A periodic problem for the Korteweg-de Vries equations, I.
A periodic problem for the Korteweg-de Vries equations, I.
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 <strong>the</strong> equationcu ′ + 6uu ′ − u ′′′ = 0,)u ′′ = 3u 2 + cu + d, u ′2 = 2(u 3 + cu22 + du + E .We obtain <strong>the</strong> elliptic function∫x =du√2u3 + cu 2 + 2du + 2E ,where u(x − ct) is a solution of <strong>the</strong> KV equation of <strong>the</strong> 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 ano<strong>the</strong>r language and by ano<strong>the</strong>r method [14]; <strong>the</strong> Sturm–Liouville equation with an elliptic potential is a special case of <strong>the</strong> Lamé equationarising from <strong>the</strong> Laplace operator on an ellipsoid where 1/2 n(n + 1)-fold ellipticfunctions also occur in <strong>the</strong> potential. (As shown in [14], <strong>the</strong>y are n-zone potentialswhich are a <strong>de</strong>generate case of <strong>the</strong> general n-zone potentials given by Theorem 2.1).In <strong>the</strong> basis ψ k , ¯ψ k <strong>the</strong> matrix Λ has <strong>the</strong> <strong>for</strong>mλ = ik2k 2 (u′′ − 2u 2 + 8k 2 ) − ick + icu2k ,µ = −u ′ + ik2k 2 (−u′′ + 2u 2 + 4k 2 u) − icu(17)2k ,and <strong>the</strong> characteristic polynomial has <strong>the</strong> <strong>for</strong>m<strong>de</strong>t Λ = 4k 6 + 2ck 4 + c2 + 4dk 2 +4cd − 2E. (17 ′ )4We see that k1 2 + k2 2 + k3 2 = −c/2. If <strong>the</strong> period T → ∞ and u(x) tends to a rapidly<strong>de</strong>creasing function, <strong>the</strong>n E → 0, d → 0. There<strong>for</strong>e, k 3 → 0, k1 2 → −κ 2 , k2 2 → −κ 2 ,and <strong>the</strong> zone contracts to <strong>the</strong> eigenvalue k 2 = −κ 2 where κ 2 = c/4.We note that <strong>the</strong> transcen<strong>de</strong>ntal equation <strong>for</strong> all <strong>the</strong> <strong>de</strong>generate points of <strong>the</strong>spectrum follows from Remark 1.2) The Case n = 2. On <strong>the</strong> basis of Remark 3 we consi<strong>de</strong>r only an equation of<strong>the</strong> <strong>for</strong>m(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 <strong>the</strong> solutions of Eq. (8 ′′ )]. The Lagrangian of <strong>the</strong>dynamical system (8 ′′ ) has <strong>the</strong> <strong>for</strong>mL = L 2 + 8cL 0 − du = L(u, u ′′ ), 8L 0 = 8u 2 , L 2 = u′′22 − 5 2 u2 u ′′ + 5 2 u4 .We <strong>de</strong>note by q <strong>the</strong> quantity q = (∂L/∂u ′′ ). The energy of a system in which <strong>the</strong>Lagrangian <strong>de</strong>pends on two <strong>de</strong>rivatives has <strong>the</strong> <strong>for</strong>mE = L − u ′′ q + u ′ q ′ .We <strong>de</strong>note 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 <strong>the</strong> Hamiltonian <strong>for</strong>mp ′ u = − ∂H∂u ,p′ q = − ∂H∂q ,u′ = ∂H∂p ,q′ = ∂H∂p q,