Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

Algebraic Discretization of the Camassa-Holm and Hun ter-Saxton Equations Rossen I. Iv ano v 1 Dep artment of Mathematics, Lund University, 2210 0 Lund, Swe den Abstract The Camassa- Holm (CH) and Hun ter-Saxto n (HS) equations hav e an int erpr etation as geo desic flow equations o n the group of diffeomor - phisms, preserving the H 1 and ˙ H 1 right-in v ariant metrics cor resp ond- ingly . There is an analogy to the Euler equations in hydrody namics, which describ e geodes ic flow for a right-in v ariant met ric on the infinite- dimensional group of diffeomorphisms preserving the v olume element of the domain of fluid flo w and to the Euler equations of rig id bo dy whith a fixed point, describing geodesics for a left-in v ariant metric o n SO (3). The CH and HS equations are integrable bi-ha miltonian equations and one of their Hamiltonian structures is as so ciated to the Viras oro alge- bra. The parallel with the in tegr able SO (3) top is made explicit b y a discr etization of bo th equatio n based on F ourier mo de s expans ion. The obtained e quations represent in tegr able tops with infinitely many momentum comp onents. An emphasis is given o n the structure of the phase space o f these equations, the momentu m map and the space of cano nical v ariable s. 1 In tro duction This geometric in terpretation of the C amassa-Holm equation [6] as a geodesic flo w e quation on the group o f diffeomorphisms, preserving the H 1 righ t- in v arian t metrics metric was noticed firstly b y Misio lek [42] and devel op ed further in many recent publications, e.g. [37, 25, 14, 15, 39, 13]. The CH equation has an in terpretation in the con text of w ater wa v es propagation [6, 34, 35, 19, 20, 32, 29]. T he sp ectral p roblem for the CH equ ation on th e line is d ev elop ed in [2, 8, 10, 11, 17, 36], the p erio dic sp ectral p roblem – in [7, 16, 48]. The CH solutions are in v estigate d in a v ariet y of recen t pap ers, e.g. in [4, 5, 9 , 12, 21, 22, 23, 26, 30, 46]. Hierarc hies of CH equations are studied in [11, 31, 33], different mo difications are s tudied in [41, 47]. There are differen t forms of the CH equation, con taining linear t erm with a first deriv ativ e u x ; with a third deriv ativ e u xxx (called sometimes Dullin-Gott w ald-Holm equ ation [19, 20, 43, 44, 49]), or without su c h terms . These terms can b e put in or remo v ed from the equation indep endent ly b y Galilea n tr ansformations. 1 Presen t a ddress: School of Mathematical Sciences, Dublin Institute of T ec hnology , Kevin Street, Dublin 8, Ireland, Email: riv ano v@dit.ie 1 W e will b e in terested in the CH equation of the form m t + au xxx + 2 mu x + m x u = 0 , m = u − u xx , (1.1) with a b eing an arbitrary constan t. It can b e wr itten in Hamiltonian form m t = { m, H 1 } , (1.2) where, assu ming that m is 2 π p erio dic in x , i.e. m ( x ) = m ( x + 2 π ), the P oisson brac k et and the Hamiltonian are { F , G } ≡ − Z 2 π 0 δ F δ m  a∂ 3 + m∂ + ∂ ◦ m  δ G δ m d x, (1.3) H 1 = 1 2 Z 2 π 0 mu d x. (1.4) The equation (1.1) is bi-Hamiltonian with a second Hamiltonian represen- tation m t = { m, H 2 } 2 , wh ere { F , G } 2 ≡ − Z 2 π 0 δ F δ m ( ∂ − ∂ 3 ) δ G δ m d x, (1.5) H 2 = 1 2 Z 2 π 0 ( u 3 + uu 2 x − a 2 u 2 x )d x. (1.6) One ca n notice that the in tegral H 0 = Z 2 π 0 m d x (1.7) is a Casimir for the second P oisson brack et (1.5). The relation of the fi rst Po isson brack et (1.3) to the Virasoro alge br a can b e seen as follo ws [18]. The 2 π -p erio d ic fu nction allo w s a F ourier de- comp osition m ( x, t ) = 1 2 π X n ∈ Z L n ( t ) e inx + a 2 , (1.8) (the realit y of m can b e ac hieve d by L − n = ¯ L n ). Th en the F ourier co effi- cien ts L n close a classical Virasoro algebra of cen tral charge c = − 24 πa w ith resp ect to the P oisson brac k et (1.3): i { L n , L m } = ( n − m ) L n + m − 2 π a ( n 3 − n ) δ n + m, 0 . (1.9) The CH equation in the form m t + 2 ω u x + 2 mu x + m x u = 0 , m = u − u xx , (1.10) can b e obtained from (1.1) via u → u + a , and apparen tly ω = 3 a/ 2. 2 Since H 0 = L 0 2 π + π a (1.11) is an inte gral of motio n (Casimir), so is L 0 . The first Hamiltonian is H 1 = 1 4 π X n ∈ Z L n L − n 1 + n 2 + a 2 L 0 + 2 π a 2 8 . (1.12) F rom (1.9) and (1.12) we obtain the ’Camassa-Holm top’ equations on the Virasoro g roup , whic h are a discretization of the Camassa-Holm equation (1.1) i ˙ L k = 1 2 π X n ∈ Z k + n 1 + n 2 L n L k − n + a 2 3 k − k 3 1 + k 2 L k , (1.13) (the dot is a t -deriv ativ e). Th is equation is analogous to the Euler top (rigid b o dy ) equation on th e Lie group SO(3) ˙ M k = 3 X p,l =1 ε k pl Ω p M l , M k ≡ I k Ω k for the quadr atic Hamiltonian H E = 1 2 3 X p =1 M p Ω p , where I k ( k = 1 , 2 , 3) are three constant s – the pr inciple inertia momen ta. The p hase space is em b edd ed in t he Lie coalgebra so( 3)* as a co adjoint orbit. The Lie-P oisson brack et, r elated to the so(3)* coalgebra is { M n , M m } = ε nmk M k . (1.14) The inertia op erator I : so(3) → so(3)* (see e.g. [1]) r elates the parametriza- tion on the so(3) algebra given by the fu nctions Ω k and the parametrization on the co-algebra so(3) * giv en by the fun ctions M k = I k Ω k . Note th at the P oisson brac k et (1.14) has a Casimir K = Ω 2 1 + Ω 2 2 + Ω 2 3 , (1.15) constraining th e p hase space on a sphere. S ince the Lie-P oisson brac k et is degenerate on so(3)*, the coadjoin t orbits (whic h are sp heres centered at the origin) are labelled by the v alue of the Casimir K . F or th e CH top (1.13) the coadjoint orbits are emb edded in th e Virasoro algebra (parameterized b y the f unctions L k ) d ue to the Lie-Poi sson brack et (1.9). 3 2 Lax represen tation for the discrete Camassa- Holm equation and in tegrals of motion The Lax pair for th e discrete CH equation (1.13) can b e obtai ned from the Lax p air for (1.10), Ψ xx =  1 4 + λ ( m + a 2 )  Ψ (2.1) Ψ t =  1 2 λ − u + a  Ψ x + u x 2 Ψ , (2.2) as follo ws. W e tak e the expansions Ψ = X n ∈ Z Ψ n 2 e i n 2 x , (2.3) u = 1 2 π X n ∈ Z u n e inx + a 2 , u n = L n 1 + n 2 . (2.4) Then (2.1) giv es 1 λ Ψ n 2 = X p ∈ Z L n 2 , n 2 − p Ψ n 2 − p , (2.5) where L n 2 , n 2 − p = − 4 n 2 + 1  L p 2 π + aδ p, 0  , or L n 2 − q , n 2 − p = − 4 ( n − 2 q ) 2 + 1  L p − q 2 π + aδ p,q  (2.6) No w from (2.2), (2.3), (2.4) and (2.5) it follo ws ˙ Ψ n 2 = X p ∈ Z A n 2 , n 2 − p Ψ n 2 − p , (2.7) where A n 2 , n 2 − p = − i 4 π  2 n p 2 + 1 n 2 + 1 + n − 3 p  u p + in  1 4 − 1 n 2 + 1  aδ p, 0 , or A n 2 − q , n 2 − p = − i 4 π  2( n − 2 q ) ( p − q ) 2 + 1 ( n − 2 q ) 2 + 1 + n − 3 p + q  u p − q + i ( n − 2 q )  1 4 − 1 ( n − 2 q ) 2 + 1  aδ p,q . (2.8) 4 Differen tiating (2.5) with resp ect to t we obtain 1 λ ˙ Ψ n 2 = X p ∈ Z ˙ L n 2 , n 2 − p Ψ n 2 − p + X p ∈ Z L n 2 , n 2 − p ˙ Ψ n 2 − p , and with the further su bstitution f rom (2 .7), 1 λ X q ∈ Z A n 2 , n 2 − q Ψ n 2 − q = X p ∈ Z ˙ L n 2 , n 2 − p Ψ n 2 − p + X p,q ∈ Z L n 2 , n 2 − q A n 2 − q , n 2 − p Ψ n 2 − p , X q ∈ Z A n 2 , n 2 − q  1 λ Ψ n 2 − q  = X p ∈ Z ˙ L n 2 , n 2 − p Ψ n 2 − p + X p,q ∈ Z L n 2 , n 2 − q A n 2 − q , n 2 − p Ψ n 2 − p , and finally , the sub stitution of (2.5) giv es X p,q ∈ Z A n 2 , n 2 − q L n 2 − q , n 2 − p Ψ n 2 − p = X p ∈ Z ˙ L n 2 , n 2 − p Ψ n 2 − p + X p,q ∈ Z L n 2 , n 2 − q A n 2 − q , n 2 − p Ψ n 2 − p , (2.9) or in matrix form, ˙ L = [ A , L ] . (2.10) After some lengthy computations one can verify that (2.10) g ive s (1 .13). The in tegrals of motion are giv en by I k = tr( L k ). F or example, I 1 = tr( L ) = X p ∈ Z L n 2 − p, n 2 − p = − 4  L 0 2 π + a  X p ∈ Z 1 ( n − 2 p ) 2 + 1 pro du ces, up to an o v erall constan t, the Casimir H 0 , (1.1 1). I 2 = tr( L 2 ) = X p,q ∈ Z L n 2 − p, n 2 − q L n 2 − q , n 2 − p = 4 π 2 X p,q ∈ Z L p − q L q − p [( n − 2 p ) 2 + 1][( n − 2 q ) 2 + 1] + 16 a π ( L 0 + π a ) X p ∈ Z 1 [( n − 2 p ) 2 + 1] 2 . (2.11) With partial fractions decomp osition with resp ect to n one can derive the iden tit y 1 [( n − 2 p ) 2 + 1][( n − 2 q ) 2 + 1] = 1 / 4 ( p − q ) 2 + 1 n ( n − 2 q ) + ( p − q ) ( p − q )[( n − 2 q ) 2 + 1] − ( n − 2 p ) + ( q − p ) ( p − q )[( n − 2 p ) 2 + 1] o . 5 F urth er, u sing the fact that all expressions that c hange sign un der p − q → − ( p − q ) are zero, due to the summation ov er all in teger n umb ers, we hav e 4 π 2 X p,q ∈ Z L p − q L q − p [( n − 2 p ) 2 + 1][( n − 2 q ) 2 + 1] = 1 π 2 X p,q ∈ Z L p − q L q − p 1 + ( p − q ) 2 n 1 ( n − 2 p ) 2 + 1 + 1 ( n − 2 q ) 2 + 1 o = 2 π 2 X p ∈ Z L p L − p 1 + p 2 X q ∈ Z 1 ( n − 2 q ) 2 + 1 . Th us , the new in tegral that app ears is P p ∈ Z L p L − p 1+ p 2 , giving H 1 , the first Hamiltonian (1.1 2). 3 Oscillator algebra, Miura transformati on and mo- men tum map Let us in tro d uce no w the oscillator alg ebra i { a n , a m } = 2 π a κ 2 nδ n + m, 0 , (3.1) where κ is an arbitrary constant. Clearly , a 0 is a Casimir due to (3.1). One ca n easily ve rify the follo wing oscillato r representat ion of the Virasoro algebra [38, 24]: L n = − κ ( n − 1) a n + κ 2 4 π a X k ∈ Z a k a n − k . (3.2) This r epresen tation is also kno wn as S uga w ara construction. F urther, it is eviden t that i { a n , L m } = na n + m + 2 π a κ n ( n + 1) δ n + m, 0 . (3.3) Since a k satisfy the ’canonical’ P oisson brac k ets they are natural candi- dates for the co ordinates in the phase-space. Th us, L n has an inte rp reta- tion of a momen tum and (3.2) giv es the momen tum map. The Suga wara construction relates to the Miur a transformation, whic h in terms o f field v ariables can b e obtain as foll ows. Defining v = 1 2 π X k ∈ Z a k e ik x + a κ (3.4) from (3 .2) and (1.8) we hav e the analog of the Miura tr ansformation: m = iκv x + κ 2 2 a v 2 + a 2 (3.5) 6 The realit y can be ac hiev ed b y taking κ pur ely imaginary , a k = ¯ a − k for k 6 = 0 and κ = 2 π ia/ ℑ ( a 0 ). Here w e n otice that the Casimir (1.7) due to (3.5) leads to the restriction Z 2 π 0 v 2 ( x, t )d x = const , (3.6) whic h r educes the ev olution o f v ( x, t ) on the L 2 -sphere. In terms of the canonical coordinates this co nd ition is X k > 0 | a k | 2 = const , (3.7) since a 0 is a constan t. It sho ws that the time ev olution of the canonical v ariables, giv en b y ˙ a n = { a n , H 1 } is constrained on the infinite-dimensional l 2 -sphere, a condition, similar to the one that w e see in the so (3) example (1.15 ). When a = 0, th e S uga w ara construction for the Virasoro mo des in the case of zero cen tral c harge is L n = 1 2 ˜ κ X k ∈ Z a k a n − k , where ˜ κ is an arbitrary constant and i { a n , a m } = ˜ κnδ n + m, 0 . (3.8) The Casimir with resp ect to the first P oisson brack et (1.3) with a ≡ 0 is Z 2 π 0 √ m d x = r π ˜ κ Z 2 π 0 v d x = 2 π r π ˜ κ a 0 , i.e. this is the Casimir a 0 of (3.8). With the expansions m = 1 2 π X n ∈ Z L n e inx , v = 1 2 π X n ∈ Z a n e inx the Su ga w ara construction tak es the form m = π ˜ κ v 2 . Since t he in tegral R 2 π 0 m d x = const is a Casimir, we hav e again Z 2 π 0 v 2 d x = const , leading to (3.7). 7 4 The Hun ter-Saxton equation The Hun ter-Saxton (HS) equation u xxt + 2 u x u xx + uu xxx = 0 describ es the propagation of w a v es in a massiv e d irector fi eld of a nematic liquid crystal [27], with the orientat ion of th e molecules d escrib ed by the field of unit 1 v ectors n ( x, t ) = (cos u ( x, t ) , s in u ( x, t )), where x is the space v ariable in a reference frame moving with the linearize d wa v e velocit y , and t is a ’slo w time v ariable’. A linear term au xxx can b e generated b y a shift u → u + a : u xxt + au xxx + 2 u x u xx + uu xxx = 0 . (4.1) The HS equation is a short-w a v e limit of the CH equation, and ca n b e obtained if one tak es m = − u xx . The Hamiltonian representat ion (1.2) – (1.4) for this equatio n is also v alid. The HS equation (4.1) is an in tegrable, bi-Hamiltonian equation with a second Hamiltonian represen tation m t = { m, H 2 } 2 , where { F , G } 2 ≡ Z 2 π 0 δ F δ m ∂ 3 δ G δ m d x, (4.2) H 2 = 1 2 Z 2 π 0 ( u − a 2 ) u 2 x d x. (4.3) The HS Lax pair is Ψ xx = λm Ψ , (4.4) Ψ t =  1 2 λ − u − a  Ψ x + u x 2 Ψ . (4.5) The analyt ic and geometric asp ects o f the HS equ ation are d iscussed in a v ariet y of r ecen t pap ers, e.g. [2 8, 3, 40] and the references therein. Again, assu ming perio dicit y and usin g the expansions Ψ = X n ∈ Z Ψ n e inx , u = 1 2 π X n ∈ Z u n e inx w e obtain the discrete form of the HS equation: in ˙ u n − an 2 u n − 1 2 X k ∈ Z k ( n + k ) u k u n − k = 0 . In a similar mann er from the Lax pair w e obtain the matrix Lax represen- tation for the discrete HS equation ˙ L H S = [ A H S , L H S ] . (4.6) 8 where L H S n,n − p = − p 2 n 2 u p , A H S n,n − p = i 2  − p 2 n − 2 n + 3 p  u p − inaδ p, 0 . The momen tum map (the Suga w ara constru ction) f or the HS equatio n remains the same as for the CH equation. Ho w ev er, it b ecomes degenerated in the case a = 0, s ince m = − u xx and the C asimir R 2 π 0 m d x = 0. Then R 2 π 0 v 2 d x = 0, whic h, for real v ariables is only p ossible when v ≡ 0, i.e. m ≡ 0. 5 Conclusions A t the examples of the CH and HS equations we ha v e sho wn th at the in- tegrable systems w ith quadr atic Hamiltonians are equiv alen t to in tegrable tops (p ossib ly with infinitely man y comp onent s), asso ciated to the algebra of their P oisson brack ets. An example for the t wo dimensional Euler equa- tions in flu id mechanics is presen ted in [50], another example for the K dV sup erequ ation in [38, 45]. Ac kno wledgmen ts The supp ort o f the G. 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