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

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6 S. P. NOVIKOVzeros (the spectral points) k n = iκ n in the upper half plane x n > 0 by the formulaa = ∏ k−iκ jj k+iκ j, whereby to a single soliton u(x − ct) there corresponds a singlespectral point k 1 = iκ 1 , where κ 2 1 = +c/4 (cf. [5]). What is the right analog of multisolitonsolutions in the periodic problem? We suppose that the periodic potentialu is such that it has only a finite number of zones; this means that starting fromsome number n > n 0 all points of both spectra a R = ±1 are doubly degenerate andthat the entire half line k 2 ≥ kn 2 0forms a single zone. The degeneracy conditionfor a spectral point is the condition b(k n , x) ≡ 0 for kn 2 ≥ kn 2 0. On multiplying theperiod T by an integer T → mT the entire zone is filled out by degenerate levelsb ≡ 0 as m → ∞. On the other hand, with correct passage to the period T → ∞,whereby the potential u T (x) with period T tends to a rapidly decreasing potentialfor T → ∞, the zones of finite size contract to isolated points of the discretespectrum. All this indicates that it is natural to suppose that the right analog ofmultisoliton solutions are the finite-zone potentials. The property that a potentialbe a finite-zone potential is conserved in time by virtue of the KV equation. Howshould one seek such solutions of the KV equation? What are finite-zone potentialslike?We consider the “higher KV equations” of order n˙u = d ( )δIndx δu(x) + c δI n−11δu(x) + · · · + c δI 0n , (6)δu(x)where I n = ∫ P n (u, u ′ , . . . , u (n) ) dx is a polynomial integral of the KV equationand c 1 , . . . , c n are arbitrary constants. Equation (6) has order 2n + 1 and by thetheorem of Lax and Gardner [6, 7] it admits the Lax representation˙L = [L, A n + c 1 A n−1 + · · · + c n A 0 ], (7)where L = −(d 2 /dx 2 ) + u and the operators A 0 , A 1 , A 2 are indicated in the introduction,A 0 = (d/dx). We shall now indicate the corollary of the Zakharov–Faddeevtheorem mentioned in the introduction which, while not noted by them, is extremelyimportant for our subsequent purposes.Proposition 2.1. The set of all fixed points (stationary solutions) of any one of thehigher KV equations is an invariant manifold also for any other of the higher KVequations (in particular, for the original KV equation) considered as a dynamicalsystem in function space.Proof. All the higher KV equations are Hamiltonian systems; the Poisson bracketsof the integrals are equal to zero [I n , I m ] = 0 for all n, m. Therefore, all the higherKV equations commute as dynamical systems in function space. Therefore, the setof fixed points for one of them is invariant with respect to all the remaining ones.This proves Proposition 2.1.□We have the following basic theorem:Theorem 2.1. 1) All the periodic stationary solutions of the higher KV equations( n)d ∑ δI n−ic i = 0 (8)dx δu(x)are potentials u(x), the number of zones of which does not exceed n.i=0
THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION 72) The equationδI n nδu(x) + ∑ δI n−ic iδu(x) = d (8′ )i=1is a completely integrable Hamiltonian system with n degrees of freedom dependingon (n+1) parameters (c 1 , . . . , c n , d), whereby the collection of n commuting integralsof this system and all the parameters (c 1 , . . . , c n , d) are expressed in terms of 2n +1nondegenerate eigenvalues of both spectra of the potentials u(x) which form theboundaries of the zones.Proof. As is known, in the case of rapidly decreasing functions u(x) from the Laxrepresentation ˙L = [A, L] it is easy to derive an equation for the eigenfunctions ψ kof the operator L: ψ k = Aψ k + λψ k . In the periodic case the analogous derivationgivesψ k = Aψ k + λψ k + µ ¯ψ k ,˙¯ψ k = A ¯ψ k + ¯µψ k + ¯λ ¯ψ k , (9)where λ + ¯λ = 0.Indeed, (L − k 2 )ψ k = 0. Therefore,0 = ˙Lψ k + (L − k 2 )ψ k = (AL − LA)ψ k + (L − k 2 )ψ k = (L − k 2 )(ψ k − Aψ k ).Since (L − k 2 )ψ k = (L − k 2 ) ¯ψ k = 0, we obtain the desired result with unknowncoefficients λ(x 0 , t), µ(x 0 , t). To determine the coefficients we make use of the factthat ψ k (x 0 , x 0 ) = ψ ′ k (x 0, x 0 ) = 0. From this for x = x 0 we have(Aψ k ) x=x0 + λ + µ = 0, ( ˙ψ k ) x=x0 = 0,( ) ddx Aψ k + ik(λ − µ) = 0, ( ˙ψ k) ′ x=x0 = 0x=x 0in the basis ψ k , ¯ψ k . (In the basis φ k , ¯φ k it is necessary to let k ↦→ 1 in the lowerequation.)( ) λ µWe consider the matrix Λ =¯µ ¯λ , where λ+¯λ = 0. This is a matrix of the Liealgebra of the group SU 1,1 to which the monodromy matrix T (k, x 0 ) belongs. Thematrix Λ, as is evident from Eq. (10), depends on u(x 0 , t), u ′ (x 0 , t), . . . , u (2n) (x 0 , t), k.In order to study the time dependence of the monodromy matrix T (k, x 0 ) byEq. (6) it is necessary to compute ψ k and ψ ′ at the point x = x 0 + T , where T isthe period. Having done this we obtain (x = x 0 )ȧ + ḃ = A(aψ k + b ¯ψ k ) + λ(aψ k + b ¯ψ k ) + µ(¯bψ k + ā ¯ψ k ),ik(ȧ − ḃ) = ddx A(aψ k + b ¯ψ k ) + λ(aψ k ′ + b ¯ψ k) ′ + µ(¯bψ k ′ + ā ¯ψ(11)k).′Substituting relation (10) into (11) and performing simple computations, we obtain(10)ȧ = µ¯b − b¯µ, ȧ + ˙ā = 2ȧ R = 0, ḃ = (λ − ¯λ)b + (a − ā)µ = 2λb + 2ia I µ. (11 ′ )Proposition 2.2. On the basis of the higher KV equation the dependence of themonodromy matrix on time is described by the equation˙ T = [Λ, T ], (12)where the matrix Λ is found from formula (10) starting from the time dependenceof the eigenfunction basis (ψ k , ¯ψ k ):( ˙ψ k , ˙¯ψk ) = A(ψ k , ¯ψ k ) + Λ(ψ k , ¯ψ k ).
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A periodic problem for the Korteweg-de Vries equations, I.

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