New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-hamiltonian structure

New Lax pair for restricted m ultiple three w a v e in teraction system, quasip erio dic solutions and bi-hamiltonian structure N. A. Kostov 1 , A. V. Tsiganov 2 1 Institute for Ele ctr onics, Bulgarian A c ademy of Scienc es, Blvd. Tzarigr adsko chausse e 72, 1784 Sofia, Bulgaria and 2 Dep artment of Mathematic al and Computational Physics St Petersb ur g State University, R ussia (Dated: Nov ember 9, 20 18) W e study restricted m ultiple three w av e intera ction system b y the in v erse scattering method . W e deve lop the algebraic approac h in terms of classical r - matrix and give an interpretation of the P oisson brac kets as linear r - matrix algebra. The solutions are expressed in terms of p olynomials of theta functions. In particular case for n = 1 in terms of W eierstrass functions. I. RESTRICTED MUL TIPLE T HREE W A VE INT ERACTION SYSTEM Several studies hav e app ear ed recently o n coupled q uadratic nonlinear o scillators [1],[2],[3] ı db j dξ + uc j − 1 2 ǫ j b j = 0 , (1) ı dc j dξ + u ∗ b j + 1 2 ǫ j c j = 0 , (2) ı du dξ + n X j =1 b j c ∗ j = 0 , (3) where ξ is the ev olution co ordinate and ǫ j corres p o nds to the no rmalized wa v e num ber misma tc hs. The system (1-3) is intro duce d in [4] to mo del the growth of a low frequency in ternal ocea n wa v e by interaction with higher frequency surface wa ves and is used in [5] as a mo del o f plasma turbulence. This system des crib e triads of w av es ( a j , b j , u ), j = 1 , . . . n evolving in ξ and in teracting with each other tr o ugh m ultiple three w av e in teraction with po ssible applicatio ns in optics. II. LAX REPRESENT A TION Let us co nsider coupled qua dratic nonlinear o scillators ı db j dξ + uc j − 1 2 ǫ j b j = 0 , (4) ı dc j dξ + u ∗ b j + 1 2 ǫ j c j = 0 , (5) ı du dξ + n X j =1 b j c ∗ j = 0 , (6) where ξ is the evolution co ordinate and ǫ j are cons tant s. The equations (4-6) can b e wr itten as La x repres e n tation dL dξ = [ M , L ] , (7) of the following linear s ystem: dψ dξ = M ( ξ , λ ) ψ ( ξ , λ ) L ( ξ , λ ) ψ ( ξ , λ ) = 0 , (8) where L, M are 2 × 2 matr ices and have the form L ( ξ , λ ) =  A ( ξ , λ ) B ( ξ , λ ) C ( ξ , λ ) D ( ξ , λ )  , (9) 2 M ( ξ , λ ) =  − ıλ/ 2 iu u ∗ ıλ/ 2  . (10) where A ( ξ , λ ) = a ( λ )   − ı λ 2 + ı 2 n X j =1  c j c ∗ j − b j b ∗ j  λ − ǫ j   , (11) B ( ξ , λ ) = a ( λ )   ıu − ı n X j =1 b j c ∗ j λ − ǫ j   , (12) C ( ξ , λ ) = a ( λ )   ıu ∗ − ı n X j =1 c j b ∗ j λ − ǫ j   , (13) where D ( ξ , λ ) = − A ( ξ , λ ) and a ( λ ) = Q n i =1 ( λ − ǫ i ). The Lax representation yields the h yp erelliptic curve K = ( ν, λ ) det( L ( λ ) − 1 2 ν I) = 0 , (14) where I is the 2 × 2 unit matrix . The moduli of the curv e (14) g enerate the in tegrals of motion J 0 , J j , K j , j = 1 , . . . , n , ν 2 = A 2 ( ξ , λ ) + B ( ξ , λ ) C ( ξ, λ ) . (15) The curve (1 5) can be written in canonical for m as ν 2 = 4 2 n +2 Y j =1 ( λ − λ j ) = R ( λ ) , (16) where λ j 6 = λ k are bra nchin g po ints. F rom (15) and explic it express ions for A ( ξ , λ ) , B ( ξ , λ ) , C ( ξ, λ ) we obtain ν 2 = a ( λ ) 2   λ 2 − 4 iJ 0 + 4 i n X j =1 K j λ − ǫ j − i n X j =1 J 2 j ( λ − ǫ j ) 2   , (17) where K j = iub ∗ j c j + iu ∗ b j c ∗ j + i ǫ j 2 ( | c j | 2 − | b j | 2 ) − i 1 2 X k 6 = j  ( | b j | 2 − | c j | 2 )( | b k | 2 − | c k | 2 ) + 2 b ∗ j c j b k c ∗ k + 2 b j c ∗ j b ∗ k c k  ǫ j − ǫ k , (18) J 0 = i | u | 2 + 1 2 i n X j ( | b j | 2 − | c j | 2 ) , J j = i ( | b j | 2 + | c j | 2 ) . Next w e dev elop a metho d which a llows to construct quasi-p erio dic and p e rio dic solutions of system (4-6). The metho d is bas ed on the application of sp ectral theory for self-adjoint one dimensional Dira c equation with quas i-(per io dic) finite gap p o tential U = − u cf. E qs. (4,5) ı d Ψ 1 j dξ − U Ψ 2 j − iλ j Ψ 1 j = 0 , (19) ı d Ψ 2 j dξ − U ∗ Ψ 1 j + iλ j Ψ 1 j = 0 , (20) with sp ectral parameter λ a nd eigenv alues λ j = i ǫ j / 2. The equation (7) is equiv alently written as dA dξ = iuC − iu ∗ B , A ( ξ , λ ) = n +1 X j =0 A n +1 − j ( ξ ) λ j , (21) 3 dB dξ = − iλB − 2 iuA, B ( ξ , λ ) = n X j =0 B n − j ( ξ ) λ j , (22) dC dξ = iλC + 2 iu ∗ A, C ( ξ , λ ) = n X j =0 C n − j ( ξ ) λ j , (23) or in different form we hav e A j +1 ,ξ = i uC j − iu ∗ B j , A 0 = 1 , A 1 = c 1 , (24) iB j +1 = − B j,ξ − 2 iu A j +1 , B 0 = − 2 u, (25) iC j +1 = C j,ξ − 2 iu ∗ A j +1 C 0 = − 2 u ∗ , (26) where c 1 is the co ns tant of integration. Differenciating Eq. (21) and using (15) we can obtain B B ξξ − u ξ u B B ξ − 1 2 B 2 ξ +  λ 2 2 − iλ u ξ u + | u | 2  B 2 = 2 u 2 ν. (27) Using ( ?? ) the eigenfunction Ψ 1 for finite-gap po ten tial U hav e the fo r m Ψ 1 ( ξ , λ ) =   U ( ξ ) U (0) n Y j =1 λ − µ j ( ξ ) λ − µ j (0)   1 / 2 exp ( − i Z ξ 0 p R ( λ ) Q n j =1 ( λ − µ ( ξ ′ )) dξ ′ ) . Spec ia l cas e o f system (4- 6) is the three wa ve system ı dA 1 dξ = ǫ A 3 A ∗ 2 , ı dA 2 dξ = ǫA ∗ 1 A 3 , ı dA 3 dξ = ǫA 1 A 2 . (28) The cor resp onding elemen ts of La x matrices a re L ( ξ , λ ) =  − ı λ 2 + ı 2 λ A − iǫA 1 − i 1 λ A 3 A ∗ 2 − iǫA ∗ 1 − i 1 λ A 2 A ∗ 2 3 ı λ 2 − ı 2 λ A  , (29) M ( ξ , λ ) =  − ı λ 2 − iǫA 1 − iǫA ∗ 1 − ıλ 2  , A = | A 2 | 2 − | A 3 | 2 . (30) T o integrate the system (28) we introduce ne w v ariable µ = ı 1 A 1 dA 1 dξ = A 3 A ∗ 2 A 1 , (31) in terms o f which o ur equations ca n b e written as dµ dξ = 2 ı p R ( µ ) , (32) where R ( λ ) = ( 1 4 λ 2 − 1 2 A ) 2 − | A 1 | 2 ( λ − µ )( λ − µ ∗ ) = (33) 1 4 λ 4 − α 1 λ 3 + α 2 λ 2 − α 3 λ + α 4 . (34) The µ v a riable and A j , j = 1 , . . . 3 o bey the equations α 1 = 0 , | A 1 | 2 − 1 2 A = α 2 , (35) −| A 1 | 2 ( µ + µ ∗ ) = α 3 , 1 4 A − µµ ∗ | A 1 | 2 = α 4 . (36) 4 which a re related to the integrals of motion o f the monomer system with α 4 = 0 .The equation o f motion is then  dµ dξ  2 = − 4  µ 4 + N µ 2 − H µ  (37) where the s ystem (28) conserves the dimensio nless v ar iable N and the Hamiltonian H N = | A 1 | 2 − 1 2 A , H = A 1 A 2 A ∗ 3 + A 3 A ∗ 1 A 2 , (38) Solving Eqs . (36) for µ v ariable we obtain µ = 1 4 ν  H + ı p P ( ν )  , (39) where P ( ν ) = 4 ν 3 − 4 N ν 2 + N 2 ν − H 2 . (40) W e seek the s olution A 1 in the following for m A 1 = p ν ( ξ ) exp iC Z ξ 0 dξ ′ ν ( ξ ′ ) ! = p ν ( ξ ) exp( iψ ( ξ )) , (41) where ν = | A 1 | 2 = ℘ ( ξ + ω ′ ) + C 1 , ℘ is the W eier strass function, and ω ′ is half p erio d.Using Eq. (39) and the following equation dν dξ = − 2 ıν ( µ − µ ∗ ) , (42) derived fro m (31) and three wa v e equa tions we obta in  dν dξ  2 = 4 ν 3 − 4 N ν 2 + N 2 ν − H 2 , (43) whose solution can b e expr essed in terms of the W eierstrass elliptic functions as ν = ℘ ( ξ + ω ′ ) + N 3 . (44) Substituting this expression in E q. (41) we o btain A 1 = r ℘ ( ξ + ω ′ ) + N 3 exp( iψ ( ξ )) , (45) where the pha se ψ ( ξ ) is given by ψ ( ξ ) = H 2 ℘ ′ ( κ )  ln σ ( ξ + ω ′ − κ ) σ ( ξ + ω ′ + κ ) + 2 ζ ( κ ) ξ  + ψ 0 , (46) and ψ 0 is initial constant pha se. II I. BI-HAMIL TONIAN ST RUCTURE In this para graph we will compute r -matrix a lgebra of restricted multiple thr e e wa ve interaction sys tem. W e note that in Lax repr e s ent ation (9) w e remov e the function a ( λ ), which is esse ntial for studying Hamiltonian dynamics of restricted multiple three wa ve in teraction sys tem. Let as consider Lax matr ix L ( λ ) = a − 1 ( λ ) L ( ξ , λ ) =  A B C − A  . ( 47) 5 and introduce standard Poisson brack et, {· , ·} 0 { f , g } 0 = − ı  ∂ f ∂ u ∂ g ∂ u ∗ − ∂ f ∂ u ∗ ∂ g ∂ u  − ı n X j =1 ∂ f ∂ b j ∂ g ∂ b ∗ j − ∂ f ∂ b ∗ j ∂ g ∂ b j ! − ı ∂ f ∂ c j ∂ g ∂ c ∗ j − ∂ f ∂ c ∗ j ∂ g ∂ c j ! The entries of L satisfy to the following w ell known equations { A ( λ ) , A ( µ ) } 0 = { B ( λ ) , B ( µ ) } 0 = { C ( λ ) , C ( µ ) } 0 = 0 , { A ( λ ) , B ( µ ) } 0 = 1 λ − µ  B ( µ ) − B ( λ )  , { A ( λ ) , C ( µ ) } 0 = − 1 λ − µ  C ( µ ) − C ( λ )  , { B ( λ ) , C ( µ ) } 0 = 2 λ − µ  A ( µ ) − A ( λ )  , which may b e rewritten as linear r -matrix algebr a { 1 L ( λ ) , 2 L ( µ ) } 0 = [ r ( λ − µ ) , 1 L ( λ ) + 2 L ( µ ) ] , (48) here 1 L ( λ ) = L ( λ ) ⊗ I , 2 L ( µ ) = I ⊗ L ( µ ) and r ( λ − µ ) is a c la ssical rationa l r -ma trix: r ( λ − µ ) = Π λ − µ , Π =    1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1    . Remind, tha t t wo Poisson brack ets { ., . } 0 and { ., . } 1 are compatible if every linear combination of them is still a Poisson br a ck et. The corr e sp o nding compatible Poisson tensor s P 0 and P 1 satisfy to the following eq uations [ [ P 0 , P 0 ] ] = [ [ P 0 , P 1 ] ] = [ [ P 1 , P 1 ] ] = 0 , (49) where [ [ ., . ] ] is the Schouten br ack et [6]. Remind that on a smo oth finite-dimensional manifold M the Schouten brack et of tw o bivectors X a nd Y is a n antisymmetric co nt rav ariant tensor of rank three and its comp onents in lo cal co ordinates z m read [ [ X , Y ] ] ij k = − dim M X m =1  X mk ∂ Y ij ∂ z m + Y mk ∂ X ij ∂ z m + cycle( i, j, k )  . The Poisson br ack e t ass o ciated with the Poisson bivector P is equal to { f ( z ) , g ( z ) } = h d f , P dg i = X i,k P ik ( z ) ∂ f ( z ) ∂ z i ∂ g ( z ) ∂ z k . (50) Here d f is cov ector with entries ∂ f / ∂ z i and h ., . i is a standard vector pro duct. There ar e a lot of the Poisson brack ets { ., . } 1 compatible with the linea r r -matrix brack et (4 8) similar to the quadratic Sklyanin algebr a [7]. Here we consider tw o examples o nly . Prop ositio n 1 If A = n X i =1 h i λ − ǫ i , B = 1 + n X i =1 e i λ − ǫ i , C = n X i =1 f i λ − ǫ i , (51) wher e h i , e i , f i ar e dynamic al variables and ǫ i ar e num eric al p ar ameters, t hen the fol lowing br ackets ar e c omp atible with line ar r -matrix br acket (48) { B ( λ ) , B ( µ ) } 1 = { A ( λ ) , A ( µ ) } 1 = 0 , 6 { A ( λ ) , B ( µ ) } 1 = 1 λ − µ  λ B ( µ ) − µ B ( λ )  − B ( λ ) B ( µ ) , { A ( λ ) , C ( µ ) } 1 = − λ λ − µ  C ( µ ) − C ( λ )  + B ( λ ) C ( µ ) , (52) { B ( λ ) , C ( µ ) } 1 = 2 λ − µ  µ A ( µ ) − λ A ( λ )  +2  1 − B ( λ )  A ( µ ) , { C ( λ ) , C ( µ ) } 1 = 2  A ( µ ) C ( λ ) − A ( λ ) C ( µ )  . Pro of : It is s ufficient to check the s tatement on an op en dense subset of the linear r -matr ix alg e br a (48) defined by the a ssumption that all the h i , e i , f i and ǫ i are differ ent . Namely , s ubstituting r a tional functions (51) int o (48 ) one gets canonica l br ack ets on the dire c t sum of n copies o f sl (2 ) { h j , e j } 0 = e j , { h j , f j } 0 = − f j , { e j , f j } 0 = 2 h j , j = 1 , . . . , n . (53) Substituting these ra tional functions (51) into the second brack ets (52) one g ets the following non-lo c a l brack ets betw een ge ner ators h i , e i , f i { h j , e j } 1 = ( ǫ j − e j ) e j , { h j , f j } 1 = − ( ǫ j − e j ) f j , { e j , f j } 1 = 2( ǫ j − e j ) h j , (54) { h i , e j } 1 = − e i b j , { h i , f j } 1 = e i c j , { e i , f j } 1 = − 2 e i a j , { f i , f j } 1 = − 2 h i c j + 2 f i a j . Now it is easy to prov e that Poisson bracket (53) is compatible with the Poisson brack e t (54). Prop ositio n 2 If A = h n λ + n − 1 X i =1 h i λ − ǫ i , B = e n + n − 1 X i =1 e i λ − ǫ i , C = f n + n − 1 X i =1 f i λ − ǫ i (55) wher e h i , e i , f i ar e dynamic al variables and ǫ i ar e num eric al p ar ameters, t hen the fol lowing br ackets ar e c omp atible with line ar r -matrix br acket (48) { B ( λ ) , B ( µ ) } 1 = { A ( λ ) , A ( µ ) } 1 = 0 , { A ( λ ) , B ( µ ) } 1 = 1 λ − µ  λ B ( µ ) − µ B ( λ )  − ρ 1 B ( λ ) B ( µ ) , { A ( λ ) , C ( µ ) } 1 = − λ λ − µ  C ( µ ) − C ( λ )  + ρ 1 B ( λ ) C ( µ ) − ρ 2 B ( λ ) , (56) { B ( λ ) , C ( µ ) } = 2 λ − µ  µ A ( µ ) − λ A ( λ )  + 2  1 − ρ 1 B ( λ )  A ( µ ) − ρ 3 B ( λ ) { C ( λ ) , C ( µ ) } = − 2 ρ 1  A ( λ ) C ( µ ) − A ( µ ) C ( λ )  + 2 ρ 2  A ( λ ) − A ( µ )  + ρ 3  C ( λ ) − C ( µ )  Her e ρ 1 = 1 e n =  1 B ( λ )  , ρ 2 = f n e n =  C ( λ ) B ( λ )  , (57) and ρ 3 = 1 − 2 h n ( λ + µ ) e n + 2 h n P n − 1 k =1 e k e n = 1 −  ( λ + µ ) A ( λ ) λ B ( λ )  −  ( λ + µ ) A ( µ ) µ B ( µ )  , (58) wher e [ X / Y ] is a quotient of p olynomia ls X and Y in variables λ and µ over a fi eld as in [7]. 7 Pro of : As ab ov e it is sufficient to chec k the statement on an op en dense subset of the linear r -ma tr ix alg e bra (48) defined by the assumption that all the h i , e i , f i and ǫ i are different. Namely , substituting r ational functions (51) int o (48) one gets lo ca l brack ets n − 1 copies of sl (2) { h j , e j } 0 = e j , { h j , f j } 0 = − f j , { e j , f j } 0 = 2 h j , j = 1 , . . . , n − 1 , (59) and degenera te brack ets { e n , f n } = − 2 h n , { h n , h i } = { h n , e i } = { h n , f i } = 0 . (60) The leading co efficients h n is the Cas imir element for these bra ckets. Substituting ra tio nal functions (51) into the second brack ets (56) one gets seco nd non-lo cal brack ets b etw e e n generator s h i , e i , f i . A t i, j = 1 , . . . , n − 1 these brack ets lo o ks like { h j , e j } 1 =  ǫ j − e j e n  e j , { h j , f j } 1 = −  ǫ j − e j e n  f j , { e j , f j } 1 = 2  ǫ j − e j e n  h j , (61) { h i , e j } 1 = − e i b j e n , { h i , f j } 1 = e i c j e n , { e i , f j } 1 = − 2 e i a j e n , { f i , f j } 1 = − 2 h i c j + 2 f i a j e n . A t e n = 1 these br ack ets coincide with the pr evious bra ck ets (54). The rema ining no n-zero brackets hav e the following for m { f i , f n } 1 = ρ 3 ( λ + µ = ǫ j ) f i , { e i , f n } 1 = − ρ 3 ( λ + µ = ǫ j ) e i , { e n , f n } = − e n . (62) Now it is e a sy to prove that Poisson brack et (59)-(6 0) is compatible w ith the Poisson bra ck et (61)-(62). This completes the pro o f. In the bo th cases we can re w r ite second Poisson brack e ts in the following r -matr ix for m { 1 L ( λ ) , 2 L ( µ ) } 1 = [ r 12 ( λ, µ ) , 1 T ( λ )] − [ r 21 ( λ, µ ) , 2 T ( µ ) ] , (63) where r 12 ( λ, µ ) =     µ λ − µ 0 0 0 0 0 µ λ − µ 0 0 λ λ − µ 0 0 0 0 0 µ λ − µ     +    0 0 0 0 0 0 ρ 1 B ( µ ) 0 − ρ 3 0 0 0 ρ 2 − ρ 1 C ( µ ) 0 0 0    (64) and r 21 ( λ, µ ) = Π r 12 ( µ, λ )Π . F or the Lax ma trix L with entries (51) we hav e ρ 1 = 1 , ρ 2 = 0 , ρ 3 = 0 . F or the Lax ma trix L with entries (55) functions ρ k are given by (57)- (58). It is ea sy to see that entries of the La x matrix (47) have the form (55 ). So , w e ca n use quadr a tic-linear algebra (63) in o rder to get bi-hamiltonian descr iption of the restricted multiple three wa ve in ter a ction system. The first part of the brackets b etw e en v ariables u , u ∗ , b i , b ∗ i and c i , c ∗ i may be dir e c tly res tored from the brackets (61)-(62). The remaining par t has to b e obtained fr o m the compa tibilit y conditions (49). As a r esult the non- zero brack ets with v aria ble s u and u ∗ lo ok like { u, u ∗ } 1 = iu, { u ∗ , b j } 1 = − i 2 b j ρ 3 ( λ + µ = ǫ j ) , { u ∗ , b ∗ j } 1 = i 2 b ∗ j ρ 3 ( λ + µ = ǫ j ) , { u ∗ , c j } 1 = i 2 c j ρ 3 ( λ + µ = ǫ j ) , { u ∗ , c ∗ j } 1 = − i 2 c ∗ j ρ 3 ( λ + µ = ǫ j ) . The lo ca l brack ets at j = 1 , . . . , n a re equal to { b j , c j } 1 = ib 2 j 2 u , , { b j , c ∗ j } 1 = iǫ j b j c j , { b j , b ∗ j } 1 = i ( b j c ∗ j − 2 uǫ j ) 2 u , 8 { b ∗ j , c j } 1 = − i ( b j b ∗ j + c j c ∗ j ) 2 u , { b ∗ j , c ∗ j } 1 = − i (2 b ∗ j ǫ j u − c j c ∗ 2 j ) 2 c j u , { c j , c ∗ j } 1 = − ib j c ∗ j 2 u . The non-lo cal brack ets at i 6 = j read as { b i , b ∗ j } 1 = − ib i c ∗ j 2 u , { b i , c j } 1 = ib i b j 2 u , { c i , c ∗ j } 1 = − ib i c ∗ j 2 u , { c ∗ i , b ∗ j } 1 = − ic ∗ i c ∗ j 2 u , { c i , c j } 1 = i ( b i c j − b j c i ) 2 u , { c i , b ∗ j } 1 = i ( c i c ∗ j + b i b ∗ j ) 2 u , { b ∗ i , b ∗ j } 1 = − ( b ∗ i c ∗ j − c ∗ i b ∗ j ) 2 u . Other brackets are equa l to zero, for instance { b i , b j } 1 = { b i , c ∗ j } = { c ∗ i , c ∗ j } 1 = 0 . Using r -matrix algebras (48) and (63) it is ea s y to prove that integrals of motion I j ∈ { J 0 , J i , . . . , J n , K 1 , . . . , K n } (18) are in the bi- in volution with res pe c t to the brack ets { ., . } 0 and { ., . } 1 : { I j , I k } 0 = { I j , I k } 1 = 0 . The o ne o f the ma in problems is that the main characteristic of the mo del, s uch as equation of motion, the Lax matrices and integrals of motion are inv ariant with r esp ect to co njugation u ↔ u ∗ , b j ↔ b ∗ j , c j ↔ c ∗ j , (65) but the s e cond brack ets { ., . } 1 do not inv ariant with r esp ect to this transfor mation. The seco nd proble m is that the s econd br ack ets { ., . } 1 are ra tional brack ets and we hav e an obvious problem with quantization o f these bra ck ets. According to [8] ther e are different bi-hamiltonian structures for a g iven in tegrable system. 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