STRUCTURE OF BINARY DARBOUX-TYPE TRANSFORMATIONS FOR HERMITIAN ADJOINT DIFFERENTIAL OPERATORS

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1 Ukrainian Mathematical Journal, Vol. 56, No. 2, 2004 SRUCURE OF BINARY DARBOUX-YPE RANSFORMAIONS FOR HERMIIAN ADJOIN DIFFERENIAL OPERAORS A. K. Prykarpats kyi 1 and V. H. Samoilenko 2 UDC For Hermitian adjoint differential operators, we consider the structure of Darboux Bäcklundtype transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux Bäcklundtype transformations. 1. Definition and Properties of Hermitian Adjoint Operators Consider the differential operator L : = n i= 1 a ( x, = i i x, (1 where a i S ( R ; End C N, i = 1, n, N Z +. Operator (1 is formally called -Hermitian adjoint if L 1 = L (or L = L 1 (2 with respect to an ordinary nonsingular bilinear form (, : = dx Sp( ( R on the product of spaces H H, where H : = L 2 ( R ; Hom ( C k ; C N, k, N Z +. Using condition (2, we easily establish that the operator l : = L 1 is formally symmetric (Hermitian because l = 1 L = 1 L 1 = L 1 = l. (3 In order that the operators l and l be well defined, it is necessary to consider the action of these operators on the corresponding spaces DH and DH, where 1 AGH University of Science and echnology, Kraków, Poland; Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. 2 Shevchenko Kiev University, Kiev. ranslated from Ukrains kyi Matematychnyi Zhurnal, Vol. 56, No. 2, pp , February, Original article submitted August 5, /04/ Plenum Publishing Corporation

2 SRUCURE OF BINARY DARBOUX-YPE RANSFORMAIONS FOR HERMIIAN ADJOIN DIFFERENIAL OPERAORS 337 k N DH : = { ϕ/ x: ϕ W2 1 ( R; Hom( C, C } and the closure is taken in the norm of the space H. hus, expression (1 can be regarded as the composition of the symmetric differential symbol and the operation of differentiation, i.e., L : = l, where, according to (3, l = l, i.e., [ n / 2] i i i= 0 l = u ( x i (4 for some functions u i S ( R ; End C N, i = 0, [ n / 2], n Z Adjointness Condition in D H D H We consider the operator expression ˆl : = ( / t L 1 defined on D H = C 1 ( R t ; DH and find a condition under which there exists the adjoint expression l ˆ* = ˆl with respect to an ordinary nonsingular bilinear form on D H D H, i.e., ( lϕ ˆ x, ψx = ( ϕx, l ˆ* ψx = ( ϕx, ˆl ψx (5 for all ( ϕ x, ψ x D H D H. Writing out condition (5 and using the Lagrange identity, we get ϕ t lϕ x, ψ x = ϕx, ψ ˆ* t l ψx + Sp Z[ ϕψ, ] + Sp( ϕt ψx ϕ xψt x = ϕx, ψt l ψx + Sp Z[ ϕψ, ] + Sp ϕt ( ψx + ϕ ψxt, Sp( ϕ ψ t x x = ϕx, ψt l ψx + Sp ( ϕ ψ x t Sp Z L [ ϕψ,, ] (6 x where Z [ ϕ, ψ ] and Z L [ ϕ, ψ ] are certain matrix bilinear forms on H H. Integrating identity (6 with respect to x R, one can easily establish that if there exists a matrix k Ω C ( R R ; End C such that x x Ω = ϕ ψx, t Ω = Z L [ ϕ, ψ ] (7 for any ( ϕ, ψ H H and if the conditions Sp ( Ω 2 Ωt, Sp Ω t S ( R ; C (8 are satisfied with respect to x R uniformly in t R, then the expression

3 338 A. K. PRYKARPAS KYI AND V. H. SAMOILENKO ˆl : = t l is a symmetric operator on the space D H D H for all t R. his is obviously equivalent to the following (1 differential-geometric condition: he matrix 1-form ω [ ϕ, ψ ] = ϕ ψ x d x + Z L [ ϕ, ψ ] d t is closed in R 2, i.e., (1 there exists a matrix function Ω [ ϕ, ψ ] such that ω [ ϕ, ψ ] = d Ω [ ϕ, ψ ] for all pairs ( ϕ, ψ H H. hus, we can formulate the following statement: Proposition 1. Let a formally -Hermitian operator of the form (1 be given on the space H. he operator ˆl = ( / t L 1 is a symmetric operator on the space D H with respect to a bilinear form on D H D H if conditions (7 and (8 are satisfied for all pairs ( ϕ, ψ D H D H and if the matrix 1- (1 form ω [ ϕ, ψ ] is closed on R 2. Note that conditions (7 and (8 are sufficient but not necessary, which follows from the construction of the k matrix Ω C ( Rx Rt; End C. Now assume that the set of functions H depends on the parameter t R as follows: where ϕ t=0 = ϕ H. ϕ t lϕ x : = 0 = : ϕ t L ϕ, (9 hen, using condition (5, one can easily establish that, for all ϕ H, we have ψ t lψ x = 0 = ψ t L ψ, (10 i.e., the spaces H and H are evolutionarily consistent with respect to the parameter t R. Furthermore, according to equalities (9 and (10, the spaces H and H canonically coincide, i.e., H H. his condition proves to be very useful for the subsequent analysis of the class of operators (1 and their informal evolution extensions (9. 3. Darboux Bäcklund ransformations and heir Structure Consider a -Hermitian adjoint differential operator L of the form (1 that acts in the space H. According to Proposition 1, a condition for the existence for the adjoint operator ˆl : = / t l, where l : = L, is the existence of a matrix k Ω ( R R ; End C such that, for all pairs (, * ϕ ψ H H, the conditions C x t x Ω = ϕ ψ x, Ω t = [, Z L ϕ ψ ], Sp( Ω 2 Ω t, Sp Ω t S ( R ; C are satisfied and the corresponding matrix 1-form ( ω 1 [ ϕ, ψ ] is closed on R 2. Assume that the last condition is satisfied and the space of the pairs of functions (, * ϕ ψ H H is evolutionarily consistent, i.e., ϕ t Lϕ = 0 = ψ L ψ. t

4 SRUCURE OF BINARY DARBOUX-YPE RANSFORMAIONS FOR HERMIIAN ADJOIN DIFFERENIAL OPERAORS 339 Now assume that the spaces H and H are isomorphic, i.e., there exists an invertible operator ˆΩ : H H whose action on a fixed pair of functions ( ϕ, ψ H H is defined by a Darboux Bäcklund-type transformation [1 3] as follows: where it is assumed that the matrix ϕ = ˆ ( ϕ: = ϕ( Ω Ω Ω0, ˆ ψ = Ω ( ψ: = ( ψx Ω Ω0, (11 Ω = Ω [ ϕ, ψ ] = 1 ( ( 1 ω [ ϕ, ψ] ( 1 + Ω 0 : = ω [ ϕ, ψ] + Ω 0, (12 P P0 k together with a constant matrix Ω 0 End C, is invertible and P0 = ( x 0, t 0 R 2 and P = ( x, t R 2 are arbitrary points in R 2. Relations (11 and (12 can be used for the construction of the isomorphisms Ω ˆ : H and ˆΩ : H H in explicit form and their subsequent interpolation onto the spaces H and H, respectively. Namely, for a pair ( ϕ, ψ H H, we get ( ϕ : = ˆ ( ϕ = ϕ( Ω Ω Ω0 = ϕ( Ω Ω ( ω ( 1 [ ϕ, ψ ] = ϕ ϕ( Ω 1 Ω ( Ω 1 1( ω ( 1 [ ϕ, ψ] = 1 ( 0 ( ϕ Ω ( ω [, ψ] ϕ, 0 0 ψ : = Ω ˆ ( ψ = 1( 1 ψ x Ω Ω = ( 1 ψx Ω Ω ω ( [ ϕ, ψ] 0 ( ( ( (13 ( = ψx ψ ( xω ( ω [ ϕ, ψ] : = 1 ( ψx Ω ( ω [ ϕ, ] ψ ( As a result of the interpolation of relations (13 onto the entire spaces with respect to the variables ϕ H and ψ H, we obtain ˆΩ = 1 ( ϕ( Ω 0 ( ω [, ψ], ˆΩ = 1 ( ψx Ω ω [ ϕ, ]. ( In order to verify that isomorphisms (14 are consistent with the evolution in the spaces H and H, and H and, we show that there exists the corresponding matrix Ω C 1 ( 1 R x R 1 k t ; End C that satisfies conditions of the types (7 and (8. Indeed, by definition, we have i.e., ( ω 1 [ ϕ, ψ ] = ϕ ψ x dx + Z L[ ϕψ, ] dt = Ω 0 Ω 1 ( d Ω Ω 1 Ω 0 = Ω 0 ( d Ω 1 1 Ω 0 = d ( Ω 0 Ω Ω0 : = d Ω[ ϕψ,, ] (15

5 340 A. K. PRYKARPAS KYI AND V. H. SAMOILENKO Ω[ ϕψ, ] : = Ω 0 Ω 1 [ ϕ, ψ ] Ω 0 = 1 ( 1 ω [ ϕ, ψ ] + Ω 0, (16 where Ω0 : = Ω 0, which follows from the existence of the limit as P P k Since the matrix Ω C ( Rx Rt; End C is assumed to be invertible, it follows from relation (16 that Ω is also an invertible matrix. By virtue of the fact that the pair of spaces H H is mapped isomorphically into the pair of spaces by mappings (14, this mapping is associated with a certain mapping of the operator l : D H D H into another operator l : D DH, which, according to (15, satisfies the identity ( ( ϕ / t l ϕ x, ψ x = ϕ, / x ψ t l ψx (17 for all ( ϕψ, H H of the type (11. As a result, it follows from relation (17 and the corresponding commutative diagram Ωˆ H / t L / t L H Ωˆ that / t L = ˆ ˆ Ω Ω Ω ˆ LΩ ˆ t 1 1, whence ˆ ˆ L = ˆLˆ ΩΩ t + Ω Ω 1 = L +[ ˆ, L] ˆ 1 Ω Ω + Ωˆ ˆ tω 1. (18 Moreover, it is easy to see that Ord L = Ord L. Since the operator L : H H is formally -Hermitian adjoint, it is necessary to verify that, under the isomorphism Ωˆ : H, operator (18 is also -Hermitian adjoint, L = L, i.e., the operator l : L is symmetric. he last property can be verified by simple calculations [2, 5]; these calculations are rather cumbersome and, therefore, we do not present them here. hus, we have proved the following statement: Proposition 2. Suppose that the conditions Sp( Ω 2 Ω t, Sp Ω t S ( R ; C are satisfied uniformly in t R. hen transformations (13 and (14 are the Darboux Bäcklund transformations for the corresponding Hermitian adjoint operators (1 and (18 defining the isomorphism of the spaces H and H. 4. Consistent Zakharov Shabat Pairs Now consider two (with respect to the variables ( t, y R 2 consistent evolutions of the spaces H and H according to the following equations for a pair of functions ( ϕ, ψ H H :

6 SRUCURE OF BINARY DARBOUX-YPE RANSFORMAIONS FOR HERMIIAN ADJOIN DIFFERENIAL OPERAORS 341 ϕ / t lϕ = 0, ψ / t lψ = 0, ϕ / y m ϕ = 0, ψ / y m ψ = 0, (19 where, by assumption, the operators l = l and m = m belong to class (5. Using relations (13 and transformation (18 for the corresponding operators L : = l and M : = m, we conclude that a system of equations of the type (19 for the pair of Darboux-transformed functions ( ϕψ, is also consistent, i.e., ϕ / t L ϕ = 0, ψ / t M ψ = 0, ϕ / y L ϕ = 0, ψ / y M ψ = 0, where ϕ = ϕ( Ω 1 Ω0, ( ψ = ( ψ x Ω Ω 0, and ω 1 [ ϕ, ψ] = ϕ ψx dx + ZL[ ϕ, ψ] dt ZM[ ϕ, ψ] dy = dω[ ϕ, ψ]. his means that the Darboux Bäcklund-transformed coefficients of the original operators L and M satisfy the same Zakharov Shabat consistency condition, which, as is known [3], is equivalent to a certain system of nonlinear evolution-differential equations for their coefficient matrix functions. In other words, in the case of a -Hermitian adjoint pair of Zakharov Shabat operators, the Darboux Bäcklund transformation (11 can be efficiently used for the regular algebraic-analytic construction of a broad class of so-called soliton and rational solutions [1, 3, 4] of the corresponding evolution-differential systems of equations. REFERENCES 1. V. B. Matveev and M. I. Salle, Darboux Bäcklund ransformations and Applications, New Springer, York ( Ya. A. Prylarpats kyi, A. M. Samoilenko, and V. H. Samoilenko, Structure of binary transformations of Darboux type and their application to soliton theory, Ukr. Mat. Zh., 55, No. 12, ( V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Soliton heory [in Russian], Nauka, Moscow ( J. C. C. Nimmo, Darboux ransformations from Reductions of the KP-Hierarchy, Preprint, University of Glasgow, Glasgow ( A. M. Samoilenko and Ya. A. Prylarpats kyi, Algebraic-Analytic Aspects of Completely Integrable Dynamical Systems and heir Perturbations [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002.