Quantized representation of some nonlinear integrable evolution equations on the soliton sector

Quantized representation of some nonlinear integr able evolution equations on the soliton sector Yair Z armi Jacob Blaus tein Institutes for Desert Research Ben-Gurion University of the Negev Midreshet Ben-Gurion, 84990, I srael Abstract The Hirota algorithm f or solving several integrabl e nonlinear evoluti on equations is sugges tive of a simple quantized representation of these equations and t heir soliton solutions over a F ock spa ce of bosons or of fermions. T he classic al nonlinear wave equation becomes a nonlinear equation f or an operator. T he solution of this equation i s constructed t hrough the operator analog of the Hirota transformation. The class ical N- solitons solution is the expec tation value of the solution operator in an N -par ticle state in the Fock space. PACS: 47. 35 .Fg; 05.45.Yv; 05.30.Jp Keywords: Nonlinear evolution equations, quantiz ed representation. Whereas the structure of the single-soliton solution s of the KdV equation [ 1 ], u t = 6 u u x + u xxx , (1) is simple, the structur e of its multiple-solitons solution i s r ather cumbersome. However, using the Hirota algori thm [2], u t , x ( ) = 2 ! x 2 log f t , x ( ) " # $ % , (2) the function f ( t , x ) may be given a simple physical interpr etation. For an N -solitons solution, with soliton wave numbers k i 1 ≤ i ≤ N , all different from one another, it is given by f t , x ( ) = 1 + ! k i ; t , x ( ) i = 1 N " + ! k i j ; t , x ( ) V k i l , k i m ( ) i l < i m # j = 1 n # $ % & ' & ( ) & * & 1 < i 1 < ! < i n " + , - . / 0 n = 2 N " ! k ; t , x ( ) = e 2 k x + v k ( ) t ( ) , v k ( ) = 4 k 2 , V k 1 , k 2 ( ) = k 1 1 k 2 k 1 + k 2 + , - . / 0 2 + , - . / 0 . (3) As one has V ( k , k ’) ≤ 1, f ( t,x ) is bounded by f t , x ( ) ! 1 + 1 n ! " k i ; t , x ( ) i = 1 N # $ % & ' ( ) n n = 1 N # ! e " k i ; t , x ( ) i = 1 N # $ % & ' ( ) . (4) Eq. ( 3) looks like a sum of Feynman diagrams co ntaining single- and multi-particle contributions of all pos sible subsets of n ≤ N of t he par ticles. The f unctions ϕ ( k i ; t , x ) may be vie wed as rea l “plane waves” , and V ( k , k ’) may be viewed as a “t wo-particle coupling coefficient”. With this obs ervation in mind, Eq. ( 3) is suggestive of the foll owing simple quantized r epresenta- tion of the solution of E q. (1 ) over a F ock space o f bosons or of fer mions, with creation and anni- hilation operator s, a k † and a k , respectively, and number operators N k , defined by N k = a k † a k , a k , a k ' † ! " # $ = % k & k ' ( ) Bos ons ( ) a k , a k ' † { } = % k & k ' ( ) Fer mi ons ) ( ) ' ( ) ) * + , , . (5) Consider the following operator F t , x ( ) = 1 + ! k ; t , x ( ) N k dk 0 " # + 1 n ! ! ! k i ; t , x ( ) N k i i = 1 n $ % & ' ( ) * V k l , k m ( ) 1 + l < m + n $ % & ' ( ) * , - . / 0 1 dk 1 dk 2 ! dk n 0 " # 0 " # 0 " # n = 2 " 2 . (6) As V(k l ,k m ) ≤ 1,int egration down to k = 0, does n ot pause any problem. T o im prove the conver- gence properties of the integrals one may multiply ϕ ( k ; t , x ) by a function of k that falls off suffi- ciently fast as k → ∞ , e.g., ! k ; t, x ( ) " ! k ; t, x ( ) e # $ k 4 $ > 0 ( ) . (7) (This amounts t o a mer e pha se shift in t he trajecto ry of a soliton.) For any state with a fini te num- ber of parti cles, the matrix element of the operator F( t , x ) i s a f inite sum of f inite ter ms. Hence, after calculating t he matrix element, one may set α to zero . As V ( k , k ’) ≤ 1, a majorant to the oper ato r F( t, x ) is the following operator, the matrix elements of which in states with a finite numbe r of parti cles ar e the upper bounds in Eq. (4): M = e ! k ; t , x ( ) N k dk 0 " # . (8) Denoting a state with r par ticles with a given w ave number, q , by q , r { } (for ferm ions, obvi- ously, only r = 1 is possible), the matrix element of F( t , x ) in a single-particle state is: q ,1 { } F t , x ( ) q ,1 { } = 1 + ! q ; t , x ( ) . (9) Eq. (9) is i dentical to the expression for f ( t , x ) of Eq. ( 2), when u ( t , x ) i s a single-soliton solution of Eq. ( 1 ) [2]. Similarly, the matrix element in a state of two particles with dif ferent wave num bers is identical to the expression for f ( t , x ) when u ( t , x ) is a two-solitons solution: q 1 ,1 { } , q 2 ,1 { } F t , x ( ) q 1 ,1 { } , q 2 ,1 { } = 1 + ! q 1 ; t , x ( ) + ! q 2 ; t , x ( ) + ! q 1 ; t , x ( ) ! q 2 ; t , x ( ) V q 1 , q 2 ( ) . (10) Extension to N > 2 is straightfor wa rd: q 1 , 1 { } , ! , q N , 1 { } F t , x ( ) q 1 , 1 { } , ! , q N , 1 { } , with q i ≠ q j (1 ≤ i , j ≤ N , i ≠ j ), is the expr ess ion for f ( t , x ) corresponding to an N -soliton solution of Eq. (1). If the particles ar e bosons, then a given momentu m state m ay be occupied by more than one par ti- cle. A matr ix element in a state, in which a given wave number is occupied by several bosons yields a soliton solution with a simple phase shift . For example, the matrix element q , n q { } F t , x ( ) q , n q { } = 1 + n q ! q ; t , x ( ) = 1 + ! q ; t , x + " ( ) " = log n q # $ % & q ( ) , (11) is equal to f ( t , x ) for a single-soliton solution wit h a pha se shift δ in the soliton tr ajectory. The same applies to a state with several distinct wav e numbers. For every wave number that is occu- pied by mor e than one boson , the co rresponding soliton is subjected to a similar phase shift. The fact that t he expression f or f ( t , x ) in the classical case is obtained as the expectation valu e of a quantum-mechanical operator leads directly to an operator-ver sion of E q. (1). Consider the opera- tor-analog of E q. (2): U t , x ( ) = 2 ! x F t , x ( ) x F( t , x ) " 1 ( ) . (12) As F( t , x ) is a diagonal op erator, U( t , x ) obeys E q. (1) on any state with a fini te number of par ticles, and the N -soliton-solution of Eq. (1) is equal to the expectation value: u t , x ( ) = q 1 , 1 { } , ! , q N , 1 { } U t , x ( ) q 1 , 1 { } , ! , q N , 1 { } , (13) Using Eq. ( 12), one can construct operators for t he inf inite sequence of conserved quantities th at characterize the soliton solutions of Eq. ( 1), as well as f or the Hamiltonian, from which Eq. (1) can be derived. F or example, the operator corresponding to the first conserved quantity, c 1 = u t , x ( ) dx !" + " # , (14) is C 1 = U t , x ( ) dx !" + " # . (15) Its action on any sta te yields C 1 q 1 , 1 { } , ! , q N , 1 { } = c 1 q 1 , ! , q N ( ) , (16) where c 1 ( q 1 ,…, q N ) is the value of c 1 for the corres ponding N -solitons solution. The same ideas apply to several othe r int eg rable equa tions. Sawada-Kotera equation [3, 4] u t = 45 u 2 u x + 15 u u xxx + 15 u x u xx + u xxxxx . (17) Eq. (17) is integr able [3, 4] . Its soliton solutions are also given by Eqs. (2) and (3), with V k , k ' ( ) = k ! k ' k + k ' " # $ % & ' 2 k 2 ! k k ' + k ' 2 k 2 + k k ' + k ' 2 " # $ % & ' . (18) mKdV equation [ 5 , 6] u t = 6 u 2 u x + u xxx . (19) Eq. (19) is integr able [5-7 ]. Its soliton solutions are given by [7] u t , x ( ) = 2 ! x tan " 1 g t , x ( ) f t , x ( ) ( ) . (20) In E q. (20) , g t , x ( ) = ! k i ; t , x ( ) i = 1 N " + ! k i j ; t , x ( ) V k i l , k i m ( ) i l < i m # j = 1 n # $ % & ' & ( ) & * & 1 < i 1 < ! < i n " + , - . / 0 n = 3 n odd N " , (21) f t , x ( ) = 1 + ! k i j ; t , x ( ) V k i l , k i m ( ) i l < i m " j = 1 n " # $ % & % ' ( % ) % 1 < i 1 < ! < i n * + , - . / 0 n = 2 n even N * , (22) V k 1 , k 2 ( ) = ! k 1 ! k 2 k 1 + k 2 " # $ % & ' 2 . (23) In this case, corresponding to the functions f ( t , x ) and g ( t , x ), there are t wo operators, F( t , x ) and G( t , x ), which contain terms with, res pectively, even and odd n in Eq. (6). Bidirectional KdV equation [8-10] u tt ! u xx ! " x 6 u u x + u xxx ( ) = 0 . (24) Eq. (24) is integrable [ 8-10]. Its soliton solutions are given by Eqs. (2) and ( 3). T he solitons may move in either dir ection along the x -axis. Hence, their velocities are given by v k , ! ( ) = ! 4 k 2 , ! = ± 1 . (25) In addition, the “coupling coefficients” V ( k , k ’) are replaced by ones, which, depend on th e wav e numbers, as well as on the velocities. For the scali ng employed in Eq. (24), they are given by: V k , ! , k ', ! ' ( ) = 12 k " k ' ( ) 2 + v k , ! ( ) " v k ', ! ' ( ) ( ) 2 12 k + k ' ( ) 2 + v k , ! ( ) " v k ', ! ' ( ) ( ) 2 . (26) These coefficients vanish in the single-particle limit ( k ’ = k , σ = σ ’). Therefore, the quantized rep- resentation described above can be constructed, with particle states characterized by t wo “quantum numbers”: k and σ . The fundamental operator s are denoted by, a k , ! † , a k , ! and N k , ! , and an N - particle state – by q 1 , ! 1 , 1 { } , ! , q N , ! N , 1 { } . The operator in Eq. (6) is replaced by F t , x ( ) = 1 + ! k , " ; t , x ( ) N k , " dk 0 # $ " = ± 1 % + 1 n ! ! ! k i , " i ; t , x ( ) N k i , " i i = 1 n & ' ( ) * + , V k l , " l , k m , " m ( ) 1 - l < m - n " l , " m = ± 1 & ' ( ) ) * + , , . / 0 1 0 2 3 0 4 0 dk 1 dk 2 ! dk n 0 # $ 0 # $ 0 # $ " i = ± 1 % i = 1 n % n = 2 # % . ( 27) The f act that σ has two values is sugges tive of a formulation in terms of spin-1/2 fermions. In class ical soliton dynamics, the single-soliton s olution plays a unique role. T here is an infi nite hierarchy of diff erential polynomials in u , the s olution of an evolution equation, which vanis h identically when u is a single-soliton solution (“special polynomials” [11, 12]). As an example, consider t he case of the KdV equation. The lowe st sc aling weight, in which sp ecial pol ynomial s exist, is 3. There are two special polynomials in this sca ling weight, given by: R 3,1 ( ) u [ ] = u x + q 1, 1 ( ) u , R 3, 2 ( ) = 3 2 q 1, 1 ( ) R 3,1 ( ) u [ ] dx !" x # ! q 1, 1 ( ) R 3,1 ( ) u [ ] dx x " # $ % & ' ( ) q 1, 1 ( ) = 1 2 u t , x ( ) dx !" x # ! u t , x ( ) dx x " # $ % & ' ( ) $ % & ' ( ) . (28) (In eac h superscript, ( W , i ) , W is the scaling weigh t, and i counts the polynomials with this s caling weight.) Replacing in E q. (28) the f unction u ( t , x ) by the operator U( t , x ) of Eq. (12 ), both special polynomials become operators, which project the full Fock space into its multi-particle subspace. The polynomials in Eq. (28) are non-local; they contain integrals over x . ( Yet, they are bounded.) A local special polynomial (containing only powe rs of u and of its spatial derivatives) first appears at scaling weight 6. It is given by [11, 12] R 6 , 1 ( ) u [ ] = u 2 ! x R 3, 1 ( ) u " # $ % & ' = u 3 + u u xx ( u x ( ) 2 . (29) (There are other special polynomials in thi s scaling weight. They are all non-local.) Using Eq. (12), one can construct the corresponding projectio n operator: R 6 , 1 ( ) U [ ] = U t , x ( ) 3 + U t , x ( ) ! x 2 U t , x ( ) " ! x U t , x ( ) ( ) 2 . (30) Again, the action of this operator on any single-pa rticle state is readily found to vanish: R 6 , 1 ( ) U [ ] q ,1 { } = 0 . (31) Thus, the specia l polynomials cor respond to an infinite hierarchy of commuting projection opera- tors. The quantized representation depends crucially on the f act that the “coupling coefficients” in Eq. (3) vanish in the limit k i = k j , i ≠ j . Hence, such a r epresentation is not pos sible if this requirement is not satisfied. T wo examples are given in the following. Kaup-Kupershmidt equation [ 13,14] u t = 180 u 2 u x + 30 u u xxx + 75 u x u xx + u xxxxx . (32) Eq. (32) is integr able [15-20 ]. Its multiple-soliton s olutions are given by u t x ( ) = 1 2 ! x 2 log f t , x ( ) " # $ % . (30) The “plane waves”, ϕ ( k ; t , x ), are defined as in Eq. (3), with soliton velocities given by v k ( ) = 16 k 4 . (33) However, the structur e of f ( t , x ) does not follow the pattern of Eq. ( 3). For the single-soliton solu- tion one has f t , x ( ) = 1 + ! q , t , x ( ) + 1 16 ! q , t , x ( ) 2 . (34) In the two-solitons solution, the expression for f ( t , x ) is: f t , x ( ) = 1 + ! q 1 ; , t , x ( ) + ! q 2 ; , t , x ( ) + 1 16 ! q 1 ; , t , x ( ) 2 + 2 q 1 4 " q 1 2 q 2 2 + 2 q 2 4 2 q 1 + q 2 ( ) 2 q 1 2 + q 1 q 2 + q 2 2 ( ) ! q 1 ; , t , x ( ) ! q 2 ; , t , x ( ) + 1 16 ! q 2 ; , t , x ( ) 2 + V q 1 , q 2 ( ) ! q 1 ; , t , x ( ) 2 ! q 2 ; , t , x ( ) + ! q 1 ; , t , x ( ) ! q 2 ; , t , x ( ) 2 ( ) + V q 1 , q 2 ( ) 2 ! q 1 ; , t , x ( ) 2 ! q 2 ; , t , x ( ) 2 V q 1 , q 2 ( ) = q 1 " q 2 2 ( ) 2 q 1 2 " q 1 q 2 + q 2 2 ( ) 16 q 1 + q 2 2 ( ) 2 q 1 2 + q 1 q 2 + q 2 2 ( ) # $ % % & ' ( ( .(36) Obviously, not all two-wave “coupling coefficients” vanish in t he limit q 1 = q 2 . Caudrey-Dodd-Gibbon equation [ 4], u t = 420 u 3 u x + 210 u 2 u xxx + 420 u u x u xx + 28 u u xxxxx + 28 u x u xxxx + 70 u xx u xxx + u xxxxxx x . (37) The i ntegrability of Eq. ( 37) is still an open question. The single- and two-solitons solutions do follow t he Hirota structure of Eqs. (2) and (3). The two-particle “coupling coefficient” is [4]: V k , k ' ( ) = k ! k ' k + k ' " # $ % & ' 2 k 2 ! k k ' + k ' 2 k 2 + k k ' + k ' 2 " # $ % & ' 2 . (38) Attempting to construct a three-solitons solution of Eq. (37), one finds that the coefficients of the second-order terms, g ( k i ; t , x )· g ( k j ; t , x ) (1 ≤ i , j ≤ 3, i ≠ j ) , ar e of t he Hirota f orm with V ( k i ,k j ) of Eq. (38). However, although the third-order term, g ( k 1 ; t , x )· g ( k 2 ; t , x ) · g ( k 3 ; t , x ), does vanish if any of the two wave numbers are equal, it cannot be fa ctorized into a product of the two-particle coeffi- cients V ( k i , k j ). The same applies to the (necessa ry) fourth-order terms. A very simple quantized represe ntation exists in the case of the Burgers equation [21], u t = 2 u u x + u xx . (39) The shock-front solutions of Eq. (39) are obtained through the Forsyth-Hopf-Cole transformation [22-24] : u t , x ( ) = ! x log f t , x ( ) " # $ % . (40) Here, f ( t , x ) has the f ollowing simple structure: f t , x ( ) = 1 + ! k i ; t , x ( ) i = 1 N " ! k ; t , x ( ) = e k x + v k ( ) t ( ) , v k ( ) = k ( ) . (41) Conseque ntly, the operator F( t , x ) is given by: F t , x ( ) = 1 + ! k ; t , x ( ) N k dk "# # $ . (42) There are many attempts in the literature at r igorous quantization procedure s of nonlinear evolu- tion equations [ 25-39]. T he quantized representa tion of equations and their solutions discussed here is somewhat dif ferent. It is characterized b y the fact that the coordinates, t and x are mer e parameters. Finally, the proposed quantum-mechanical representation opens a new vista for adding perturba- tions to a nonlinear wave equ ation. I n classical systems, the perturbation is a f unctional of the un- known solution, typically, a differential polynomia l in the latter. A common way for analyzing the effect of the p er turbation is through a Normal Form expansion [40-45, 11]. 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