Integration of the Finite Toda Lattice with Complex-Valued Initial Data

Transcription

1 Integration of the Finite Toda Lattie with Complex-Valued Initial Data Aydin Huseynov* and Gusein Sh Guseinov** *Institute of Mathematis and Mehanis, Azerbaijan National Aademy of Sienes, AZ4 Baku, Azerbaijan ( **Department of Mathematis, Atilim University, 6836 Inek, Ankara, Turkey ( Abstrat: We show that the finite Toda lattie with omplex-valued initial data an be integrated by the inverse spetral method For this goal spetral data for omplex Jaobi matries are introdued and an inverse spetral problem from the spetral data is solved The time evolution of the spetral data for the Jaobi matrix assoiated with the solution of the Toda lattie is omputed Using the solution of the inverse spetral problem with respet to the time-dependent spetral data we reonstrut the time-dependent Jaobi matrix and hene the desired solution of the finite Toda lattie Keywords: Toda lattie, Jaobi matrix, differene equation, spetral data, inverse spetral problem INTRODUCTION There are three kinds of the Toda latties; finite, semiinfinite, and doubly-infinite Toda latties There is a huge number of papers devoted to the investigation of the Toda latties and their various generalizations, from whih we indiate here only [ 2] For some tehniques of investigation of nonlinear disrete equations (differene equations) see [3 5] In the present paper, we deal with the finite Toda lattie subjet to the omplex-valued initial onditions The (open) finite Toda lattie is a Hamiltonian system whih desribes the motion of N partiles moving in a straight line, with exponential interations Adjaent partiles are onneted by strings Let the positions of the partiles at time t be q (t), q (t),, q N (t), q n = q n (t) is the displaement at the moment t of the n-th partile from its equilibrium position We assume that eah partile has mass The momentum of the n-th partile at time t is therefore p n = q n The Hamiltonian funtion is defined to be H = 2 N n= The Hamiltonian system q n = H, p n N 2 p 2 n + n= e qn qn+ ṗ n = H q n is q n = p n, n =,,, N, ṗ = e q q, ṗ n = e qn qn e qn qn+, n =, 2,, N 2, ṗ N = e q N 2 q N Let us set a n = 2 e(qn qn+)/2, n =,,, N 2, b n = 2 p n, n =,,, N Then the above system an be written in the form ȧ n = a n (b n+ b n ), ḃ n = 2(a 2 n a 2 n ), () n =,,, N, with the boundary onditions a = a N = (2) If we define the N N matries J and A by b a a b a a b 2 J =, (3) b N 3 a N 3 a N 3 b N 2 a N 2 a N 2 b N a a a a A =, (4) a N 3 a N 3 a N 2 a N 2 then the system () with the boundary onditions (2) is equivalent to the Lax equation d J = [J, A] = JA AJ (5) The system (), (2) is onsidered subjet to the initial onditions a n () = a n, b n () = b n, n =,,, N, (6) a n, b n are given omplex numbers suh that a n (n =,,, N 2), a N =

2 Existene of the Lax representation (5) allows to solve the nonlinear initial-boundary value problem (), (2), (6) by means of the inverse spetral problem method 2 SPECTRAL DATA FOR COMPLEX JACOBI MATRICES AND INVERSE SPECTRAL PROBLEM An N N omplex Jaobi matrix is a matrix of the form b a a b a a b 2 J =, (7) b N 3 a N 3 a N 3 b N 2 a N 2 a N 2 b N for eah n, a n and b n are arbitrary omplex numbers suh that a n is different from zero: a n, b n C, a n (8) If the entries of the matrix J are real, then the eigenvalues of J are real and simple However, if the matrix J is omplex, then its eigenvalues may be nonreal and multiple Let R(λ) = (J λi) be the resolvent of the matrix J (by I we denote the identity matrix of needed dimension) and e be the N-dimensional vetor with the omponents,,, The rational funtion w(λ) = ((R(λ)e, e ) = ( (λi J) ) e, e (9) we all the resolvent funtion of the matrix J Denote by λ,, λ p all the distint eigenvalues of the matrix J and by m,, m p their multipliities, respetively, as the roots of the harateristi polynomial det(j λi), so that p N and m + +m p = N We an deompose the rational funtion w(λ) into partial frations to get w(λ) = p m k β kj (λ λ k ) j, () β kj are some omplex numbers uniquely determined by the matrix J Definition The olletion of the quantities {λ k, β kj (j =,, m k, k =,, p)}, () we all the spetral data of the matrix J For eah k {,, p} the (finite) sequene {β k,, β kmk } we all the normalizing hain (of the matrix J) assoiated with the eigenvalue λ k Let us indiate a onvenient way for omputation of the spetral data of omplex Jaobi matries For this purpose we should desribe the resolvent funtion w(λ) of the Jaobi matrix Given a Jaobi matrix J of the form (7) with the entries (8), onsider the eigenvalue problem Jy = λy for a olumn vetor y = {y n } N n=, that is equivalent to the seond order linear differene equation a n y n + b n y n + a n y n+ = λy n, (2) n {,,, N }, a = a N =, for {y n } N n=, with the boundary onditions y = y N = Denote by {P n (λ)} N n= and {Q n (λ)} N n= the solutions of equation (2) satisfying the initial onditions P (λ) =, P (λ) = ; (3) Q (λ) =, Q (λ) = (4) For eah n, P n (λ) is a polynomial of degree n and is alled a polynomial of first kind and Q n (λ) is a polynomial of degree n and is known as a polynomial of seond kind The equality det (J λi) = ( ) N a a a N 2 P N (λ) (5) holds so that the eigenvalues of the matrix J oinide with the zeros of the polynomial P N (λ) In [6] it is shown that the entries R nm (λ) of the matrix R(λ) = (J λi) (resolvent of J) are of the form { Pn (λ)[q R nm (λ) = m (λ) + M(λ)P m (λ)], n m, P m (λ)[q n (λ) + M(λ)P n (λ)], m n, n, m {,,, N } and M(λ) = Q N (λ) P N (λ) Therefore aording to (9) and using initial onditions (3), (4), we get w(λ) = R (λ) = M(λ) = Q N(λ) P N (λ) (6) Denote by λ,, λ p all the distint roots of the polynomial P N (λ) (whih oinide by (5) with the eigenvalues of the matrix J) and by m,, m p their multipliities, respetively: P N (λ) = (λ λ ) m (λ λ p ) mp, is a onstant We have p N and m + + m p = N Therefore by (6) deomposition () an be obtained by rewriting the rational funtion Q N (λ)/p N (λ) as the sum of partial frations We an also get another onvenient representation for the resolvent funtion as follows If we delete the first row and the first olumn of the matrix J given in (7), then we get the new matrix b () b () b () 2 J () =, b () N 4 a() N 4 N 4 b() N 3 a() N 3 N 3 b() N 2 n = a n+, n {,,, N 3}, b () n = b n+, n {,,, N 2} The matrix J () is alled the first trunated matrix (with respet to the matrix J) Theorem 2 The equality holds w(λ) = det(j () λi) det(j λi) Proof Let us denote the polynomials of the first and the seond kinds, orresponding to the matrix J (), by P n () (λ) and Q () n (λ), respetively It is easily seen that P () n (λ) = a Q n+ (λ), (7)

3 Q () n (λ) = {(λ b )Q n+ (λ) P n+ (λ)}, a n {,,, N } Indeed, both sides of eah of these equalities are solutions of the same differene equation n y n + b () n y n + n y n+ = λy n, n {,,, N 2}, N 2 =, and the sides oinide for n = and n = Therefore the equalities hold by the uniqueness theorem for solutions Consequently, taking into aount (5) for the matrix J () instead of J and using (7), we find det(j () λi) = ( ) N a() N 3 P () N (λ) = ( ) N a a N 2 a Q N (λ) Comparing this with (5), we get Q N (λ) () P N (λ) = det(j λi) det(j λi) so that the statement of the theorem follows by (6) The inverse problem is stated as follows: (i) To see if it is possible to reonstrut the matrix J, given its spetral data () If it is possible, to desribe the reonstrution proedure (ii) To find the neessary and suffiient onditions for a given olletion () to be spetral data for some matrix J of the form (7) with entries belonging to the lass (8) This problem was solved reently by the author in [6] and the following results were established Given the olletion () define the numbers p m k ( ) l s l = j ) ( is a binomial oeffiient and we put l β kj λ l j+ k, (8) ( l j j ) = if j > l Using the numbers s l let us introdue the determinants s s s n s s 2 s n+ D n =, n =,,, N (9) s n s n+ s 2n Theorem 3 Let an arbitrary olletion () of omplex numbers be given, λ,, λ p ( p N) are distint, m k N, and m + + m p = N In order for this olletion to be the spetral data for some Jaobi matrix J of the form (7) with entries belonging to the lass (8), it is neessary and suffiient that the following two onditions be satisfied: (i) p k= β k = ; (ii) D n, for n {, 2,, N }, and D N =, D n is the determinant defined by (9), (8) Under the onditions of Theorem 3 the entries a n and b n of the matrix J for whih the olletion () is spetral data, are reovered by the formulas a n = ± (D n D n+ ) /2 D n, (2) n {,,, N 2}, D =, b n = n D n n D n, (2) n {,,, N }, =, = s, D n is defined by (9) and (8), and n is the determinant obtained from the determinant D n by replaing in D n the last olumn by the olumn with the omponents s n+, s n+2,, s 2n+ Remark 4 It follows from the above solution of the inverse problem that the matrix (7) is not uniquely restored from the spetral data This is linked with the fat that the a n are determined from (2) uniquely up to a sign To ensure that the inverse problem is uniquely solvable, we have to speify additionally a sequene of signs + and Namely, let {σ, σ 2,, σ N } be a given finite sequene, for eah n {, 2,, N } the σ n is + or We have 2 N suh different sequenes Now to determine a n uniquely from (2) for n {,,, N 2} we an hoose the sign σ n when extrating the square root In this way we get preisely 2 N distint Jaobi matries possessing the same spetral data The inverse problem is solved uniquely from the data onsisting of the spetral data and a sequene {σ, σ 2,, σ N } of signs + and 3 THE TIME EVOLUTION OF THE SPECTRAL DATA AND SOLVING PROCEDURE FOR THE TODA LATTICE Theorem 5 Let {a n (t), b n (t)} be a solution of (), (2) and J = J(t) be the Jaobi matrix defined by this solution aording to (3) Then there exists an ivertible N N matrix-funtion X(t) suh that X (t)j(t)x(t) = J() for all t (22) Proof Let A(t) be defined aording to (4) Then the Lax equation (5) holds Denote by X = X(t) the matrix solution of the initial value problem Ẋ(t) = A(t)X(t), (23) X() = I (24) Suh solution X(t) exists and is unique From the Liouville formula ( det X(t) = exp ) tra(τ)dτ it follows that det X(t) so that the matrix X(t) is invertible Using the formula dx = X dx X, equation (23), and the Lax equation (5), we have d ( X JX ) = dx dj JX + X X + X J dx dj = X AXX JX + X X X JAX ( ) dj = X [JA AJ] X = Therefore X (t)j(t)x(t) does not depend on t and by initial ondition (24) we get (22) The proof is omplete It follows from (22) that for any t the harateristi polynomials of the matries J(t) and J() oinide Therefore we arrive at the following statement Corollary 6 The eigenvalues of the matrix J(t), as well as their multipliities, do not depend on t

4 Remark 7 From the skew-symmetry A T = A of the matrix A(t) defined by (4) it follows that for the solution X(t) of (23), (24) we have X (t) = X T (t), T stands for the matrix transpose Indeed, by equation (23), d ( X T X ) = dxt X + dx XT = X T A T X X T AX = X T AX X T AX = Therefore X T (t)x(t) does not depend on t and by initial ondition (24) we get X T (t)x(t) = X T ()X() = I Theorem 8 The dynamis of the solution of the Toda equations (), (2) with the arbitrary omplex initial data (6) orresponds to the following evolution of the resolvent funtion w(λ; t) of the matrix J(t) : w(λ; t) = e2λt w(λ; ) 2 S(u)e 2λ(t u) du, (25) = w(z; )e 2zt dz (26) Γ in whih Γ is any losed ontour that enloses all the eigenvalues of the matrix J() By Corollary 6 the eigenvalues of the matrix J(t) and their multipliities do not depend on t However, the normalazing numbers of the matrix J(t) will depend on t The following theorem desribes this dependene Theorem 9 For the normalizing numbers β kj (t) (j =,, m k ; k =,, p) of the matrix J(t) the following evolution holds: β kj (t) = e2λ kt = p k= m k s=j m k e 2λ kt j= β ks () (2t)s j (s j)!, (27) β kj () (2t)j (j )! (28) Proof Aording to (), we have p m k β kj (t) w(λ; t) = (λ λ k ) j (29) We will dedue (27), (28) from (25), (26) First we ompute the funtion in (26) We use the known formula f (n) (z) = f(ζ) dζ, n! γ (ζ z) n+ γ is the boundary of the disk D = {ζ C : ζ z δ} with δ > and f(z) is analyti in a domain of C ontaining D For any k {,, p} we set Γ k = {z C : z λ k = δ} and hoose δ > so small that Γ,, Γ p are disjoint and all inside Γ Then we have, by (26) and (29), Further, = Γ p m k β kj () Γ e 2zt (z λ k ) j dz e 2zt (z λ k ) j dz = e Γk 2zt (z λ k ) j dz = (2t)j (j )! e2λ kt Therefore for we have the formula (28) Now we rewrite (25) in the form, using (29), p m k β kj (t) (λ λ k ) j = e2λt 2 p m k β kj () (λ λ k ) j S(u)e 2λ(t u) du (3) Take any l {,, p} and s {,, m l }, multiply (3) by () (λ λ l ) s and integrate both sides over Γ l to get m l dλ β lj (t) j= Γ l (λ λ l ) j s+ = m l β lj () j= Γ l e 2λt dλ (λ λ l ) j s+ We have used the fat that the integral of the seond term of the right side of (3) gave zero beause of the analytiity of that term with respet to λ Hene beause Γ l β ls (t) = e2λ lt Γ l m l j=s dλ (λ λ l ) j s+ = e 2λt dλ (λ λ l ) j s+ = Now, (3) means that (27) holds β lj () (2t)j s (j s)! { if j = s, if j s, if j < s, (2t) j s (j s)! e2λ lt if j s (3) We get the following proedure for solving the problem (), (2), (6) From the initial data {a n, b n} we form the Jaobi matrix J() and ompute the spetral data {λ k, β kj () (j =,, m k, k =,, p)} (32) of the matrix J() Then we make up the time-dependent spetral data {λ k, β kj (t) (j =,, m k, k =,, p)}, (33) β kj (t) is obtained from (32) by (27), (28) Finally, solving the inverse spetral problem with respet to (33) we onstrut a Jaobi matrix J(t) The entries {a n (t), b n (t)} of the matrix J(t) give a solution of problem (), (2), (6) 4 EXAMPLES In the ase N = 2, equations (), (2) beome ȧ = a (b b ), ḃ = 2(a 2 a 2 ), ȧ = a (b 2 b ), ḃ = 2(a 2 a 2 ), a = a = This system in turn is equivalent to ȧ = a (b b ), ḃ = 2a 2, ḃ = 2a 2 (34) with respet to a, b, and b Let us take initial onditions in the form a () = i, b () = λ +, b () = λ,

5 λ and are arbitrary omplex numbers and We have [ ] λ + i J() = i λ Hene det(j() λi) = (λ λ ) 2, w(λ; ) = det(j () () λi) det(j() λi) = λ λ (λ λ ) 2 = + λ λ (λ λ ) 2 Therefore λ is a double eigenvalue of J() and the spetral data of J() is the olletion {λ, β () =, β 2 () = } By (27), (28) we find β (t) =, β 2 (t) = + 2t Solving the inverse spetral problem we get λ + J(t) = + 2t i + 2t Hene the desired solution is a (t) = i + 2t λ i + 2t, b (t) = λ + b (t) = λ + 2t + 2t + 2t, 2 Now we take for system (34) initial onditions in the form a () = ( ), b () =, b () =, is an arbitrary omplex number suh that and We have [ ] ( ) J() = ( ) Hene det(j() λi) = λ(λ ), w(λ; ) = det(j () () λi) det(j() λi) = λ λ(λ ) = λ + λ Therefore λ = and λ = are the eigenvalues of J() and the spetral data of J() is the olletion {λ =, λ 2 =, β () =, β 2 () = } By (27), (28) we find β (t) = + ( )e 2t, β ( )e2t 2(t) = + ( )e 2t Solving the inverse spetral problem we get that the desired solution is a (t) = et ( ) + ( )e 2t, b ( )e2t (t) = + ( )e 2t, b (t) = + ( )e 2t REFERENCES [] M Toda, Waves in nonlinear lattie, Progr Theoret Phys Suppl 45 (97) 74 2 [2] M Henon, Integrals of the Toda lattie, Phys Rev B 9 (974) [3] H Flashka, The Toda lattie, I, Phys Rev B 9 (974) [4] Flashka H The Toda lattie, II, Progr Theoret Phys 974;5:73 76 [5] S V Manakov, Complete integrability and stohastization in disrete dynami media, Zh Exper Teor Fiz 67 (974) English translation in Soviet Physis- JETP 4 (975) [6] M Ka and P van Moerbeke, On an expliitly soluble system of nonlinear differential equations related to ertain Toda latties, Adv Math 6 (975) 6 69 [7] J Moser, Three integrable Hamiltonian systems onneted with isospetral deformations, Adv Math 6 (975) [8] M Toda, Theory of nonlinear latties, Springer, New York, 98 [9] Yu M Berezanskii, The integration of semi-infinite Toda hain by means of inverse spetral problem, Rep Math Phys 24 (986) 2 47 [] G Teshl, Jaobi operators and ompletely integrable nonlinear latties, Math surveys monographs, vol 72 Amer Math So Providene (RI); 2 [] A I Aptekarev and A Branquinbo, Padé approximants and omplex high order Toda latties, J Comput Appl Math 55 (23) [2] Yu M Berezanskii and A A Mokhonko, Integration of some differential-differene nonlinear equations using the spetral theory of normal blok Jaobi matries, Funtional Anal Appl 42 (28) 8 [3] A Huseynov, On solutions of a nonlinear boundary value problem for differene equations, Transations Natl Aad Si Azerb Ser Phys-Teh Math Si 28(4) (28) 43 5 [4] H Bereketoglu and A Huseynov, Boundary value problems for nonlinear seond-order differene equations with impulse, Appl Math Meh (English Ed) 3 (29) [5] A Huseynov, Positive solutions for nonlinear seondorder differerne equations with impulse, Dynamis of Continuous, Disrete and Impulsive Systems, Series A: Mathematial Analysis 7 (2) 5 32 [6] G Sh Guseinov, Inverse spetral problems for tridiagonal N by N omplex Hamiltonians, Symmetry, Integrability and Geometry: Methods and Appliations (SIGMA), 5 (29), Paper 8, 28 pages