Generalized Burgers equations and Miura Map in nonabelian ring. nonabelian rings as integrable systems.

Transcription

1 Generalized Burgers equations and Miura Map in nonabelian rings as integrable systems. Sergey Leble Gdansk University of Technology

2 Table of contents 1 Introduction: general remarks 2 Remainders Linearization Left division 3 Darboux transformation and evolution. Generalized Burgers equations 4 Schrödinger equation 5 Quantum scattering 6

3 General purposes Theory development Physics: Scattering and multi-center problems; Time-dependent nonsingular potentials of quantum mechanics, quantum nonlinear equations. Math.: Dressing scheme development, noncommutative Burgers- and Miura-like nonlinear linearizable equations multiple scattering series approach.

4 History Matveev, V.B. (1998) Darboux transformations in associative rings and functional-difference equations, in J. Harnad and A. Kasman (eds.), The Bispectral Problem, AMS series CRM Proceedings and Lecture Notes, Montreal, 14, pp S. Leble, A. Zaitsev Division of differential operators, intertwine relations and Darboux transformations Rep. Math. Phys. 40 (2000) Evgeny V. Doktorov, Sergey B. Leble Dressing Method in Mathematical Physics ( Springer-Verlag, 2007)

5 Algebra Remainders Linearization Left division Let K be a differential ring of the zero characteristics with unit e and with an involution denoted by a superscript *. The differentiation is denoted as D. The differentiation and the involution are in accordance with operations in K, i.e (i) (a ) = a, (a + b) = a + b, (ab) = b a, a, b K ; (ii) D(a + b) = Da + Db, D(ab) = (Da)b + adb ; (iii) (Da) = Da ; (iv) operators D n with different n form a basis in a K-module Diff(K) of differential operators. The subring of constants is K 0 and a multiplicative group of elements of K is G; (v) s K ϕ K such that Dϕ = sϕ,

6 Differentiating Remainders Linearization Left division it also means the existence of a solution of the equation Dφ = φs, (1) There are lots of applications of the rings of square matrices in the theory of integrable nonlinear equations, as well as in classical and quantum linear problems. In this case matrices are parameterized by a variable x and D can be a derivative with respect to this variable or a combination of partial derivatives that satisfies the conditions (i) and (ii). If D is the standard differentiation, then the involution * may be the Hermitian conjugation. In the case of a commutator the operator D acts as Da = [d, a] and (Da) = [d, a].

7 Differential operator Remainders Linearization Left division Let L = N a n D n, a n K (2) n=0 be a differential operator of the order N. We shall study a right and left division of L by the operator L s = D s, L = ML s + r, L = L s M + + r +, (3) where M and M + are the results of right and left division, r and r + being the remainders.

8 Miura equations Remainders Linearization Left division If the representation (27) is valid, then the remainder r and the result of division M are written via differential Polynomials r = N a n B n (s), n=0 where M = N a n H n 1 = n=1 N 1 n=0 b n D n, (4) N b n = a k B k 1,n (s), n = 0, 1,..., N 1. (5) k=n+1

9 Remainders Linearization Left division As a corollary we get Theorem For the linear operator L to be a right divisible by L s without remainder, it is necessary and sufficient that s be a solution of the Miura equation N a n B n (s) = 0. (6) n=0 If this condition holds, the operator L factorizes as L = ML s where M is given by (4) and (5).

10 Bell polynomials Remainders Linearization Left division The left and right non-abelian Bell polynomials B n (s) are defined by the recurrence relations for left BPs and B n (s) = DB n 1 (s) + B n 1 (s)s, n = 1, 2,... (7) B n + (s) = DB n 1 + (s) + s, n = 1, 2,... (8) for right BPs with the initial condition B 0 (s) = e. For example B 0 (s) = e, B 1 (s) = s, B 2 (s) = Ds + s 2,... (9) r = a 0 + a 1 s + a 2 (Ds + s 2 ) +...,

11 Remainders Linearization Left division Equation (6) is nonlinear. For N = 2 it is the Riccati type equation known in the theory of the KdV equation as the Miura map. Therefore, it is natural to term it as a generalized right Miura equation. It links the function s and coefficients of the operator L. Theorem Let an invertible function ϕ be a solution to the linear differential equation N a n D n ϕ = 0. (10) n=0 Then the operator L, defined by (2), is right divisible by L s, where s = ϕ ϕ 1 and ϕ Dϕ. Moreover, s is a solution of the right Miura equation (6).

12 Left division problem Remainders Linearization Left division For the solution of a left-division problem let us write a result of division in the form M + = N 1 n=0 b + n D n. (11) Now we should determine b + n, n = 0, 1,..., n 1. To this aim we substitute the representation (11) into the right-hand side of the second equation of (27). Following the lines of the Proposition, we obtain b + N 1 = a N, (12) b n + = a n+1 L s b n+1 +, n = 0, 1,..., N 2, (13) r + = a 0 L s b + 0. (14)

13 Solving subsequently Remainders Linearization Left division equations (13) and (14), we arrive at b + n = N k=n+1 ( 1) k n 1 L k n 1 s a k, n = 0, 1,..., N 1, (15) r + = N ( 1) k L k s a k. (16) k=0 The entities b + n, n = 0, 1,..., N 1 and r + can be expressed in terms of the right Bell polynomials L k s a = L k s ea = ( 1) k B + k (s)a.

14 Left division problem solution Remainders Linearization Left division Formulas (27), (11), (15), and (16) give a solution of the left division of L by L s. So, the following Theorem For the operator L to be left divisible by the operator L s it is necessary and sufficient that s be a solution of the differential equation N B k (s) + a k = 0. (17) k=0 If this condition holds, the operator L factorizes as L = L s M +, where M + is given by (11). For the reminder r + and the result of division M + equations and (11) take place.

15 Eigenvalue problem. Remainders Linearization Left division The case of right Miura equation N B k (s)a k = 0. (18) yields k=0 L = ML s, (19) that may be directly applied to the eigenvalue problem that leads to and may be read as Lψ = ML s ψ = ψλ (20) L s ML s ψ = L s ψλ, (21) L s M ψ = L ψλ, (22)

16 The DT of solution Remainders Linearization Left division ψ = L s ψ, (23) If s = (Dφ)φ 1, then, formally L s = φdφ 1. (24) L 1 s = φd 1 φ 1. (25)

17 Left Miura equation. Remainders Linearization Left division The case of left Miura equation N r + = ( 1) k L k s a k = 0. (26) k=0 that is equivalent to (18) yields hence the eigenvalue problem may factorize as L = L s M +, (27) Let L 1 s Lψ = L s M + ψ = ψλ (28) L s M + L s φ = L s φλ (29) M + L s φ = φλ (30)

18 Remainders Linearization Left division B + polynomials and Miura in N=2.. for left BPs and B n + (s) = DB n 1 + (s) + s, n = 1, 2,... (31) Let us take the case N=2 for illustration r + = a 0 L s a 1 + L 2 s a 2, acting by the DT operators, Da = a a 2 2sa 2 + (s 2 s )a 2 a 1 + sa 1 + a 0 = 0. (32) let us recall that it is linearized by anti-cole-hopf substitution s = φ 1 φ.

19 Recurrence Remainders Linearization Left division. let us compare with the right one (18) 2 B k (s)a k = 0. (33) The KdV case: the right Miura is The left one is equivalent k=0 it differs by the only sign at s. a 2 (s + s 2 ) + a 1 s + a 0 = 0. (34) a 2 = 1, a 1 = 0, a 0 = u(x, t) µ, (s + s 2 ) + u = µ. (35) s s 2 + u = µ, (36)

20 Stationary ZS Remainders Linearization Left division Another situation is in a nonabelian case. E.g. go down to the ZS problem. Compare the right a 1 s + a 0 = 0. (37) and the left one case sa 1 + a 0 = 0. (38)

21 One more differentiation The problem of the operator division is directly connected with the Darboux transformation. To clarify this point, suppose that in the ring K there exists one more differentiation D 0 which commutes with the operator D. For example, it may be a differentiation in a parameter t.

22 Intertwine relation Let us introduce an auxiliary commutation relation Indeed, L s r = rl s + Dr + [r, s]. (39) L s r rl s = (D s)r r(d s) = Dr sr rd + rs = rd + Dr sr rd + rs = Dr + [r, s]. Taking into account the equalities (52) and (27), we arrive at the relation L s (D 0 L) = (D 0 L)L s + D 0 s Dr [r, s], (40) where L = L s M + r. (41)

23 Burgers and DT As the result, the important conclusion takes place: Theorem If a function s satisfy the equation D 0 s = Dr + [r, s], (42) the operator L s intertwines the operators D 0 L and D 0 L, L s (D 0 L) = (D 0 L)L s. (43) The explicit expression for L could be obtained in terms of (63) and (27) and has the form N L = (a nh n 1 + a n H n sa n H n 1 ) + a 0. (44)

24 No factorization! Remark: No necessity of factorization (r=0), the element s should only solve the generalized Burgers equation (42). Namely: Let us write (42) explicitly using (27). It is established that for the intertwine relation (43) to be valid, it is necessary and sufficient that s be a solution of the equation D 0 s = N (Da n B n (s) + a n B n+1 (s) sa n B n (s)). (45) n=0 or, using (7), one obtain the form convenient for analysis D 0 s = N D(a n B n (s)) + [a n, s]b n (s)s). (46) n=0

25 On linearization Remark The equation (46) is nonlinear but linearizable. This equation (in a different form) was introduced in [Leble, Darboux Transforms in Rings with Differentiation (1993) preprint KGU (1993)], [Schimming, Rida [1996]]. The form we suggest is convenient for a further investigations, e.g., in the framework of bilinearization technique of Hirota.

26 Cole-Hopf In the case of commuting coefficients and L = D 2 equation (65) is known as the Burgers equation. By this reason and due to the integrability of (46) by the Cole-Hopf transformation, it is natural to refer to (46) as a generalized Burgers equation. Theorem Suppose an invertible function ϕ be a solution to linear differential equation D 0 ϕ = Lϕ. Then the function s satisfy the generalized Burgers equation (46).

27 Corollary The obvious corollary of the intertwine relation (43) and the Proposition 5 is Theorem Let functions ψ and ϕ be solutions of the equations D 0 ψ = Lψ, D 0 ϕ = Lϕ (47) for an invertible function ϕ. Then the function ψ = L s ψ = Dψ sψ, s = (Dϕ)ϕ 1 (48) is a solution of the equation D 0 ψ = L ψ. (49)

28 Matveev theorem The last statement accomplishes the proof of Matveev theorem for differential polynomials (see, e.g. [Matveev]) in its non-abelian version. The equality (64) gives a representation of the transformed operator in terms of the generalized Bell polynomials. The explicit expressions for the transformed coefficients are a N [1] = a N, a k [1] = a k + (50) N [a n B n,n k + (a n sa n )B n 1,n 1 k ], (51) n=k k = 0,..., N 1. The functions B m,n are introduced in [Zaitsev,Leble].

29 Final formule We reproduce here the definition and some statements about them. Definition B n, 0 (σ) = 1, n = 0, 1, 2,..., and recurrence relations B n, k (σ) = B n 1, k (σ)+db n 1, k 1 (σ), k = 1, n 1, n = 2, 3,.... (2.6) B n, n (σ) = DB n 1, n 1 (σ) + B n (σ), n = 1, 2,.... define the generalized Bell polynomials B m,n. The functions B m,n are introduced in [Zaitsev, Leble ROMP 2000]. The following formula is extracted, B n, n k+1 (σ) = n ( i k i=k Sergey Leble Gdansk University n of Technology ) B n, n i (σ) D i k σ, k = 1, n, n = 0, 1, 2,... ;

30 Bell polynomials it gives the link between standard (nonabelian) Bell polynomials and the generalized ones: B n+1 (σ) = n B n, i (v) D n i σ, n = 0, 1, 2,.... i=0 Evaluation of the first three generalized Bell polynomials by the definition gives B n, 1 (σ) = σ; B n, 2 (σ) = σ 2 +n Dσ; B n, 3 (σ) = σ 3 +n σ σ+(n 1) σ Dσ

31 Schrödinger equation Left DT Let us introduce again an auxiliary commutation relation Indeed, L s r + = r + L s + Dr + + [r +, s]. (52) L s r + r + L s = (D s)r + r + (D s) = Dr + sr + r + D + r + s = r + D + Dr + sr + r + D + r + s = Dr + + [r +, s]. Taking into account the equalities (52) and (27), we arrive at the relation L s (D 0 L) = (D 0 L)L s + D 0 s Dr + [r +, s], (53) where L = L s M + + r +. (54)

32 Schrödinger equation On proof The corresponding GBE leads to the intertwine relation defines the equation if with its solution D 0 s Dr + [r +, s], (55) L s (D 0 L) = (D 0 L)L s, (56) D 0 φ = (Ls M + + r + ) φ (57) (D 0 L)ψ = 0 (58) ψ = L s φ. (59)

33 Schrödinger equation Quantum 1D evolutionn The simplest quantum evolution is defined by: the right Burgers is a 2 = 1, a 1 = 0, a 0 = u(x, t), iψ t = Hψ = ( D 2 + u(x, t))ψ. (60) 2 D 0 s = D(a n B n (s)) + [a n, s]b n (s)s). (61) L = a 0 + n=0 s t = D( (s + s 2 ) + u). (62) L = L s M + r. (63) 2 (a nh n 1 + a n H n sa n H n 1 ). (64) n=1

34 Schrödinger equation Applications Its applications are well-known in the KP equation theory. Recent results for multi-rouge waves see in Matveev presentation (NEEDS-2015) and (P Dubard and V B Matveev) Multi-rogue waves solutions: from the NLS to the KP-I equation. Nonlinearity 26 (2013) R93R125. Solutions of the NLS equation above provide 2n-parametric family of the smooth rational solutions to the KP-I equation:. Obviously the function ; where (4u t + 6uu x + u xxx ) = 3u yy. f (k; x; y; t) := exp(kx + ik 2 y + k 3 t + φ(k)) satisfies the system φ(k) := Φ(k) ϕ 3 k 3 ;

35 Schrödinger equation Illustrations The explicit solutions of the dressed second equation u(x; y; t) = 2 xx logw (f 1 ;...; f 2n ) = 2( v 2 B 2 ) where v is a solution of NS equation are also interesting for 1D quantum mechanics. Link to NS theory was found in the mentioned paper [Matveev]. The potential is restricted from both sides. The corresponding solution of the temporal Schrödinger and Burgers equation is given by ψ := W (f 1,..., f n, f ) W (f 1,..., f n )

36 Schrödinger equation Figure: Amplitude of the solution to the NLS equation for n = 2 Zoo of Burgers Eq. The left DT leads to the GBE+ equation

37 Quantum scattering Applications Much more interesting the noncommuting coefficients of L, even if one take the stationary solution of the evolution equation σ t = φ xt φ 1 φ x φ 1 φ t φ 1 = 0. (2.5) Hence, showing the first term, we write Da 0 + a 0 s sa 0 + N (D(a n B n (s)) + [a n s]b n (s)s) = 0. (65) n=1

38 Quantum scattering Applications Let us take an example of a 0 (Â). Then the function s should be also a function of the operator  and, a 0 commutes with s. Generally the relation (42) means, that if all a n commute with s, we arrives at Dr = 0; the equation therefore is r = C, DC = 0. (66) The expansion s = S n  n (67) leads to the recurrence for S n. Am example:

39 Quantum scattering Scattering problem Consider a scattering problem for a non-spherical potential Û: ( r 2 1 r r + ˆL ) 2 2r 2 + Û E ψ( r) = 0, (68) where ˆL 2 is square of the angular momentum operator, E is the energy of particle. The asymptotic of wave function ψ( r) looks like ψ( r) r exp(i k r) + f (θ) eikr r, (69) where f (θ) is scattering amplitude. The operator ˆL 2 commutes with all radial derivatives, in particular with = / r. In three-dimensional space the DT can be reduced to the one-dimensional Heisenberg matrix (or operator) problem with appropriate variable with a basis of the rest variables.

40 Quantum scattering The radial DT In our case r, functions a 2 and a 1 are the same as in (??) and a 0 = ˆL 2 2r 2 + Û. The radial DT for any solution of the Schrödinger equation, but s should be treated as a function of the operator variable ˆL 2. The transformation of the potential: Û Û (1) = Û + 1 r 2 s. (70) To find the operator s, we use the covariance principle for (68), that yields explicit constraint for s which looks like a 0 + [a 0, s] + (a 1 s) + [a 1, s]s + {a 2 (s + s 2 )} + [a 2, s](s + s 2 ) = a 0 + (a 1 s) + {a 2 (s + s 2 )} = 0.

41 Quantum scattering Expansions Integrating over r, we obtain the operator equation for s which in our case can be written as s + 2 r s + s2 = ˆL 2 r 2 + 2Û + C(ˆL 2 ). (71) The constant of integration C(ˆL 2 ) is a function of the operator variable ˆL 2 which does not depend on r [?]. The sense of this constant can be understood from the asymptotic behavior of s at infinity r, where the equation (71) goes to s + s 2 = C(ˆL 2 ). The general solution of this (Riccati) equation for the asymptotic in r at infinity gives either oscillations or s( ) = K(ˆL 2 ), then C(ˆL 2 ) = K(ˆL 2 ) 2.

42 Quantum scattering Expansions The operator s may be found as a series n=0 s nˆl 2n, where coefficients s n depend only on r. It is easy to show that (71) leads to the recursion relations for coefficients s n : s r s 0 + s 2 0 = 2Û + C(0), s r s 1 + s 0 s 1 + s 1 s 0 = ˆL 2 r 2 + C (0), s n + 2 n r s n + s k s n k = C (n) (0), n 2. k=0 For example, the first equation in the region where U = 0 looks like s r s 0 + s 2 0 = K 0, where K 0 is zeroth coefficient in the expansion K = n=0 K nˆl 2n.

43 Quantum scattering Equation (70) gives nonlocal (with respect to angles) potential which depends on ˆL 2. Thus, we have the algorithm that determines the operator s and the dressed potential via the operator K. To evaluate cross section, we need only partial phases or scattering amplitude related to the operator K. In order to find the partial phases for the dressed potential we should apply the DT to the wave function. However, a trouble occurs: the DT in general modifies the plane wave exp( k r). Thus, the DT applied to wave function ψ( r) with the asymptotic (69) gives another asymptotic. In some particular cases, special choice of the operator K allows us to avoid such a problem. We would consider this choice as a condition in the formulation of a scattering problem.

44 Quantum scattering Expansions ndeed, consider the partial wave asymptotic for a non-spherical potential [?] ψ J ( r) 1 2kr (kr+ıδ J Λ J ( n) kr δ J Λ J ( n)), (72) where n is the unit vector directed as r, δ J denotes partial shifts, and Λ J ( n) are normalized eigenvectors of the S-matrix operator (partial harmonics). The most simple formulas for the shifts δ (1) J for the potential Û (1) result when partial harmonics Λ J are also eigenvectors of operator K. For example, suppose all partial harmonics Λ J are eigenvectors of K but only Λ 0 has nonzero eigenvalue κ, KΛ 0 ( n) = κλ 0 ( n). The asymptotic dressing is reduced to the action of the operator KSergey onleble asymptotic Gdansk University (72). of Technology

45 Quantum scattering Expansions It is easy to show by using expression ( ) κ k ln = 2 arctan(k/κ) κ + k for real-valued variables k and κ that the DT changes only the partial shift δ 0 : δ (1) 0 = δ 0 arctan(k/κ). (73) In this special case we add only one additional parameter. In the region k κ the second term of the equation (??) practically does not contribute to the partial cross section σ J = 4π k 2 sin2 δ J. (74) It yields an essential contribution to the cross section when k κ and so it can be considered as a correction at low energies.

46 Left division introduce new versions of generalized Miura and Burgers equations in non-commutative case. Nonabelian version allows to incorporate matrix and operator problems. QM application; the generalization yields multi-center and multi-channel phenomena description/interpretation. Open problems: Operator problems. D= [, quantum evolution in Heisenberg picture. Discrete equation, DT, Polynomials, etc. Modifications quantum, statistical physics edt