A periodic problem for the Korteweg-de Vries equations, I.

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8 S. P. NOVIKOVThis proposition has been proved above. It implies:Corollary 2.3. The equationΛ ′ − ˙Q = [Q, Λ], (13)holds and is the compatibility condition for the two-parameter family of matricesT (k, x, t) satisfying the equations T ˙ = [Λ, T ], T ′ = [Q, T ], where the matrix Q isthat of formula (4).Proof. Since T ˙ = T ˙ ′ , it follows that [Λ ′ , T ] + [Λ, T ′ ] = [ ˙Q, T ] + [Q, T ˙ ]. Using theJacobi identity, we obtain [Λ ′ − ˙Q − (Λ, Q), T ] = 0.This proves the corollary (it now follows easily from elementary properties of theLie algebra of the group SU 1,1 ).□We now proceed to the basic Theorem 2.1.If u(x) is a stationary solution of Eq. (6), then the corresponding monodromymatrix T (k, x) is also stationary T ˙ ≡ 0 and ˙Q ≡ 0. From Corollary 2.3 we obtainthe equationΛ ′ = [Q, Λ], (14)where Sp Λ = λ+¯λ = 0. The eigenvalues of the matrix Λ are integrals of the system(8). Since Sp Λ = 0 for the eigenvalues of the matrix Λ we have α ± (k) = ± √ det Λ,where det Λ = |λ| 2 − |µ| 2 .Further, det Λ is a polynomial in k 2 of degree 2n+1, as is easily verified from theform of the matrix Λ, whereby the leading coefficient is a nonzero constant. Theroots of the polynomial det Λ = 0 are numerical integrals of the system Λ ′ = [Q, Λ];formally the coefficients of the polynomial det Λ in the variable k 2 are written in theform of polynomial expressions in u(x) and its derivatives with constant coefficients.We remark that the equation Λ ′ = [Q, Λ] is equivalent to the original equationddx( ∑ciδI n−iδu(x))= 0 for the function u(x). Thus, we have 2n + 1 integrals of thissystem (the roots of the equation det Λ = 0 or the coefficients of the polynomialdet Λ), the first n + 1 of which are formed from the constants (c 1 , . . . , c n , d).We shall prove that these integrals are commutative and define all the zoneboundaries of the potential u(x). Using Eq. (12) [Λ, T ] = 0, we can obtain thefollowing relations:1) (a − ā)µ = 2λb, 2) ¯bµ = ¯µb. (15)From this we immediately obtaine 2iφ = b/¯b = µ/¯µ, where φ = arg b, ia I /b = λ/µ. (15 ′ )At the (nondegenerate) points of the spectrum k 2 = k n∣ ia I∣∣∣ ∣ b ∣ = λµ ∣ = 1, k = k n, (16)or |λ| 2 − |µ| 2 = det Λ = 0. Thus, the zone boundaries are the zeros of the polynomialdet Λ = 0. Passage to the limit of infinite period whereby u tends to a rapidlydecreasing function shows that the last n coefficients of the polynomial det Λ arealgebraically independent integrals which are polynomials in u and its derivativesfor almost any values of the first n + 1 constants (c 1 , . . . , c n , d). We thus have nindependent integrals of the Hamiltonian system (8 ′ ) with n degrees of freedom.From the previously mentioned theorem of Zakharov and Faddeev it can be seen
THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION 9also that these integrals commute. Indeed, the manifold of functions [the stationarypoints of Eq. (6)] lies in a complete function space where all 2n + 1 integralsare commutative. On the finite-dimensional manifold of functions in question thesymplectic form is degenerate. The integrals (c 1 , . . . , c n , d) after restriction to themanifold have zero Poisson brackets with all functions on this manifold. It is easyto verify that the symplectic form on this manifold is in fact obtained by restrictionof a form from the entire function space. This implies the complete integrability ofEq. (8 ′ ) and all the assertions of the basic Theorem 2.1.□Remark 1. It is evident from Eqs. (15 ′ ) that φ = arg µ + mπ, φ = arg b, andthus ± sin φ is expressed in terms of u and its derivatives. Combining this fact withProposition 1.1, we find that also the degenerate points of finite-zone potentialsu(x) can be obtained from the transcendental equation(5):where ± sin φ = sin(arg µ).2πn =∫ T0(2k − u k ± u k sin φ )dxRemark 2. Periodic and conditionally periodic solutions are obtained when thelevel surface of all the commuting integrals found of Eq. (8 ′ ) for whatever valuesof the constants (c 1 , . . . , c n , d) is compact (i.e., is an n-dimensional torus). Ingeneral, for randomly chosen eigenvalues or zone boundaries (or, what is the samething, values of the constants and integrals), periodic functions are obtained withn incommensurable periods. It is thus more natural to solve the inverse scatteringproblem in almost-periodic functions rather than in periodic functions only.Remark 3. The matrix Q itself has the form of the matrix Λ for the “zero-order KVequation” { ˙u = u ′ }, where the operator A 0 = d/dx, [A 0 , L] = u ′ . The stationarysolutions in this case are constants constituting 0-zone potentials. Further, forany n the transformation u → u + c takes an n-zone potential into an n-zonepotential, and by means of this transformation it can be achieved that the functionu = v + const satisfies the equationδI nδu(x) + c δI n−22δu(x) + · · · + c δI 0nδu(x) = d,where c 1 = 0 (it is assumed that c 0 = 1). By studying the form of the matrixΛ = Λ R + iΛ I starting from the the formulas (10), it is possible to show that thematrix Λ R has the form (n ≥ 1)Λ R =( n−1 ∑( dγ qdxq=0δI qδu(x)) ) (0 )k 2(n−1−q) 1,1 0where the constants γ q are expressed in terms of the constants c 1 , . . . , c n .Remark 4. The stationary solutions of the KV equation of order q are containedamong the stationary solutions of the KV equation of order n > q as degeneratetori of Eq. (8 ′ ), where the n + 1 constants (c 1 , . . . , c n , d) and the values of the nintegrals I 1 , . . . , I n are chosen such that these integrals depend on the entire levelsurface. We shall now consider stationary solutions of the KV equations for n = 1and n = 2.
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A periodic problem for the Korteweg-de Vries equations, I.

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