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

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10 S. P. NOVIKOV1) The Case n = 1. In this case we have the equationcu ′ + 6uu ′ − u ′′′ = 0,)u ′′ = 3u 2 + cu + d, u ′2 = 2(u 3 + cu22 + du + E .We obtain the elliptic function∫x =du√2u3 + cu 2 + 2du + 2E ,where u(x − ct) is a solution of the KV equation of the type of a simple wave.According to Theorem 2.1 u(x) is a 1-zone potential. This fact was first provedby E. Ince in 1940 in another language and by another method [14]; the Sturm–Liouville equation with an elliptic potential is a special case of the Lamé equationarising from the Laplace operator on an ellipsoid where 1/2 n(n + 1)-fold ellipticfunctions also occur in the potential. (As shown in [14], they are n-zone potentialswhich are a degenerate case of the general n-zone potentials given by Theorem 2.1).In the basis ψ k , ¯ψ k the matrix Λ has the formλ = ik2k 2 (u′′ − 2u 2 + 8k 2 ) − ick + icu2k ,µ = −u ′ + ik2k 2 (−u′′ + 2u 2 + 4k 2 u) − icu(17)2k ,and the characteristic polynomial has the formdet Λ = 4k 6 + 2ck 4 + c2 + 4dk 2 +4cd − 2E. (17 ′ )4We see that k1 2 + k2 2 + k3 2 = −c/2. If the period T → ∞ and u(x) tends to a rapidlydecreasing function, then E → 0, d → 0. Therefore, k 3 → 0, k1 2 → −κ 2 , k2 2 → −κ 2 ,and the zone contracts to the eigenvalue k 2 = −κ 2 where κ 2 = c/4.We note that the transcendental equation for all the degenerate points of thespectrum follows from Remark 1.2) The Case n = 2. On the basis of Remark 3 we consider only an equation ofthe form(d δIndx δu(x) + 8c δI )0= 0, (8 ′′ )δu(x)where 8I 0 = ∫ (8u 2 ) dx, I 2 = ∫ ( )u ′′22 − 5 2 u2 u ′′ + 5 2 u4 dx [we obtain all 2-zone potentialsby adding a constant to the solutions of Eq. (8 ′′ )]. The Lagrangian of thedynamical system (8 ′′ ) has the formL = L 2 + 8cL 0 − du = L(u, u ′′ ), 8L 0 = 8u 2 , L 2 = u′′22 − 5 2 u2 u ′′ + 5 2 u4 .We denote by q the quantity q = (∂L/∂u ′′ ). The energy of a system in which theLagrangian depends on two derivatives has the formE = L − u ′′ q + u ′ q ′ .We denote u ′ by p q and q ′ by p u . Then E = H(u, q, p u , p q ) = V (u, q) + p u p q , andEqs. (8 ′′ ) assumes the Hamiltonian formp ′ u = − ∂H∂u ,p′ q = − ∂H∂q ,u′ = ∂H∂p ,q′ = ∂H∂p q,
THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION 11where V = L − u ′′ q = − q22 − 5 2 qu2 − 5 8 u4 + 8cu 2 − du. However, the study of thisHamiltonian system with two degrees of freedom is not so simple. We thereforecompute its remaining integral by using the Lax representation Λ ′ = [Q, Λ] of thisdynamical system. The operator A has here the formA = A 2 + 16cA 0 == d5dx 5 − 5 2 u d3dx 3 − 15 d2u′4 dx 2 + 15u2 − 25u ′′ d8 dx + 15 8)(uu ′ − u′′′+ 16c d2 dx .Computing the matrix Λ by Eqs. (10), we obtain (in the basis ψ k , ¯ψ k )−16λ = ik ()k 2 − u(4)2 + 4uu′′ + 3u ′2 − 3u 3 + 2k 2 u 2 + 16k 4 + 16ick − 8 icuk ,16µ = (6uu ′ − u ′′′ + 4k 2 u ′ ) ++ ik ()k 2 − u(4)2 + 4uu′′ + 3u ′2 − 3u 3 + 2k 2 u ′′ − 4k 2 u 2 − 8k 4 udet Λ = k 10 + 2ck 6 − d (16 k4 + k 2 c 2 + E )+ I + 16cd32 16 2 ,whereI = p 2 u − 2up u p q + (2q − 16c)p 2 q + D,D = u 5 + 16cu 3 − 4uq 2 + 32cuq − 2dq.− 8icuk ,The integrals I and E = H are commutative, and the Hamiltonian system (8 ′′ ) isthus completely integrable. Let p 2 u = α1,2{(2q − 16c)p 2 α2 2 for q ≥ 8c,q =−α2 2 for q < 8c.Thenα 2 1 ± α 2 2 = I − D(u, q) + u(E − V ), ±α 2 1α 2 2 = (2q − 16)(E − V ) 2 ,±2α 2 j = A ± √ a 2 − 4B, j = 1, 2,(18)(19)(19 ′ )where A = I − D + 2u(E − V ), B = (2q − 16c)(E − V ) 2 . We find that for givenvalues of the constants c, d, E, I we must seek regions of the (q, u)-plane whereA ≥ 0, A 2 − 4B ≥ 0, q ≥ 8c. (20)Compact regions satisfying inequality (20) give tori in the phase space (u, q, p u , p q ).It is evident from the equations that compact regions are possible only in the halfspace q ≥ 8c. Indeed, for q < 8c the expression A ± √ A 2 − 4B always has roots ofdifferent signs which coalesce only at the isolated points A = 0, B = 0.For these potentials (periodic and conditionally periodic with two periods) therelation ∑ j k2 j = 0 is satisfied.As noted in Remark 3, the remaining 2-zone potentials found in Theorem 2.1are obtained by adding a constant u → u + const, where this constant is the sum ofthe eigenvalues — the zone boundaries — of the new potential v(x). The equationsgiving the singular points of the Hamiltonian system, as is easily seen, reduce tothe conditionsp u = p q = 0, q = − 5 2 u2 , 10u 3 − 16cu + d = 0
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A periodic problem for the Korteweg-de Vries equations, I.

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