The universal families of one dimensional commuting hamiltonians which generate all two dimensional isospectral deformation equations

Volume 80A, number 1

PHYSICS LETTERS

10 November 1980

THE UNIVERSAL FAMILIES OF ONE DIMENSIONAL COMMUTING HAMILTONIANS WHICH GENERATE ALL TWO DIMENSIONAL ISOSPECTRAL DEFORMATION EQUATIONS D.V. CHUDNOVSKY’ Division de Physique Théorique, CEN, Saclay, 91190 Gif-sur- Yvette, France

and G.V. CHUDNOVSKY1

University of Paris Sud, 91405 Orsay, France Received 28 August 1980

We naturally associated with every linear spectral problem on a Riemann surface r, a completely integrable nonlinear hamiltonian. Any two dimensional isospectral deformation is then decomposed into the common action of two commuting hamiltonians from the universal family of one dimensional hamiltonians determined only by r.

1. We incorporate the isospectral deformation equation for linear differential operators, where the dependence on a spectral parameter is rational on a Riemann surface F, into the concept of the generalized operator Russian chain [4,9,11]. Let us consider this general linear differential equation first of all in the case F = P1. In this case the equation can be represented as follows: (d/dx) q~= U(X;x)J2~, with m

U(X;x)=

~)

(1.1)

ri

re,. ‘~

i=1 /1 (X—a1)’

tions I~.Since there are many ways of achieving this representation, we prefer now to follow the simplest pattern that is presented in (1.4). It should be noted that the problem of representation of the potential in terms of squares of eigenfunctions was considered for the first time by Borg [1], and general results for the Schrodinger equation belong to Levitan [2]. His method is quite general and works in other circumstances as well. All the matrices in (1.1) and (1.2) are of the size n X n, but their elements are themselves operators on the Hubert space H. The problem formally adjoint

U

+

0,, 1X’,

(1.2)

/0

where a1, am, °° are the positions of singularities and the A = aj is of multiplicity r~i = 1, m, o°. Following our general procedure [4,5,9] we convert the linear equation (1.1) into a non-linear hamiltonian system possessing a large family of first integrals, Moreover we consider the situation when the change from the linear problem (1.1) into a nonlinear system is achieved by representing each of the coefficients U~,1as a certain quadratic function of the eigenfunc-

to(l.l)has the form

(d/dx) ~t

=

—~~U(X;x).

(1.3)

...,

...,

On leave from Dept. of Mathematics, Columbia University, New York, NY 10027, USA.

Henceforth we adopt nQtations such that the eigenfunctions of (1.1) are denoted as 4 while those of the adjoint problem (1.3) are denoted by CI’ [9]. Now the potential U,,~has to be represented n terms of the product of eigenfunctions ~ and cI’?~with respect to a certain operator (matrix for the classical problems) valued measure ~ This measure ~ is essentially the “spectrum” of the problem (1.1). Naturally, ~ can be absolutely continuous with respect to the Lesbegue measure or may have a singular part of the form dE~= 1 ~ (A Xk)dA. We refer the —

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PHYSICS LETTERS

reader to the discussion in [5,9]. Finally we are given the representation of the potential in the following

general form:

10 November 1980 ..

CIA]

=

1~T~x

4~f~ir(x,ILL) d

(1.8)

,

for a rational function ir(X, p) in A and ji.

u~ ~

(1.4)

x for scalar functions p~

Lemma 1.9. In order for system (1.5) to have the

1(X):1 = 1,..., r1 i = 1, m. These scalar functions are not defined yet and will be defined in terms depending on the structure of (1.2) in the A-plane. Now the equations (1.1), (1.3) can be ...,

rewritten as the coupled system of non-linear equa-

first integrals C[A] ofthe form (1.8), the following functional relations between the rational functions u(A, ji) and ir(X, p) are necessary and sufficient: —u(X,p1)ir(A,p2) + a(p2,p1)ir(A,p2)

tions in terms of elements of I’x and I~: +

(d/dx)~~ = f~a(X, p) ~

ir(X, p1)u(A, ~i2)—ir(A,ji1)a(p1,

1 10

1~2)—0,

identically in A, ~ (1.5)

(d/dx)CI~= —b~f4~u(A~p) ~

,

for all A, where the rational function u(X, p) is defined asfollows: m rj a(X, JI) = —~ . (1.6) z1 J1 (A — a.)’

~ ~

The choice ofp~1(p)has not yet been made, and it is not obvious how to make it. The choice of a(X, ~) should be such that the system (1.5) will have many first integrals. More precisely, it should have oo~~ .2 first integrals (where oo stands for the parametrization

Remark 1.11. The functional relation (lAO) can be rewritten in the following short form: 7r(X,p2) a(A,p2)—u(~i1,p2) = (1.12) —-

ir(A,~i1) u(A,p1)—a(p2,p1)

The solution of the functional equation (1.10) or (1.12) depends on singularities a1, .,am and multi. olicities r1,..., rm, r,,,. Here is the example considered in [9] when m = 1 and a1 = and r ~ 1. Then . .

00

rco—1

a(A, ~) =

~ x’~=—’,.~+ ~

(1.13)

1=0

for an arbitrary function ~(p) and

in A). First of all there are the obvious first integrals of the system 1A1 (1.5) 11 7~ b X rI’X =K / Obviously the quantity K [Alis the first integral of

ir(X,The /1) gauge = (A transformation ii) of eq. (1.l)is performed (1A4) in the following way. Let g = g(x) be a non sin~gular n X n matrix. Then the transformation 1 -~g~I =

(1.5), and the canonical choice of F~may correspond

changes the potential U(A, x) into a new one:

to the choice K[X] = II (the identity matrix) when ~ —1 = (4~) We need 00 more first integrals. In our

U(A,x)=g~g~ +gU(A,x)g

-

.

-

,

—

-

previous papers [4,5,9] we found these integrals in the special case when m = 1, r = 1 and = 00• Ina(X, this p) case 1(x) + U0(x)1 and the a1 function x)p) = = AU isU(A; u(A, A + p. In this case the first integrals of (1 .5) have the form [9,11]: -

C[A]

=

~

d~ A ‘~ ~

...

We follow this pattern and we now look for the first integrals of (1.5) having the very special form of being quartic in cI’. These integrals should have the following form inspired by (1.5) 2

On more gauge transformation is allowed. We can

multiply ~ froma multiplication the right by anof x-independent C~ and perform c1~from the matrix left by an x-independent matrix D~.In this case the system of non-linear equations (l.5)is unchanged provided we change the spectral measure ~ by C~1~

X D~.This transformation is especially useful since for the absolutely continuous measure ~

one can

always take d~ to be simply dAn. The most general solution of the functional equation (1.10), (1.12) for an arbitrary form of the potential

(1.2) is given below in sect.2. Naturally, there are

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PHYSICS LETTERS

several possible forms of the rational function ~(A, ~) that determine the universal family of integrals C[A] in (1.8). However, if we want the system (1.5) to be generated by the functions C[X] (1.8), then IT(A, 7r) is unique up to a change of variables. In this case the simplest form of ir(A, p) is simply the Cauchy kernel IT(A, ~) = (A py~. This is exactly the situation of (1. 13)—(1 .14). In general for the Riemann surface I’ = P1, the function IT(X, ii) and, consequently, the first integrals C[A], are universal functions independent of the structure of the potential and depend only on the Riemann surface. Here is one example:

10 November 1980

our paper, any two dimensional completely integrable equation is obtained in this way. We are happy that the quantities C[A] introduced for the first time in [4,9, 11] are universal quantities generating all two dimensional isospectral deformation equations. General two dimensional isospectral deformation equations, according to the ideology of [6,8,10], are

—

presented as the consistency condition for two linear

problems of the type (1.1) or (1.3). Namely, we introduce two variables x and t, and two linear problems rationally depending on A: v

—

~X,x

~

—

-

4~x,x— _4)xU0~),

V(A)4~~ ~

=

Lemma 1.15. For arbitrary singulanties a 1, and corresponding multiplicities r1, .., rm, r we have the following rational function u(A, p) correspond. ing to the potential (1.2): m 1 1

-

x’

—

=

(2.1) (2.2)

—4)xV(X).

. .

r00

j1 /1 (A

—

a1)J (ii

—

ai)’iJ~

1 l6~ ‘

+,4~A1p’°°J.

Then the functional equations 1. (1.10), (1.12) are satisfledThough with 7r(X, p) = (A p) the systems (1.5) corresponding to (1.1) and (1.3) are different, their first integrals defmed in (1.8) are the same and depend only on Riemann surface F. In other words, all the isospectral deformation equa—

tions rationally depending on a spectral parameter A

1 are equivalent to the universal for the of case ofF = P generated by the functionals family hamiltonians C[A]. 2. In the previous chapter we considered one dimensional equations (1.1) or (1.3) and represented them as nonlinear hamiltonian completely integrable systems (1.5). Moredverwe showed that for F = P1, all the system (1.5) canonically constructed by (1.1)— (1.3) admit the universal family of first integrals C [A] (1.8) depending on the Cauchy kernel ir(X, p) = (A —

p)1. Now we follow our proposal put forth in [4,5,

9,11,151 and consider simultaneously two commuting hamiltonians of the form (2.5). These two commuting one dimensional hamiltonians naturally generate two

Here U(A) and V(X) are rational functions in A with coefficients depending on x and t: r

-

U(X) = i=1 /=1 ~ (A ~ s~ E ~ i=1 /=1 (A

V(A) =

—

~“ -

U

u

~

(2.3)

÷ V,~-A/ + V~

(2.4)

~“ a~)I. + j=1

V —

r,0

~

+

~.

b~)J 1=1

Then consistency condition for eqs. (2.1)—(2.2) takes thethe form of a two dimensional matrix system of equations: U(A) V(A) —

+

[U(A) V(A)]

=

0

(2 5)

X

Eq. (2.5) is understood to be the equation on and V It is21.well known [5,8,9,10] that (2.5) contains a very wide class of physically important two dimensional equations such as the sine-Gordon, principal chiral fields, different u-models, the operator non-linear Schrödinger equation, the Gross-Neveu model, the massive Thirring model etc. Following the pattern of.sect. 1, we replace the linear systems (2.1)—(2.l’) añd(2.2)—(2.2’)by the corresponding nonlinear equations of the form (1.5). These systems have the form

~

=

f~i~ui (A, p) d~CI~ (2.6)

‘

f4~ a

cI~~ = —CI~

1 (A, p) dE.~cI’~

dimensional completely isospectral deformation equations [4,5,9]. Moreover, and this is the main result of

and 3

Volume 80A, number 1

PHYSICS LETTERS

10 November 1980

(2.7) =

f~ u2(A, p) d~

~

~

(2.14) = fu(A, p)~ ~ In particular, according to (2.9), it follows from

-

—~

J.1

Here u1(A, p) and a2(A, ii) are constructed in the canonical way from U(X) (2.3) and V(X) (24) follow-

lemma 2.12 that the equations (2.6) are hamiltonian systems with the hamiltonian and the equations

ing the recipe of lemma 1.17. Then by lemma 1.9 the universal quantitiesC[A] 1 C[A] = A ~ (2.8) p are first integrals both of(2.6) and of(2.7). It should be noted that due to the construction o1(A, p), u2(X, we have the following symmetry relation that is

(2.7) are hamiltonian systems having the hamiltonian ~02 - Moreover, and this is the most important point, these two hamiltonians commute (theorem 2.18).

~

—

Lemma 2.15. Let u(X,p)be a function such that A u(A, ~) r()r(p) (2.16) —

important for the introduction of the hamiltonian struc-

— p Then the corresponding hamiltonian ~JC0 is reduced

ture

to functionals C[X]:

a1(A,p)=u1(p,A);

a2(A,p)o2(p,A).

(2.9)

First of all we introduce the simplest hamiltonian structure that corresponds to the non reduced case, when there are no restrictions on•U1,1, V1,1. In this case we have the canonical variables ~ and (E2)~, where the variable conjugate to a coordinate (4~)~ is (I’~)~[9,11]. With such a definition of the hamiltonian structure, the hamiltonian equations for the given hamiltonian X can be written in the following form d

Fx

4~=

-------;

Here for a matrix we denote ~

)

~ ~

CI’

=

—

j.

(2.10)

(CIa) and a functional !C of

(2.11) ~

fr(A)C[A]dE~

-

(2.17)

Theorem 2.18. Let u1(A, p) and a2(X, ji) be two functions which satisfy the condition (2.17) for some functions r1(X) and r2(X). Then the corresponding hamiltonian and ~C02commute. This description of a(X, p) in (2.16) gives us the possibility to define u(X, p) “canonically” in terms of a1 and r1. For fixed a1, ~.., am and r1,..., rm, r0, we call u(A, p) a ‘canonical rational function if u(X, p) has the form (2.16), and has singularities only at a1, -.., am, cc of multiplicities r1, ..., Tm, r,~,precisely. As one understands, the “canonical” u(A, p) is not necessarily unique since residues at singularities are not specified. In other words, for given a1,.., am and r1 , - -, rm, r,,, there are different completely integrable two dimensional systems. .

In the notations of (2.10) we can represent each of the systems (2.6), (2.7) as a hamiltonian system. Lemma 2.12. Let u(A, p) be a rational function such that a(A, p) = a(p, A). Then for the hamiltonian

~c ~ tr(f

~0=tr

~

Mi /.L2

Definition 2.19. For fixed singularities a1,~.,am, with multiplicities r1, - . -, rm, r,0, we define a “canonical” function o(A, p) by a formula cc

u(X,p)~T),

(2.20)

(2J3)

the hamiltonian system of equations (2.10) for ~JC0 has the form: 4

with the rational function r(A) having singularities at ..., am, 00 ofmultiplicities T1, ., Tm, F,,, + L

a1,

Volume 80A, number 1 m =

PHYSICS LETTERS

rj

E E c~,1(A i1 /“l

of the first and of the second order. In all these cases

r,,,+1 —

a1)/

+

c,,, A! .

J0

for arbitrary constants c11:j = 1, and c,,,1: / = 0, r,,, + 1.

- . -,

10 November 1980

(2.21)

“

r1 i = 1,

- . -,

m

-..,

Corollary 2.22. For any two functions 01 (A, p) and o2(X, p) satisfying the conditions of definition 2.19 two hamiltonians and commute. Consequently, any two systems of equations (2.6) and (2.7) with the “canonical” o1(A, p) and o2(A, ~) represent commuting Harniltonian flows. This explains why the two dimensional completely integrable systern of equations (2.5) is decomposed into two onedimensional commuting Hamiltonians (2.6) and (2.7). The main result of this paper is the decomposition of any two dimensional isospectral deformation eqUation into two commuting Hamiltonian flows from a universal family of functionals C[X]. Our program, briefly outlined in [4,5,9,11] needs to be presented in further detail, and is the subject of a series of our papers which has been prepared for publication. First of all, we need the canonical way to construct the measured d~~for any given linear problem (1.1). It is very easy to show that the measure d~ tint transforms the linear equations (2.1) into non-linear systems (2.6) exists (cf. [2,9,131). This is a rather simple consequence of the general spectral theory. It is much more non-trivial to choose ~ in “the best” way. We made such a choice in [5,9,14,15] for the case of an arbitrary matrix linear differential equation

the corresponding measure ~ is easily expressed in terms of the corresponding spectral measure of the linear problems and in terms of the scattering data (2.1) in the linear problems (2.l)—(2.2). References [1] G. Borg, Acta Math. 81(1949) 3. [2] Levitan, Dokl.G.V. Acad. Sci. USSR 83 (1952) [3] B.M. D.V. Chudnovsky, Chudnovsky, Phys. Lett.349. 73A (1979) 292. [4] DV. Chudnovsky, Phys. Lett. 74A (1979), 185; C.R. Acad. Sci. Paris 289 A (1979), A-731. [5] DV. Chudnovsky, CargIse lectures, june 1979, in: Bifurcation phenomena in mathematical physics and related topics (Reidel, Dordrecht, 1980) pp. 440—510; Lecce Conference on Non-lineai evolution equations, June 1979, in: Lecture Notes in Physics, Vol. 120 (Springer, Heidelberg, 1980), pp. 103—150. [6] HF. Baker, Proc. Royal Soc. London 118 (1928) 584. [7] Drach, C.R. Acad. Sci.Chudnovsky, Paris 168 (1919) 47, 337. [8] JD.V. Chudnovsky, G.V. Z. Phys. D5 (1980) [9] D.V. Chudnovsky, Les Houches Lectures August 1979, in: Lecture Notes in Physics, Vol. 126 (Springer, Heidelberg, 1980) pp. 352—417. [10] V.E. Zakharov, A.B. Shabat, Fund. Anal. App!. 13 (1979) 3. [11] D.V. Chudnovsky, Proc. Nat!. Acad. Sd. USA, to appear. [12] R. Jost, RG. Newton, Nuovo Chnento 1(1955)590. [13] D. Kaup, J. Math. Anal. Appl. 59 (1976) 849. [14] D.V. Chudnovsky, G.V. Chudnovsky, Phys. Lett. 72A (1979) 291. [15] D.V. Chudnovsky, J. de Physique, to appear.

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