgers’ hierarc hy app ear only f or the case n = 1, namely the Bur gers’ equation (3.1). F r om the ab o ve patterns in the symm etry generators we allo w ourselv es the follo wing Supp osition 3.1: Ther e exist no se c ond-de gr e e p otential symmetries for the Bu r g e rs’ hier ar chy (3.4) for n > 1 and th e maximum set of first-de gr e e p otential symmetries for the hier ar chy (3.4) for al l natur al numb ers n > 1 , is g iven by the fol lowing Lie symmetry gener ator s: Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (3.33a ) Z 4 = x ∂ ∂ x + ( n + 1) t ∂ ∂ t − u ∂ ∂ u (3.33b) Z ∞ = e − v  ∂ f ∂ x − uf ( x, t )  ∂ ∂ u + f ( x, t ) e − v ∂ ∂ v , (3.33c ) wher e f t − f ( n +1) x = 0 . 3.3 Recipro cal-B¨ ac klund transformations of the Burgers’ hierarch y With the general expressions of th e conserve d cu r rent , (3.12b), and flux, (3.12c), together with the r elation (3.13) for the the Bu rgers’ hierarch y (3.4), w e use the opp ortunit y to transform th e hierarc h y b y a r ecipro cal-B¨ ac klund transformation (see e.g. [17] and [6]) and hence present a transformed Burgers’ hierarc hy . The f ollo wing t wo Prop ositio n s giv e the result for b oth the Burgers’ h ierarc hy (3.4) and the p oten tial Bur gers’ hierarc hy (3.5 ): Nonlo cal symmetries, conserv ation la ws and transform ations 11 Prop osition 3.2: Under the r e cipr o c al-B¨ acklund tr ansform ation R :          dy ( x, t ) = Φ t 1 dx − Φ x 1 ,n dt dτ ( x, t ) = dt U ( y , τ ) = u (3.34) with Φ t 1 = u, Φ x 1 ,n = − P n [ u ] u (3.35) the Bur gers’ hier ar chy u t = R n [ u ] u x , wher e R [ u ] = D x [ u ] + u + u x D − 1 x ◦ 1 , (3.36) tr ansforms to the hier ar chy U τ = n U D y [ U ] − U y o n P n [ U ] U o , (3.37) wher e P [ U ] = U D y [ U ] + U. (3.38) Prop osition 3.3: Under the r e cipr o c al-B¨ acklund tr ansform ation R :          dy ( x, t ) = Φ t 2 dx − Φ x 2 ,n dt dτ ( x, t ) = dt V ( y , τ ) = v (3.39) with Φ t 2 = e v , Φ x 2 ,n = − D n x [ v ] e v , (3.40) the p otential Bur gers’ hier ar chy v t = P n [ v x ] v x , wher e P [ v x ] = D x [ v x ] + v x , (3.41) tr ansforms to the hier ar chy V τ = n D y [ V ] − V y o  e V D y [ V ]  n e V  . (3.42) W e giv e some exp licit examples of the equations (3.37) and (3.42): Under the r ecipro cal-B¨ ac klund transformation (3.34)–(3.35) the Bur gers’ hierarch y (3.36) with n = 1 transform s to U τ = U 2 U y y + U 2 U y (3.43) 12 N Euler and M Euler and for n = 2 w e obtain U τ = U 3 U 3 y + 3 U 3 U y y + 3 U 2 U y U y y + 3 U 2 U 2 y + 2 U 3 U y . (3.44) Under the recipro cal-B¨ ac klund transformation (3.39)–(3.4 0 ) the p oten tial Burgers’ hier- arc hy (3.41) with n = 1 transforms to V τ = e 2 V  V y y + V 2 y  (3.45) and for n = 2 w e obtain V τ = e 3 V  V 3 y + 6 V y V y y + 4 V 3 y  . (3.46) 4 The Calogero-Dega sp eris-Ibragimo v-Shabat hierarc h y The th ird-order evol u tion equation u t = u 3 x + 3 u 2 u xx + 9 uu 2 x + 3 u 4 u x (4.1) is kn o wn as the Caloge r o-Dega s p eris-Ibragimo v-Shabat equation and is a well-kno wn C - in tegrable ev olution equation which can b e linearised by a nonlo cal transformation ([19 ], [4], [16]). In [16] we deriv ed a second-order nonlo cal recursion op erator for (4.1), namely R [ u ] = D 2 x + 2 u 2 D x + 10 uu x + u 4 +2  u xx + 2 u 2 u x + 2 ue − 2 R u 2 dx Z e 2 R u 2 dx u 2 x dx  D − 1 x ◦ u − 2 ue − 2 R u 2 dx D − 1 x ◦   u xx + 2 u 2 u x  e 2 R u 2 dx + 2 u Z e 2 R u 2 dx u 2 x dx  (4.2) and also rep orted some nonlo cal symmetries that follo w from this recursion op erato r . In terms of th e recur sion op erator (4.2) a lo cal Calogero-Deg asp eris-Ibragimo v-Shabat hierarc hy of C -in tegrable ev olution equations can b e presen ted in the f orm u t = R n [ u ] u x , n = 1 , 2 , . . . . (4.3) Equation (4.1) then corr esp onds to (4.3) with n = 1. F or n = 2 the second member of hierarc hy (4.3) is u t = u 5 x + 5 u 2 u 4 x + 40 uu x u 3 x + 25 uu 2 xx + 50 u 2 x u xx + 10 u 4 u 3 x +120 u 3 u x u xx + 140 u 2 u 3 x + 10 u 6 u xx + 70 u 5 u 2 x + 5 u 8 u x . (4.4) W e no w inv estigate the nonlo ca l s ymmetry structure in the sense of its first- and second- degree p ote ntial symmetries and obtain the corr esp onding nonlo cal conserv ation la w s. W e sho w that the linearisations of (4.1) and (4.4 ) follo w directly fr om the second p oten tiali- sation of (4.1) and (4.4), r esp ectiv ely . Nonlo cal symmetries, conserv ation la ws and transform ations 13 4.1 Nonlo cal conserv ation laws and linearisation of the Calogero-Degas- p eris-Ibragimo v-Shabat hierarc h y The results for the fir st and second memb ers of the h ierarc hy (4.3) are giv en b y the follo wing t wo prop ositions: Prop osition 4.1: The Calo ger o-De gasp eris-Ibr agimov-Shab at e quation (4.1 ), u t = u 3 x + 3 u 2 u xx + 9 uu 2 x + 3 u 4 u x , admits a first p otentialisation of the form v t = v 3 x − 3 4 v 2 xx v x + 3 v x v xx + v 3 x (4.5) and se c ond p otentialisation of the form w t = w 3 x , (4.6) wher e v x = u 2 (4.7a) v t = 2 uu xx − u 2 x + 6 u 3 u x + u 6 (4.7b) w x = e v v 1 / 2 x (4.7c) w t = e v  1 2 v − 1 / 2 x v 3 x − 1 4 v − 3 / 2 x v 2 xx + 2 v 1 / 2 x v xx + v 5 / 2 x  . (4.7d) The c orr e sp onding nonlo c al c onserve d cu rr ent, Φ t , and flux, Φ x , ar e Φ t = ue R u 2 dx (4.8a) Φ x = −  u xx + 4 u 2 u x + u 5  e R u 2 dx (4.8b) and the line arising tr ansformatio n that tr ansforms (4.1) to (4.6) is w x = ue R u 2 dx . (4.9) F or th e second memb er of th e C alogero- Degasp eris-Ib ragimo v-Shabat hierarch y we hav e Prop osition 4.2: The se c ond Calo ger o-D e gasp eris-Ibr agimov-Shab at e quation (4.4), u t = u 5 x + 5 u 2 u 4 x + 40 uu x u 3 x + 25 uu 2 xx + 50 u 2 x u xx + 10 u 4 u 3 x +120 u 3 u x u xx + 140 u 2 u 3 x + 10 u 6 u xx + 70 u 5 u 2 x + 5 u 8 u x , 14 N Euler and M Euler admits a first p otentialisation of the form v t = v 5 x + 5 v x v 4 x − 5 2 v − 1 x v xx v 4 x + 10 v 2 x v 3 x + 5 v xx v 3 x + 5 v − 2 x v 2 xx v 3 x − 5 4 v − 1 x v 2 3 x − 35 16 v − 3 x v 4 xx − 5 2 v − 1 x v 3 xx + 25 2 v x v 2 xx + 10 v 3 x v xx + v 5 x (4.10) and se c ond p otentialisation of the form w t = w 5 x , (4.11) wher e v x = u 2 (4.12a ) v t = 2 uu 4 x − 2 u x u 3 x + u 2 xx + 10 u 3 u 3 x + 50 u 2 u x u xx + 20 u 5 u xx + 70 u 4 u 2 x +20 u 7 u x + u 10 (4.12b) w x = e v v 1 / 2 x (4.12c ) w t = e v  1 2 v − 1 / 2 x v 5 x − v − 3 / 2 x v xx v 4 x + 3 v 1 / 2 x v 4 x − 3 4 v − 3 / 2 x v 2 3 x + 9 4 v − 5 / 2 x v 2 xx v 3 x +2 v − 1 / 2 x v xx v 3 x + 7 v 3 / 2 x v 3 x − 15 16 v − 7 / 2 x v 4 xx + 15 2 v 1 / 2 x v 2 xx + 8 v 5 / 2 x v xx + v 9 / 2 x  . (4.12d) The c orr e sp onding nonlo c al c onserve d cu rr ent, Φ t , and flux, Φ x , ar e Φ t = ue R u 2 dx (4.13a ) Φ x = −  u 4 x + 26 uu x u xx + 6 u 2 u 3 x + 8 u 3 x + 44 u 3 u 2 x +14 u 4 u xx + 16 u 6 u x + u 9  e R u 2 dx (4.13b) and the line arising tr ansformatio n that tr ansforms (4.4) to (4.11) is w x = ue R u 2 dx . (4.14) In order to derive the auxiliary systems for th e Calogero-Dega sp eris-Ibragimo v-Shabat hierarc hy , (4.3), w e need the in tegrating factors of this hierarc hy and the in tegrating factors of the corresp onding p oten tial hierarch y . F or the first tw o members of the hierarc hy the in tegrating factors are giv en by the follo wing Lemma 4.1: The thir d-or der Calo ger o-De gasp eris- Ib r agimov-Shab at e quation, (4.1), and the fifth-or der Calo ger o-De gasp eris-Ibr agimov-Shab at e quation, (4.4), ad mit only on e in- te gr ating factor, Λ , namely Λ( x, u, u x , . . . ) = u. (4.15) Nonlo cal symmetries, conserv ation la ws and transform ations 15 F or the first p otentialisation of (4.1), namely for the thir d-or der p otential e quation (4.5), the c omplete set of inte gr ating factors of se c ond-or der ar e Λ( x, v , v x , v xx ) = a ( x ) e v v − 3 / 2 x v xx + 2 a ( x ) e v v 1 / 2 x − 2 e v v − 1 / 2 x da dx , (4.16) wher e d 3 a dx 3 = 0 , (4.17) and for the first p otentialisa tion of (4.4), namely for the fifth-or der p otential e quation (4.10), the c omplete set of inte gr ating factors of se c ond-or der ar e Λ( x, v , v x , v xx ) = a ( x ) e v v − 3 / 2 x v xx + 2 a ( x ) e v v 1 / 2 x − 2 e v v − 1 / 2 x da dx , (4.18) wher e d 5 a dx 5 = 0 . (4.19) T o pr o ve Lemma 4.1 we just v erify (2.6). Lemma 4.1 tempts us to make the follo wing Supp osition 4.1: Al l first p otentialisations of the Calo ger o-De gasp eris-Ibr agimov-Shab at hier ar chy, (4.3), for al l n ∈ N admit the fol lowing c omplete set of inte gr ating factors of se c ond or der: Λ( x, u, u x , u xx ; n ) = a ( x ) e v v − 3 / 2 x v xx + 2 a ( x ) e v v 1 / 2 x − 2 e v v − 1 / 2 x da dx , (4.20) wher e d n a dx n = 0 . (4.21) Remark on the pro of of Pr op osition 4.1 and Pr op osition 4.2: F or the linearisations of (4.1) and (4.4) in (4.6) and (4.11 ), resp ectiv ely , we mak e use of the in tegrating factor Λ = e v v − 3 / 4 x v xx + 2 e v v 1 / 2 x , (4.22) whic h corresp onds to the case a ( x ) = 1 in Lemma 4.1. If one uses instead the explicitly x -dep end ent in tegrating factors, the resulting linear equations also dep end explicitly on x . 16 N Euler and M Euler 4.2 P otential symmetries of the Calogero-Degasp eris-Ibragimo v-Shabat hierarc h y W e no w stud y the symmetry prop erties of th e auxiliary systems for the C alogero-D egasp eris- Ibragimo v-Shabat hierarch y (4.3). Case n = 1 : The fi rst-degree p otent ial symmetries of the first mem b er of the hierarc hy (4.3), i.e. (4.1), are giv en by th e Lie p oin t symmetries of the firs t auxiliary system (4.7a)– (4.7b). W e obtain Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (4.23a ) Z 4 = 1 3 x ∂ ∂ x + t ∂ ∂ t − 1 6 u ∂ ∂ u , Z 5 = ue − 2 v ∂ ∂ u − e − 2 v ∂ ∂ v . (4.23b) The second-degree p oten tial sym m etries of the fi rst mem b er of the hierarc hy (4.3) are giv en b y th e Lie p oin t s y m metries of the com bined auxiliary system (4.7 a )–(4.7d). W e obtain Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ w (4.24a ) Z 4 = 1 3 x ∂ ∂ x + t ∂ ∂ t − 1 6 u ∂ ∂ u + 1 6 w ∂ ∂ w , Z 5 = ∂ ∂ v + w ∂ ∂ w (4.24b) Z 6 =  1 2 e − v − uw e − 2 v  ∂ ∂ u + w e − 2 v ∂ ∂ v + 1 2 x ∂ ∂ w (4.24c ) Z 7 = ue − 2 v ∂ ∂ u − e − 2 v ∂ ∂ v . (4.24d) Case n = 2 : The firs t-degree p oten tial symmetries of the second member of the hierarch y (4.3), i.e. (4.4), are giv en by the Lie p oint symmetries of th e fir st auxiliary system (4.12a)– (4.12b). W e obtain Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (4.25a ) Z 4 = 1 5 x ∂ ∂ x + t ∂ ∂ t − 1 10 u ∂ ∂ u , Z 5 = ue − 2 v ∂ ∂ u − e − 2 v ∂ ∂ v . (4.25b) The complete set of Lie p oin t sym metries of the auxiliary system (4.12a)–(4.1 2d ) are Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ w (4.26a ) Z 4 = 1 5 x ∂ ∂ x + t ∂ ∂ t − 1 10 u ∂ ∂ u + 1 10 w ∂ ∂ w , Z 5 = ∂ ∂ v + w ∂ ∂ w (4.26b) Z 6 = ue − 2 v ∂ ∂ u − e − 2 v ∂ ∂ v . (4.26c ) Nonlo cal symmetries, conserv ation la ws and transform ations 17 W e note that the second member of the h ierarc hy (4.4) do es not admit second-degree p oten tial sym metries. W e allo w ourselve s th e follo wing Supp osition 4.2: Ther e exist no se c ond-de gr e e p otential symmetries for the Calo ger o- De gasp eris-Ibr agimov-Shab at hier ar chy, (4.3), f or n > 1 and the maximum set of first- de gr e e p otential symmetries for the hier ar chy (4.3) is give n by the fol lowing Lie symmetry gener ator s: Z 1 = ∂ ∂ t , Z 2 = ∂ ∂ x , Z 3 = ∂ ∂ v (4.27a ) Z 4 = 1 n x ∂ ∂ x + t ∂ ∂ t − 1 2 n u ∂ ∂ u , Z 5 = ue − 2 v ∂ ∂ u − e − 2 v ∂ ∂ v , (4.27b) for al l natur al numb ers n > 1 . 5 Concluding remarks W e ha v e in tro duced second-degree p oten tial symmetries in Definition 2.1 and studied the Burgers’ hierarc hy and the Calogero-Deg asp eris-Ibragimo v-Shabat h ierarc hy . W e ob- tained second-degree p oten tial symm etries only f or the first m em b ers of the hierarc hies. Nonlo cal conserv ation la ws and nonlocal transform ations whic h linearise the hierarchies w ere obtained th rough the second p oten tialisat ions. It would b e in teresting to inv estigate higher-degree p oten tial symmetries fu rther for other sym metry-in tegrable hierarc hies and linearisable hierarc h ies as w ell as for systems of evolutio n equations. Preliminary calcula- tions sh o w th at systematic p otentia lisation of some symm etry-in tegrable equations, such as the Kr ic heve r -Novik o v equ ation, lead to in teresting auto-B¨ ac klund transformations for the equations. A complete description of the p otent ialisation for a class of Krichiv er- No vik ov equ ations an d its connection with higher-degree p otenti al symmetries is currently in p reparation [9] and is planned for pub lication as a f ollo w -up pap er. 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