объединенный ИНСТИТУТ ядерных исследований cut OQ6'0 ДУШ Е V.S.Gerdjikov, M.LIvanov

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1 cut OQ6'0 объединенный ИНСТИТУТ ядерных исследований ДУШ Е V.S.Gerdjikov, M.LIvanov THE QUADRATIC BUNDLE OF GENERAL FORM AND THE NONLINEAR EVOLUTION EQUATIONS. HIERARCHIES OF HAMILTONIAN STRUCTURES Submitted to "Bulgarian Journal of Physics" 1982

2 1. INTRODUCTION The application of the inverse scattering method (ISM) to quadratic bundles of special type allowed one the integrate some physically important nonlinear evolution equation (NLEE). The first in this direction was the paper /B/, where starting from a bundle of the type*: L,(A)<fr<» - [io s 4- + i- u? +AQ< 1 > -#]* (1) <ж,л)-0. QZ Z Q?> i 0 u t \ (i.d an exhaustive study of the massive Thirring model has been presented. In the papers /3,4/ it has been shown, that the bundle: L e (\)*<» E[ia s -± + AQ<»-A 2 ]0< 8 >(*,A) = O. Q (»=(j j\l.2) allows one to solve the modified nonlinear Schrodinger equation (NLS eq.): i«(lq 8 lq) - 0. p - <q*. (1.3) The complete integrability and the construction of the hierarchy of Hamiltonian structures 'W for the NLEE, related to (1.2), has been proved in' 7/. The considerations there have been based on the method of expansions over the "squared" solutions /7 * J0/. Another NLEE, related to (1.2) is the Mikhailov model: hut - 2ihihgh lx + m 8 h 1-0. h t -q x, h 2xt + 2 t h i t l e h Bx + ш 2 h g - 0, hg «p, a relativistically invariant field theory model, equivalent to the massive Thirring model'' 7 ''. In refs/ 1I ' 18/ the following bundles have been considered: "Here we have used characteristics variables; thus the bundle L, (A) equals the summ T(A) + X(A) of the bundles, introduced in ref/ 8/. 1

3 L s (\W (a),[i e r 8 A. + \Q ( 2 ) -(Л 8 +l)],fr (8) (x. Л) -0. (1.5) L 4 (A)0 {4) s [ia 8 +GA + /5)Q ) +(5A 2 +2iaV)]^(4) («.A)-0. (]. 6 ) They allow one to obtain the Lax representation for another variant of the NLS eq.: iq,+ q +2/8% 8 q _ijjl( q 2 q ) x -0. (1.7) Note, that both variants of the modified NLS equation are applicable in plasma physics' 1 *" 1 ". On the other hand, the standard ways of the ISM have been applied to the polynomial bundle of maximally general type 3, which in the 2x2 case has the form: ЬЛК)ф (Ъ) ш [iff. + 2 A k U k (x) (6) (x. A) = 0. (1.8) dx k=o The analysis, based on the study of the central extensions of Lie algebras (see refs/ 18 ' 1 " and the references therein) lead to explicitly Hamiltonian form of the NLEE. Unfortinately the corresponding Kirillov-Kostant syplectic form is degenerated. The natural solution of the problem consists, roughly speaking, in the following: one should somewhat restrict the form of the linear problem (1.8) so, that the phase space of the corresponding NLEE coincides with the orbit of the co-adjoint action in our algebra (more rigorously - see ref/ 18/ ). In particular, for the quadratic ina bundles this restriction, together with an appropriate choice of the gauge' 80 '' and the co-adjoint action, leads to: L(A)^r s[iff 3 A + Q o + хо 1 + г 0 -А г^(х,а)=0, Qj-f J 1 J. (1.9) Therefore, from general arguments it follows, that the NLEE, related to (1.9) possess a hierarchy of Hamiltonian structures, the corresponding symplectic forms Q m, ro =0, _+l, +2,... being nondegenerate. But generically both the NLEE and even the simplest of the forms 0. depend nonlocally on q j, Pj, r Q. As a conjecture, allowing one to obtain local NLEE we propose the following. It is well known, that the solution of the inver- 2

4 se scattering problem for a large number of linear problems L(A) is equivalent to the solution of a Riemann problem /l/. This last problem has an unique solution with fixed norm, i.e., with fixed value of the solution for Л-» «. Obviously the normalization of the Riemann problem should correlate with the asymptotics of the solutions LUty/ - 0 of the linear problem for К -m. Our conjecture consists in that one should require canonical normalisation for the Riemann problem. In particular, for the system (1.9) the corresponding Riemann problem is written down in ref. /el/, and the requirement of canonical normalization is fulfilled if: r o- -! g-*i p r (1.Ю) As we shall see below, the "restriction" (1.10) leads to local NLEE, and the simplest symplectic form 0 Q becomes canonical. The same conjecture, applied to the 2x2 polynomial bundles of power N > 3 also allows one to obtain local NLEE and simple 2-form fl 0 / /. In this case from the 4N initially independent elemets of L 6 (A) (1.8) there remain only 2N-2 independent ones, i.e., this conjecture has the sence of reduction for the NLEE / г я /. Let us now explain why the condition (1.10) is not essentially a restriction for the system (1.9). Really, applying the gauge transformation /e0/ '; L -. L4A) - e^ w l». l,»» w, *x)- f dy(i'o-to). O.ID the system (1.9) goes into the system (1.9'), with Q{ and r' 0 instead of Q ( and t g, where: (1.12) Obviously the transformation (1.11) is equivalent to the change of variables (1.12) in the NLEE. In the present paper the expansions of w (::)-(::) and v.jt, <7. - dia (<7., «О over the "squared" solutions If I of (1.9), (1.10) obtained in ref. /E1/ are applied for the study of corresponding NLEE. The main result consists in the explicit construction of the Hamiltonian structures of these NLEE and the calculation of their action-angle variables. The appropriate choice of the system of "squares"ivl * in ref. makes all the proofs in Sec.3 analogical to the simpler cases /7, ' /. This once more leads up to the conjecture, that the expansion over the 3

5 "squares" method is of more universal character and may be applied to a much larger class of linear problems L(A),see^e/ln Sec.4 we display a new 2-component variant of the modified NLS eq. Equations (1.3) and (1.7) may be obtained from it by special reductions, after which the symplectic forms 0 k with even indices become degenerate. Therefore the corresponding actionangle variables are calculated using one of the formsftgt+l > e.g., n_j or Q t. We also give a generalization of the Mikhailov model (1.4). In Sec. 2 we formulate some of the results in ref., which are needed in Sees. 3 and 4. The authors are grateful to Academicians Kh.Ja.Khristov and I.T.Todorov for their support. We thank P.P.Kulish, A.G.Reyman and M.A.Semenov-Tian-Shanskii for numerous stimulating discussions. 2. PRELIMINARIES Let the potentials Qi(x) in L(\) (1.9), (1.10) be complexvalued functions of Schwartz type such, that the discrete spectrum A of the operator L(A) is simple and finite. The Jost solutions are given by their asymptotics: о lim <Ks. A) e * "** - 1, lim ф(.г,\)в Л "** «?, x-~,-,-» ( 2, } ф(х, A)- \\ф-,ф + \\, ф(х, A) «^+,^ their (ЬпА* columns ф, ф (ф,ф ) being analytic in A for una > О ia 8 < 0), О), The The transi transfer matrix is introduced by ф(х, A)-0(x.A)S(X), <»- ь +.- J (2.2) dets(a)-l. '-(** ' "'--") The diagonal elements a~(a) are also analytic in A for ImA 2 < 0, and satisfy the dispersion relations: P DW--L f -^i-ln[l + p+p-( M )] + 2 In " a +, и Г (i-a a-1 X-A D(A) = ±1па г (А), ImA z 0. Г =R»iR, (2 ' 3 > where p- (A) = Ъ ± /а.*> A e + 6Д are the discrete eigenvalues of L(A) and the contour Г is introduced in fig. I of. 4

6 . /Й1/ The mam result of ref. consists in the_proof of the completeness relations for the systems M> and (Ф) of "squared" solutions of LA), where: Я\ *»*(*. А), Л 6 Г, **(!). в Т(х), a-1 N1, 1Ф 3 {Ф ± (х,х), ЛСГ, Ф*(х). Ф^(х). a=l N1, ** - М "Ч^ж. Л), Ф* - «Г>*<*. А), ^(xj-i-*^. A)L, Ч ± -ф ± * а A da a + «*(х, А), Ф ± -ф ± *ф ± (х,\), (Ф Ф\ **+-{*.'*). (Ф Х ФЛ ФОФ '\Ф\Ф\)' -1 / ' \ у Z l 0, 7 +Z u y, Here we shall write down only the symplectic variant of the completeness relation: S(x-y) - /da[q(x. A) P T (y.a) - P(x,A)Q T (y,a)]a I [< (x)p +T (y) - P e +(x)q+ T <y) + Q~ a (x)?l\) -P^DQ^IA,,, (2.4) where P(X. A) - A-(p p-»-)(x.a) - -1-(а + Ф + +а-ф-)(х. A), ir it i 1 (2.5) Q(x. A) - i (<7 + Ф + -р + У 4 ")(х. A) 1_(р-*-_<,-ф-)( х,а), 2b + b _ 2b+b" p*«- xae*f± Wt о*(ж)-х [«* 2 a в?±-dji*(di, a a a / 0 -Цд / 0-1 Ь* : *" (x) - b-*~ (x). A 0 - (. "o = [ 5

7 т These relations allow one to_exp3pd any function f - (t.,...,t t ) cs(c 4 l over the systems V, {Фt or IP, Qi. The corresponding expansion coefficients have the form:»*,«. ", 4 IP. 4 to. n. where f, ] is the following skew-scalar product in S(C*J: T -*U\ It. g] - / dxf '(ж)а 0 В( х ). А 0 -. (2.6) \-to 8 0 / In particular, for f - w(x) and f «o^5w(x) the corresponding expansion coefficients are expressed through the scattering data of the system L(A) and their variations: [Ф~, w] - +ia"b", [Ф", о г 8т] --ia" 5p ", _ (2.7) -, w] - + U" b ". -, o 3 8w] - -ia &r-, The expansions themselves have the form: and w(x) --L /йл(р + Ф + +р~ ") (x, X) + "Г ( » S [c+ 4«+(x) - c-*-(x)] - i f d\ P(x, Л) +i 1 [P+(«)+P a (x)], a a a a a a a - l p ami ' 8 ) + J. Sw(x) - - -L/ d\(s P i + - 5p~ Ф~)(х,А) -2 2 Ш*(х) + й~(х)]- - / d\[q(x, Л)8р(Л) - P(x. \)SqW] + 2 [V* (x) + V ~(x)], (2.9) Г. a _ 1?«- e c ± «f «W + W«i «' v;(x) -Q*(X)S P ; -pjws*. In (2.9) we have introduced the following notations (compare with (2.5) and (2.7)): 8p(X) - [P. a s Sv], Sq(\) -. a 9 Sw], P(A) - -i-ta[l + P + p-wl, qw - 4-ln W, X 6 Г, >r 2 b~ ( 2 ' Ю )

8 Ai AX + v a ~ - *" o+ ' ч а а Note, that the set of expansion coefficients for w(x) over the system f l coincides with the set of scattering data T introduced in ref. / B l /, formula (2.7). Thus the formulae (2.8) and (2.9) allow one to interprete the transfer from w"(x) to T and the ISM itself as a generalized Fourier transform. Let us give also the explicit form of the operatorsл+:. " Z io', - z n I - i d i ^"'в-ч. -Ч, I ^ ff-xv.- ) are eigen» and adjoint- for which the elements of 1 and Ж functions: (Л + -л)* ± (х.л) -о. легид; (л + -л а + )**(х).*;(!), (Л_ -А)ф-(х.А) = 0. ЛеГ^Л; (2.12) (Л_-А а ± )Ф; (х) - Ф; (х). Analogically the operator * Л - ) is naturally related to the system fp, Ql, since: Z + - (A-A)P(x, X) -0. (A -A)0(x, A) «0. А С Г и Д. (2.13) The operators Л+and Л satisfy conjugation-like relations with respect to the skew-scalar product (2.6): [f. A + g] = [Л_Г. g], ff, Ag] - [Af, g]. f. g $(C*). (2.14) With the help of the operators Л+ and Л one is able to obtain compact expressions for the expansion coefficients D^ of D(A) (2.3): WA) - A" m D ( m ), A» 1; D(A) A m D ( - m ), A «1, and their variations as functionals of w(x): D< m > ---i /*dx f*dyv5 T (y)a n A? +1 w(y) + -~ i (2.15) m ±- fdx (< ^Тх). 0) A m w(x). «The authors are gratefull to S.V.Manakov, who called their attention to the operator A. 7

9 8D (m) - -G A _ 1 w]. (2.16) It is also easy to check, that from the expansion of w( x ) over the sympletic basis IP, Ql and from (2.12), (2.13) there follows: F(A + )w - F(A_) w -F(A)w. (2.17) Thus in the formulae (2.14) and (2.15) one may replace the operator Л + Ьу Л_ or Л. This we shall use below, and here we write down the explicit form of the first few D^: бе D«---JL /"*&,,.,-, Л ж ). (2-18) 3. THE DESCRIPTION OF THE NLEE AND THEIR HAMILTONIAN STRUCTURES Starting from the results, listed in Sec.2, it is not difficult to construct the theory of the NLEE related to LXA) (1.9), (1.10).In the proofs below we shall follow the ideas of ref. /e ' 7/, but will prefer to use the expansions over_ the symplectic basis IP, Ql rather than those over the system (V, because the operator A is "selfadjoint" with respect to the skew-scalar product, see (2.14). Theorem. Let the potential of the linear problem (1.9), (1.10) w(x, t) and the meromorphic function F(X) are such that 0 t P(A_ + )^«and the integrals / 44-4^ and fd\[iis L -F(A)] Г «К Г ** are absolutely convergent for all 0<t<». Then «(ж, С) satisfies the NLEE: S

10 ie 8 i* + F(A)w -0 (3.1) (2.10) satisfies the follo if and only if the set fp(\), q(a) } wing linear equations: *-._ 0. i *. - F(A). *-.* -e (3-2) 1-0, i ± - F(* a+ ). dt dt - a The proof is obtained directly, incerting in the l.h.s. of (3.1) the expansions (2.8) for w(x, t) and (2.9) for the variation of the form ff e Sw«a, St +0(8t) a over the system P,Q. 3 dt 8 Remark 1. If F(A) --F 8 (A)/F, (A),, where F 8 (A) and F X (A) are polynomials in A> then the NLEE (3.1) should be understood as i< A >»» -+', e (A)» -о- (з.з) Remark 2. From (2.16) it is obvious, that the NLEE (3.1) may be written in two more forms, equivalent to (3.1): ia_ * dt + F(A.)w «.0, + ia,5l+f(a )w = 0. 8 dt o v e r t n e Remark 3. From the expansions of w(x,t) and о а *г system 1*1 there follows, that the NLEE (3.1) is equivalent to the following set of linear equations for the scattering data T: i^l» ±F (A)p±(A,t). ji i =±F(A a+ )c±(t). i^l=0. (3.4) at dt - a dt From (3.2) and (2.3) it follows, that: ^ - 0. (3.5) at i.e., the quantities D ( m ) are integrals of motion for the NLEE (3.1). Let us go now to the Hamiltonian structures of the NLEE (3.1). The corresponding phase space is naturally parametrized by the

11 independent elements of the potentials of «. The Hamiltonian H should be constructed as a linear combination of the integrals of motion D^m^. One should also find a symplectic form О such, that the Hamiltonian equations of motion, given by H andfl : coincide with the NLEE (3.1). Let F(A)-2F k A k over the positive and negative powers of Л, be a polynomial H F -i2f k D< k + 1 > (3. 7) and 0-0 0, where OQ is a canonical 2-form: "о "- тм*? "s** 1 - (3.8) 2 8 Here the sign A means exterior product. From (3.7), (2.16) and (2.17) there immidiately follows that fih f --i-lffgs*. F(A)w]. Thus it is easy to check that (3.6) with H-H r and П-П. coincides with the NLEE (3.1). Note, that the choice of the pair ilg, H F leading to eq.(3.1) or to an eq. equivalent to (3.1) is by no means unique / *' 8/. Thus the choice: П-О т '- -1аз8^Л% а «;]. (3.9) H " H F " l 2 F k D. (3.10) leads to equations, differing from (3.1) by a left multiplication with the operator Л т. The corresponding linear equations for the scattering data may differ from (3.2) only for A» 0, i.e., on a manifold with measure zero. Thus the pairs П т, Н^ m = 0,+l^+2,... give us a hierarchy of Hamiltonian structures for the NLEE (3.1)' Sometimes for the NLEE in the form (3,3) it is convenient to choose: Гг -~ -^з 5 *^ F f la);,aw], (3.11) H F i.i» l i k D«k + 1 > 1 r,w-ir i ( /. (3.12) The pairwise compatibility of the 2-formsfi is most easily established by recalculating them in terms of the scattering data variations. This is conveniently done with the help of the ie

12 symplectic completeness relation (2.4). Incerting in into (3.8) and (3.9) and using the first line in (2.10) we readily obtain: 0 m - i /dw m 8pUl)*«qW + w (3.13) + i 2 [X 8p + A8q + + X" в? _ лв5"] Thus in terms of fp(a), q(a) all О-simultaneously become canonical, and therefore allq m are pairwise compatible between themselves. Let us express now the Hamilton!an H r in terms of the scattering data. From (3.9) and (2.3) we obtain, that: H LfttMFU)p(^-i 1 [Fi^-)-F {.lsz.)}, FW - / A 4\'F(A'), (3.14) i.e.^ H F depends only on the half of tue canonical variables fp, ql (2.10). Thus the complete integrability of all the NLEE (3.1) is established; the corresponding action-angle variables are given by (2.10). Let us finish this paragraph by the remark, that the symplectic basis IP, Ql allows one to define explicitly the Lagaran^e manifold of the NLEE Ж by: *. f fi It. Pl-0. [f, P* ) - 0 I. (3.15) We list without proofs all the important properties of %,see refs. / e - 7 / : i) If fcl, then \i _ Л f-л ГсЖг ii) we. Ж, and therefore F(A + ) i - F(A_)w - F(A) v С Я, iii) if Щ*. t) satifies the NLEE (3.1), then <r 8 i«с Я. iv) dimlr - codlmft. Let us only verify that Я is indeed the Lagrange manifold, that OJfl a 0. Indeed, from а 9 8тсШ and (3.15) we have i.e., \я % Ъ*. P] - 8p(X) - 0,, А СГиД, and from (3.13) it is obvious, that Q % i0, II

13 4. EXAMPLES OF NLEE Here we shall consider some interesting from our point of view NLEE (3.1). Choosing F(A) 4\«in (3.1) we obtain the system: la 8 w u + W l l x +1B 1 +U 1 * 1-240*0-0. *»8 *0t + *0жж -1В 0 -аио» 8 "11 + U l w 0 *«lpl u 0*l - в -(4:::)' "-Mu 0 -q 0 Pi +Ч1Р0' which after the involution u i-5-4?pf-4 p o' P 0 - e o4s. Pi-'i 4»' «""«I*' 'Г"'I (4-2) goes into the following 2-component modified NLS eq.: i q it + <Чи + l f i«?4!ж + v t q, - 8v 0 q 0-0. i4 Ct + Vx-^O^fflJ, -2lv 0 q, x + + 'l l 4 l' 2 v o4i + v i (4.3) This system contains as particular cases the NLS equation, and also the modified NLS eqs. (1.3) and (1.7). Indeed the NLS equation iq 0t + q 0 x x -2t 0 q 0 q 0-0 is obtained from (4.3) with qj«= 0. This is to be expected, since for q t «0 the linear problem (1.3) goes into the Zakharov-Shabat system, which has been investigated in detail earlier, see refs. /1,e " 10/. For q 0 * 0 (4.3) is reduced to the NLEE: 2 iq lt + Ч1ж,- + *«Х Ч?Ч*, +^14,1*4,-0. (4.4) which after the change of variables (1.12) with f(x) - L /dy qf 2 x goes into the modified NLS eq. (1.3). This change of variables is closely related to the gauge transformation (1.11), after 12

14 which the linear problem (1.9), (1.10) with Q 0 «0 becomes equal to L 8 (A) (1.2). This may be used to check, that in this way all the results related to the system (1.2) are reproduced, see ref.. Another possible reduction, leading to equations of NLS'type has the form: '0 - e Q ap i. a--na*, q (д %5 ) Then (4.3) becomes: and after the change of variables (1.12) with f(x) m ~ /dy q 8 one obtains the following modified NLS eq.: x iu,+u + it, ( u 2 u) -2o E f 0 u 8 u -0, u - Ч1(ж)е-»» (4.7) This with e.m a fl, «-- 8 comsides with (1.7). a* The involution (4.2) imposes the following restrictions on the set of scattering data T: P + (X) p-*tt* 4 ). b+ f Q b-* V -1 ra*, «1 (4.8) Analogically (4.5) leads to P ± W - -P *W), Л Г a*(x) - **(-*), ImA 8 > 0 (4.9) Note, that the restrictions (4.9) do not depend one and coincide with the properties of the scttering data for L g (A) (1.2); the former is gauge equivalent (see (1.11)) to LW withq 0» 0. The Hamiltonian structure of the NLS eq. (4.3) is most simply given by Q-Q., H --4D (e) which after taking into consideration the involution (4.2) are equal to: Q 0 -i /dxu 0 8q* 0 A«q r^sq o ASq*]- 13

15 - i / dxsp'tt) Sq'U) + Г N i + 81(1 + ч) а 2 1 [8Х» + л 8/3 -вх^лвэ*] + (4.10) i N l l _ л И Т* О ИТ в Р'Ш - -i-lntl - в р*(мр + *(ЧА*)], 5'(х) - yintb* (Х)/ь + *(х* ч )1, х е г, х «+ - *S+ + i k L tab * - ^ е + *i a m l N + -^n«i(. e 4 1,Q* 0 -t 1 «e 4 1 ;> V i! <* "> - 8i/d *p'(,x) + -J^S^(Л.*)' -(»Х* + )Ч. From (4.3) it is easy to obtain the explicit form of the actionangle variables. As it has been noted, the modified NLS eq. (4.7) is obtained from (4.1) by imposing the two involutions (4.2) and (4.5). From (4.9) and (3.13) it follows, that all 2-forms Q g k 0;analogically D( tt+1 i 0, see (3.14). The Hamiltonian structure of (4.7) may be given, e.g., by П -ff_!, H --4b (4 > or by О - fij, H - 4D('), where the sign - here means, that in the expressions for O k and D* m ' one should impose the involutions (4.2) and (4.5). The explicit calculation of 0, m>0in terms of the potential W(x, t) is related to the calculation of the inverse operator A""' 1, which for аф О is difficult. In this case we can make use of the corresponding Foisson brackets:. а1 ш -[VFa 8. A 0 A m a 8 A 0 VQ ], / V, F \ T ST Sf (4.12) l VF - \, (V k F) T -(!-, ), k-0.1. \V F / * Qk *"* Incerting F-Q«w ii (4.12) we obtain the corresponding Poisson brackets between the elements of the potentials Q, in 14

16 . Iff 8 w, e J" 8 A 0 wl m»л ш 5(х-у). (4.13) where the i,j-th element of the matrix IF Q! m is equal to ifj, Qj 1 ш. Now it is easy to check, that the Hamiltonian equations of motion: with H»-4D (4) directly lead to the system (4.1). In order to impose the involutions (4.2), (4.5) on the Foisson brackets I, 12 one should calculate the corresponding Dirack brackets, which_in our case is difficult. Therefore we shall only write down flj " l - ^T~ Hx[Sq* л J_ 8 q l 2-n. d x - t 1 q 2 Sq* A Sql, 1 1 l 1 which together with H» generates (4.3). In terms of the scattering data the expression for Q_j has the same form as that of 0 0, given in ref. 7 /, and we omit it. Thus the NLS eqs. (1.3), and (4.7) have an equivalent sets of action-angle variables. Let us give one more example of a NLEE, generalizing the Mikhailov system: w lit ^Mi's'it + 3 a 2 "l I " 2 c i w i - - I x + (ЧЛ),- 0. (4.15) This system is obtained from (3.3) with F 1 (A)-A 8 and F 2 (A) * с = const, after imposing the involution (4.5); (4.14) survives also the involution (4.2). The Hamiltonian structure of (4.14) may be given, e.g., by fl = 0 3 and H = сб (8). In conclusion letus make a few remarks. i) Considering block bundles of the form (1.2), (1.3), (1.6) it is possible to solve the multicomponent (vector and matrix) analog of the NLEE's, considered above, see refs. / *«1S/. The corresponding expansions over the "squared" solutions may be derived analogously to ref. /86/. The difficulties with the explicit calculation of the operator Л have been noted in ref/ 28 '. ii) The polynomial bundles (1.8) are easily written down as eigenvalue problems of the type [J }-\]ф-0,, where ф = dx = (qfr,a^fr,..., X ф ),1 is a degenerate constant matrix and the potential U is a specific 2Nx2N matrix, whose matrix 15

17 elements are expressed by those of U k, see, e.g., ref.. Thus the study of the polynomial bundles and their possible involutions is directly related to the reduction problem of the NLEE, ref. / 8 8 /. iii) Other variants of the modified NLS equations have been considered in refs. /83-2 * /. REFERENCES 1. Zakharov V.E. et al. Soliton Theory: The Inverse Problem Method. Nauka, M., (1980). 2. Mikhailov A.V., Kuznetsov E.A. TMF, 30, 1977, p Каир D.J., Newell A.C., J.Math.Phys., 19, 1978, p Morris H.C., Dodd R.K. Physica Scripta, 20, 1979, p Magri F. J.Math.Phys., 19, 1979, p Kulish P.P., Reyman A.G. Zapiski LOMI, 77, 1978, p Gerdjikov V.A., Ivanov M.I., Kulish P.P. TMF, 44, 1980, p Каир D.J., Newell A.C. Adv. Math., 31, 1979, p Gerdjikov V.S., Khristov E.Kh. Mat. zametki, 28, 1980, p.501; Bulgarian J.Phys., 7, 1980, pp.28, Builough R.K., Dodd R.K. Physica Scripta 20, 1979, p Shen L.Y., Wei Z.D. Preprint of the University of Science and Technology of China, Heifei, China, Wadati M., Konno K., Ichikava Y.-H. J.Phys.Soc.Japan, 46, 1979, p Konopelchenko B.G. J.Phys.A; Math, and Gen., 14, 1981, p Rogister A., Phys.Fluids, 14, 1971, p Mj^lhus E. J.Plasma Phys., 16, 1976, p.321; 19, 1978, p Spatschek K.H., Shukla P.K., Yu M.Y. Nucl. Fusion, 18, 1977, p Mio K. et al. J.Phys.Soc. Japan, 41, 1976, p Reyman A.G., Semenov-Tian-Shanskii M.A. DAN USSR, 251, 1980, p.1310; Inventiones Math., 54, 1979, p.81; 63, 1981, p Reyman A.G. Zapiski LOMI, 95, 1980, p Zakharov V.E., Taktadjan L.A. TMF, 38, 1979, p.26; Zakharov V.E., Mikhailov A.V. JETF, 74, 1978, p Gerdjikov V.S., Ivanov M.I. JINR, E , Dubna, Gadjiev I.Т., Gerdjikov V.S., Ivanov M.I. Zapiski LOMI,120, 1982, p Mikhailov A.V. Physica D, 3D, 1981, p Chen H.H., Lee Y.C., Liu C.S. Physica Scripta, 20, 1979,p Gerdjikov V.S., Ivanov M.I. TMF, 52, 1982, p.89. Received by Publishing Department on August с

18 Герджиков B.C., Иванов М.И. Е Квадратичный пучок общего вида и нелинейные эволюционные уравнения. Иерархия гамильтоновских структур При использовании метода разложения по "квадратам" решений описан класс нелинейных эволюционных уравнений, связанных с квадратичным пучком общего вида. Доказано, что эти уравнения являются вполне интегрируемыми гамильтоновскими системами и обладают иерархией гамильтоновских структур. Работа выполнена в Лаборатории теоретической физики ОИЯИ. Препринт Объединенного института ядерных исследований. Дубна 1982 Gerdjikov V.S., IvanovM.I. E The Quadratic Bundle of General Form and the Nonlinear Evolution Equations. Hierarchies of Hamiltonian Structures Using the method of expansions over the "squared" solutions of the auxiliary linear problem, the class of nonlinear evolution equations related to the quadratic bundle of general form is described. It is proved, that these equations are completely integrable Hamiltonian systems, possessing hierarchies of Hamiltonian structures. The investigation has been performed at the Laboratory of Theoretical Physics, JINR. Preprint of the Joint Institute for Nuclear Research. Dubna 1982

19 Редактор Э.В.Ивашкевич. Макет Р.Д.Фоминой. Набор Е.М.Граменицкой. Подписано а печать Формат 60x90/16. Офсетная печать. Уч.-изд.листов 1,37. Тираж 580. Заказ Издательский отдел Объединенного института ядерных исследований. Дубна Московской области.