Generalized Baecklund-Darboux transformation: conservation laws, rational extensions and bispectrality
aa r X i v : . [ m a t h . A P ] M a y Generalized B¨acklund-Darbouxtransformation: conservation laws,rational extensions and bispectrality
Alexander Sakhnovich
Abstract
B¨acklund-Darboux transformations are closely related to the in-tegrability and symmetry problems. For the generalized B¨acklund-Darboux transformation (GBDT), we consider conservation laws, ra-tional extensions and bispectrality. We use the case of the nonlinearoptics equation (and its auxiliary linear system) as an example.
B¨acklund-Darboux transformations and related commutation methods con-stitute one of the most fruitful approaches to the construction of explicitsolutions of linear and nonlinear equations and to the explicit spectral theo-retic results (see, e.g., [1,3,4,8–10] and numerous references therein). Similarto various symmetries studied in group analysis, B¨acklund-Darboux transfor-mations (BDTs) transform solutions of linear differential equations of somefixed class into solutions of differential equations of the same (or anotherfixed) class and potentials into potentials, whereas potentials are often solu-tions of the corresponding integrable nonlinear equations. Some connectionsbetween Darboux transformations and potential symmetries are discussed inan interesting paper [12].Though the idea of BDTs and commutation methods goes back to B¨ack-lund, Darboux and Jacobi, the notion of the so called
Darboux matrix hasa much later origin. According to J. Cie´sli´nski [1], ”all approaches to theconstruction of Darboux matrices originate in the dressing method”. In1rder to explain the notion of the Darboux matrix, we consider some initiallinear system u x = G ( x, z ) u, u ∈ C m , (1.1)where u x := ddx u ( u x = ∂∂x u in the case of several variables), G ( x, z ) is some m × m matrix function (i.e., G ( x, z ) ∈ C m × m ) and C stands for the complexplain. The Darboux matrix (matrix function) b w satisfies the equation b w x = e G b w − b wG, e G ( x, z ) ∈ C m × m . (1.2)It is easy to see that if (1.1) and (1.2) hold, the product e u = b wu satisfies the transformed system e u x = e G ( x, z ) e u. The cases, where the Darboux matrix b w can be constructed explicitly, if u and G are given explicitly, are of special interest. Clearly, in those cases weconstruct explicitly e u and also e G .We are interested in a generalized version of the B¨acklund-Darboux trans-formation for which we use the acronym GBDT. In GBDT (for the case of onespace variable) Darboux matrix is presented as the transfer matrix functionin Lev Sakhnovich form: b w ( z ) = w A ( z ) = I m − Π ∗ S − ( A − zI n ) − Π ; Π k ∈ C n × m , S ∈ C n × n ;(1.3)where A S − SA = Π Π ∗ , (1.4) I m is the m × m identity matrix. We note that an additional variable x appears in w A , Π , Π and S , so that we deal with matrix functions Π k ( x ), S ( x ) and Darboux matrix w A ( x, z ). GBDT is applicable to all systems u x = G ( x, z ) u , where G rationally depends on z , that is, G ( x, z ) = (cid:16) r X k =0 z k q k ( x ) + j X s =1 r s X k =1 ( z − c s ) − k q sk ( x ) (cid:17) (see the paper [13], the reviews [14, 15] and Chapter 7 in [16] as well asnumerous references therein). The coefficients e q k and e q sk of the transformed2rational) matrix function e G ( x, z ) are expressed via the coefficients q k and q sk of G ( x, z ) and matrix functions S ( x ), Π ( x ) and Π ( x ). Matrix functionsΠ k ( x ) are constructed as solutions of systems related to the initial system.Here, we shall consider in detail GBDT for the system u x = G ( x, z ) u, G = i zD − [ D, ̺ ] , D = diag { d , d , . . . , d m } = D ∗ ; (1.5) ̺ ∗ = B̺B, B = diag { b , b , . . . , b m } ( b k = ± D, ̺ ] := D̺ − ̺D, (1.6)where diag denotes a diagonal matrix. System (1.5) is an auxiliary systemfor the well-known N -wave (nonlinear optics) equation:[ D, ̺ t ] − [ b D, ̺ x ] = (cid:2) [ D, ̺ ] , [ b D, ̺ ] (cid:3) , (1.7) b D = diag { b d , . . . , b d m } = b D ∗ . (1.8)More precisely, equation (1.7) is equivalent to the compatibility condition G t − F x + [ G, F ] = 0 (1.9)of the auxiliary systems u x = G ( x, t, z ) u, G = i zD − [ D, ̺ ] , (1.10) u t = F ( x, t, z ) u, F = i z b D − [ b D, ̺ ] . (1.11)We shall study conservation laws, rational extensions and bispectrality forthe case of GBDT for system (1.5) and also for the N -wave equation (1.7).The integrable model describing interaction of three wave packages [18]is the most well-known subcase of the N -wave equation (1.7). Putting, forinstance, m = 3, B = I and d > d > d , and using transformations [11, Ch.3] ψ = ( b d − b d ) / ( d − d ) , ψ = ( b d − b d ) / ( d − d ) ,ψ = ( b d − b d ) / ( d − d ); (1.12) φ = − i p d − d ̺ , φ = − i p d − d ̺ ,φ = − i p d − d ̺ , (1.13)3e rewrite (1.7) in the standard form of the corresponding 3-wave interaction:( φ ) t + ψ ( φ ) x = i εφ φ , ( φ ) t + ψ ( φ ) x = i εφ φ , ( φ ) t + ψ ( φ ) x = i εφ φ , (1.14)where φ ( x, t ) stands for the function, which takes values complex conjugateto φ ( x, t ), and ε =( d b d − d b d + d b d − d b d + d b d − d b d ) × (cid:0) ( d − d )( d − d )( d − d ) (cid:1) − / . (1.15)If α is a scalar value or matrix, the notation α stands for the scalar, whichis complex conjugate to α , or the matrix with entries complex conjugate tothe entries of α , respectively. GBDT for the system (1.5) and for the N -wave equation was described in [13](see also [16, Subsection 1.1.3 and Section 7.1] and references therein). Inthis section, we give some necessary definitions and results from [13] and [16,Subsection 1.1.3 and Section 7.1]. We consider the case x ≥ x ≥ , t ≥ N -wave equation. We fix n × n matrices A and S (0) = S (0) ∗ and an n × m matrix Π(0) such that AS (0) − S (0) A ∗ = iΠ(0) B Π(0) ∗ . (2.1)We introduce the n × m matrix function Π( x ) and the n × n matrix function S ( x ) via initial values Π(0) and S (0) and differential equationsΠ x = − i A Π D + Π[ D, ̺ ] , S x = Π DB Π ∗ . (2.2)Relations (2.1) and (2.2) yield AS ( x ) − S ( x ) A ∗ = iΠ( x ) B Π( x ) ∗ . Then thefollowing proposition describes GBDT determined by A , S (0) and Π(0). Proposition 2.1
Let system (1.5) ( such that (1.6) holds ) be given. Then ( in the points of invertibility of S ( x )) the matrix function w A ( x, z ) = I m − i B Π( x ) ∗ S ( x ) − ( A − zI n ) − Π( x ) (2.3)4 s a Darboux matrix of (1.5) and satisfies the equation ddx w A ( x, z ) = (cid:0) i zD − [ D, e ̺ ( x )] (cid:1) w A ( x, z ) − w A ( x, z ) (cid:0) i zD − [ D, ̺ ( x )] (cid:1) , (2.4) where e ̺ = ̺ − B Π ∗ S − Π , e ̺ ∗ = B e ̺B. (2.5) Moreover, if det S ( x ) = 0 for x ≥ , a normalized fundamental solution e w ( x, z ) of the transformed system e w x = (i zD − [ D, e ̺ ]) e w, e w (0 , z ) = I m (2.6) is given by the equality e w ( x, z ) = w A ( x, z ) w ( x, z ) w A (0 , z ) − , (2.7) where w is the fundamental solution of the initial system (1.5) normalized by w (0 , z ) = I m . Next, we add the variable t and consider Π( x, t ), S ( x, t ) and w A ( x, t, z ) whichare determined by A , Π(0 ,
0) and S (0 ,
0) = S (0 , ∗ via equations (2.2), (2.3)and Π t = − i A Π b D + Π[ b D, ̺ ] , S t = Π b DB Π ∗ . (2.8)Instead of (2.1) we assume that AS (0 , − S (0 , A ∗ = iΠ(0 , B Π(0 , ∗ , (2.9)which implies that AS ( x, t ) − S ( x, t ) A ∗ = iΠ( x, t ) B Π( x, t ) ∗ . Proposition 2.1yields: Proposition 2.2
Let an m × m matrix function ̺ ( ̺ ∗ = B̺B ) be contin-uously differentiable and satisfy the N -wave equation (1.7) . Then e ̺ of theform e ̺ ( x, t ) := ̺ ( x, t ) − B Π( x, t ) ∗ S ( x, t ) − Π( x, t ) (2.10) satisfies ( in the points of invertibility of S ) the equality e ̺ ∗ = B e ̺B and the N -wave equation. Using [16, Therem 6.1] on wave functions, we easily obtain the next state-ment. 5 emark 2.3
If the conditions of Proposition 2.2 hold, det S ( x, t ) = 0 on thesemi-band ≤ x < ∞ , ≤ t ≤ ε and w ( x, t, z ) is the initial wave function ( i.e., w x = Gw , w t = F w and w (0 , , z ) = I m ) , then the transformed wavefunction e w ( x, t, z ) is given by the equality e w ( x, t, z ) = w A ( x, t, z ) w ( x, t, z ) w A (0 , , z ) − . (2.11) Example 2.4
The real-valued case is of special interest [11, Section 3.4].It is immediate from Proposition 2.2 ( and formulas defining e ̺ considered inthis proposition ) that equalities ̺ = ̺, A = − A, Π(0 ,
0) = Π(0 , , S (0 ,
0) = S (0 , , (2.12) yield the equality e ̺ = e ̺ . Assume that (2.12) holds, and so e ̺ = e ̺ . Then, in thesubcase m = 3 , B = I , d > d > d of the 3-wave equation, we can rewrite (1.14) in the real-valued form and obtain solutions of the corresponding exactresonance equations. Namely, we set ϕ = − i φ = − p d − d e ̺ , ϕ = − i φ = − p d − d e ̺ ,ϕ = − i φ = − p d − d e ̺ , and, taking into account (1.14) , derive ( ϕ ) t + ψ ( ϕ ) x = εϕ ϕ , ( ϕ ) t + ψ ( ϕ ) x = εϕ ϕ , ( ϕ ) t + ψ ( ϕ ) x = − εϕ ϕ . (2.13) When ̺ is differentiable with respect to t (to x ), we can differentiate bothsides of the first relation in (2.2) (in (2.8)) with respect to t (to x ). If ̺ isdifferentiable with respect to x and t and ̺ t is continuous, then, accordingto the so called Clairaut’s (or Schwarz’s) theorem, we have Π xt = Π tx . ThusΠ xt = Π tx is the necessary condition of the compatibility of the first relationsin (2.2) and (2.8). In a similar way, S xt = S tx is a necessary condition of the6ompatibility of the second relations in (2.2) and (2.8). In other words, thenecessary compatibility conditions are( − i A Π D + Π[ D, ̺ ]) t = ( − i A Π b D + Π[ b D, ̺ ]) x , (3.1)(Π DB Π ∗ ) t = (Π b DB Π ∗ ) x , (3.2)where Π x should be substituted by − i A Π D + Π[ D, ̺ ] and Π t should be sub-stituted by − i A Π b D + Π[ b D, ̺ ]. Proposition 3.1
Let ̺ be differentiable and satisfy the N -wave equation (1.7) . Then the equalities (3.1) and (3.2) hold. Proof.
Substituting expressions for Π t and Π x we obtain( − i A Π D + Π[ D, ̺ ]) t = − i A ( − i A Π b D + Π[ b D, ̺ ]) D + ( − i A Π b D + Π[ b D, ̺ ])[
D, ̺ ] + Π[
D, ̺ t ] , ( − i A Π b D + Π[ b D, ̺ ]) x = − i A ( − i A Π D + Π[ D, ̺ ]) b D + ( − i A Π D + Π[ D, ̺ ])[ b D, ̺ ] + Π[ b D, ̺ x ] . Therefore, taking into account (1.7) we derive( − i A Π D + Π[ D, ̺ ]) t − ( − i A Π b D + Π[ b D, ̺ ]) x = Π (cid:0) [ D, ̺ t ] − [ b D, ̺ x ] − (cid:2) [ D, ̺ ] , [ b D, ̺ ] (cid:3)(cid:1) = 0 , and so (3.1) is proved. Using the same substitutions as before, we have also(Π DB Π ∗ ) t = (cid:0) − i A Π b D + Π[ b D, ̺ ] (cid:1) DB Π ∗ + Π DB (cid:0) i b D Π ∗ A ∗ + [ ̺ ∗ , b D ]Π ∗ (cid:1) , (3.3)(Π b DB Π ∗ ) x = (cid:0) − i A Π D + Π[ D, ̺ ] (cid:1) b DB Π ∗ + Π b DB (cid:0) i D Π ∗ A ∗ + [ ̺ ∗ , D ]Π ∗ (cid:1) . (3.4)Equality (3.2) follows from (3.3), (3.4) and B̺ ∗ = ̺B . (cid:3) The sufficiency of the proved above equalities Π xt = Π tx and S xt = S tx for the compatibility of (2.2) and (2.8) is not self-evident but may be provedin a way similar to the proof of [16, Theorem 6.1]. Remark 3.2
We note that relations (3.1) and (3.2) may be considered asconservation laws for Π( x, t ) . .2 Rational extensions According to Propositions 2.1 and 2.2, in the case ̺ ≡ , σ ( A ) = { } ( σ stands for spectrum) , (3.5)the transformed potentials e ̺ are rational matrix functions ( rational exten-sions in the terminology of [6]). More precisely, if ̺ ≡ A is nilpotent,formulas (2.2), (2.3), (2.5), (2.7), (2.8), (2.10) and (2.11) imply the followingproposition. Proposition 3.3
Let (3.5) hold. Then Π( x ) and S ( x ) are matrix polynomi-als with respect to x , Π( x, t ) and S ( x, t ) are matrix polynomials with respectto x and t ; e ̺ ( x ) = p ( x ) / det S ( x ) and e ̺ ( x, t ) = p ( x, t ) / det S ( x, t ) , where p ( x ) is a matrix polynomial with respect to x , p ( x, t ) is a matrix polynomial withrespect to x and t , det S ( x ) is a polynomial with respect to x and det S ( x, t ) is a polynomial with respect to x and t . Moreover, for Darboux matrices w A we have w A ( x, z ) = P ( x, z ) / det S ( x ) and w A ( x, t, z ) = P ( x, t, z ) / det S ( x, t ) ,where P ( x, z ) ( P ( x, t, z )) is a matrix polynomial with respect to x and /z ( x, t and /z ) . Thus, the transformed fundamental solutions e w ( x, z ) andwave functions e w ( x, t, z ) are expressed via matrix polynomials and exponentsof diagonal matrices: e w ( x, z ) = w A ( x, z )e i zxD w A (0 , z ) − , e w ( x, t, z ) = w A ( x, t, z )e i z ( xD + t b D ) w A (0 , , z ) − . (3.6) Proof.
Since ̺ ≡
0, the first equation in (2.2) yields that the i th column f i of Π is given by f i ( x ) = e − i d i xA f i (0) = P n − k =0 1 k ! ( − i d i xA ) k f i (0), and so Π( x )is a matrix polynomial. (Here we used the equality A n = 0, which holds forall n × n nilpotent matrices.) Hence, the second equation in (2.2) yields that S ( x ) is a matrix polynomial. In the same way we prove the polynomial formof Π( x, t ) and S ( x, t ), and the required properties of e ̺ follow. Accordingto formula (1.84) from [16, p. 24] we have w A ( z ) Bw A ( z ) ∗ = B and, inparticular, the equalities w A (0 , z ) − = Bw A (0 , z ) ∗ B, w A (0 , , z ) − = Bw A (0 , , z ) ∗ B are valid. Finally, the representations of w A ( x, z ) and w A ( x, t, z ) follow fromthe definition (2.3), properties of Π and S and the expansion ( A − zI n ) − = − z − P n − k =0 ( z − A ) k . (cid:3) emark 3.4 System (1.5) includes important subclasses: self-adjoint andskew-self-adjoint Dirac-type systems ( which are also called Zakharov-Shabator AKNS systems ) . Namely, putting D = j = (cid:20) I m − I m (cid:21) ( m + m = m ) , V = (cid:20) vv ∗ (cid:21) (3.7) and B = j , we rewrite (1.5) , (1.6) in the form of the self-adjoint Dirac-typesystem u x = i( zj + jV ) u, v = 2i ̺ , (3.8) where ̺ is the upper right m × m block of ̺ . Assuming (3.7) and B = I m ,we rewrite (1.5) , (1.6) in the form of the skew-self-adjoint Dirac-type system u x = (i zj + jV ) u, v = − ̺ . (3.9) The notion of bispectrality was introduced in [2] (see also [7] and referencestherein). Matrix bispectrality for system (1.5) was studied in [17]. Namely,it was shown that for e ̺ constructed in Proposition 3.3 and, correspondingly,for the solution W ( x, z ) = w A ( x, z )e i zxD of system W x = (i zD − [ D, e ̺ ]) W, (3.10)there is a non-degenerate matrix linear differential operator B with respectto the variable z such that B ( z ) W = 0. It was suggested in [7] (and insome earlier works by F. Gr¨unbaum and coauthors) that the bispectralityrequirement W B ( z ) = Θ( x ) W is more meaningful than the requirement B ( z ) W = Θ( x ) W . Unfortunately, the approach from [17] does not workproperly for that case. Therefore, we should consider the system W x = (i zD − [ D, e ̺ ]) W − i zW D (3.11)instead of system (1.5). In view of (2.4), the matrix function w A satisfiesthis system in the case ̺ ≡
0. Thus, we can deal in the same way as in [5],where Dirac systems were treated. Namely, if σ ( A ) is concentrated at onepoint λ ∈ C we write down the resolvent ( A − zI n ) − in the form( A − zI n ) − = − n X k =1 ( z − λ ) − k ( A − λI n ) k − . (3.12)By virtue of (2.3) and (3.12) the next proposition is immediate.9 roposition 3.5 Assume that ̺ ≡ , the spectrum of A is concentrated atsome point λ ∈ C and e ̺ is given by (2.5) . Then system (3.11) is bispectralin the sense of [7]. Indeed, it is easy to choose some coefficients c s so that the operator B ( z ) = P n +1 s =1 c s ( z − λ ) s ∂ s ∂z s is non-degenerate and satisfies the equalities B ( z )( z − λ ) − k = 0 (1 ≤ k ≤ n ) . (3.13)According to formula (2.4) and identity ̺ ≡
0, the matrix function W = w A ( x, z ) satisfies (3.11). Therefore, definition (2.3) of w A and equalities(3.12) and (3.13) imply B ( z ) W = 0.If we want to consider system (3.10), where ̺ ≡ σ ( A ) = { λ } , λ ∈ R , we rewrite the first relation in (2.2) in the form (cid:0) Π( x )e i λxD (cid:1) x = − i( A − λI m ) (cid:0) Π( x )e i λxD (cid:1) D. (3.14)Hence, taking into account the fact that σ ( A ) is concentrated at λ , we see thatΠ( x )e i λxD = P ( x, λ ) = P ( x ), where P ( x ) is a matrix polynomial with respectto x . Substituting Π( x ) = P ( x )e − i λxD into (2.2) and using the equalities D = D ∗ , λ = λ , we obtain S x = P ( x )e − i λxD DB e i λxD P ( x ) ∗ = P ( x ) DBP ( x ) ∗ , i . e ., S ( x ) = e P ( x ) , (3.15)where e P ( x ) a matrix polynomial with respect to x . Introduce an operator B = diag {B , . . . , B m } such that B k has the form f B k = N X ℓ =1 c ℓk ( z ) ∂ ℓ ∂z ℓ (cid:0) ( z − λ ) m f (cid:1) , N ∈ N , (3.16)and notice that the equality N X ℓ =1 c ℓk ( z ) ∂ ℓ ∂z ℓ (cid:0) ( z − λ ) m w A ( x, z )e i zxD e k (cid:1) = 0 , (3.17) e k := { δ ik } mi =1 (1 ≤ k ≤ m )is equivalent to the relation N X ℓ =1 c ℓk ( z ) (cid:18) ∂∂z + i d k x (cid:19) ℓ × (cid:16) ( z − λ ) m e i λxD (cid:16) I m − i BP ( x ) ∗ e P ( x ) − ( A − zI n ) − P ( x ) (cid:17) e − i λxD e k (cid:17) = 0 , N X ℓ =1 c ℓk ( z ) (cid:18) ∂∂z + i d k x (cid:19) ℓ (3.18) × (cid:16) (det e P ( x ))( z − λ ) m (cid:16) I m − i BP ( x ) ∗ e P ( x ) − ( A − zI n ) − P ( x ) (cid:17) e k (cid:17) = 0 . Open Problem.
Are there some cases, where matrices D , A , S (0) andΠ(0) generate nontrivial e ̺ such that W B ( z ) = 0 for the solution W ( x, z ) = w A ( x, z )e i zxD of (3.10) and for B 6 = 0 of the form given above? In otherwords, are there D , A , S (0), Π(0) and { c lk } satisfying (3.18)? Acknowledgement.
The research was supported by the Austrian Sci-ence Fund (FWF) under Grant No. P24301.
References [1] Cieslinski J L 2009 Algebraic construction of the Darboux matrixrevisited
J. Phys. A :40 404003[2] Duistermaat J J and F.A. Gr¨unbaum 1986 Differential equations inthe spectral parameter Commun. Math. Phys.
J. Funct. Anal. :2 401[4] Gesztesy F and Teschl G 1996 On the double commutation method
Proc. Amer. Math. Soc. 124 :6 1831[5] Gohberg I, Kaashoek M A and Sakhnovich A L 1998 Pseudocanon-ical systems with rational Weyl functions: explicit formulas andapplications
J. Differential Equations :2 375[6] Grandati Y 2011 Solvable rational extensions of the isotonic oscil-lator
Ann. Physics :8 2074[7] Gr¨unbaum F A 2014 Some noncommutative matrix algebras arisingin the bispectral problem
SIGMA Symmetry Integrability Geom.Methods Appl. Paper 078118] Gu C H, Hu H and Zhou Z 2005
Darboux transformations in inte-grable systems. Theory and their applications to geometry (Mathe-matical Physics Studies vol 26) (Dordrecht: Springer)[9] Kostenko A, Sakhnovich A and Teschl G 2012 Commutation Meth-ods for Schr¨odinger Operators with Strongly Singular Potentials
Math. Nachr. :4 392[10] Matveev V B and Salle M A 1991
Darboux transformations andsolitons (Berlin: Springer)[11] Novikov S, Manakov S V, Pitaevskii L P and Zakharov V E 1984
Theory of solitons. The inverse scattering method , ContemporarySoviet Mathematics (New York: Consultants Bureau [Plenum])[12] Popovych R O, Kunzinger M and Ivanova N M 2008 Conservationlaws and potential symmetries of linear parabolic equations
ActaAppl. Math. :2 113[13] Sakhnovich A L 1994 Dressing procedure for solutions of nonlinearequations and the method of operator identities
Inverse Problems J. Math. Anal.Appl.
Mathematical Modelling ofNatural Phenomena :4 340[16] Sakhnovich A L, Sakhnovich L A and Roitberg I Ya 2013 Inverseproblems and nonlinear evolution equations. Solutions, Darboux ma-trices and Weyl–Titchmarsh functions (de Gruyter Studies in Math-ematics vol 47) (Berlin: De Gruyter)[17] Sakhnovich A L and Zubelli J P 2001 Bundle bispectrality for ma-trix differential equations
Integral Eguations Operator Theory
472 1218] Zakharov V E and Manakov S V 1975 Theory of resonance inter-action of wave packages in nonlinear medium.
Soviet Phys. JETP ,69