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

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2 S. P. NOVIKOVnumber of other equations have subsequently been found which admit the “Lax representation”˙L = [A, L] for a certain pair of operators A, L. P. Lax [7] and Gardner[6] have shown that the known polynomial integrals I n (u) = ∫ ∞−∞ P n(u, . . . , u (n) ) dxof the KV equation (the I n are expressed in terms of the spectrum of the operatorL) all determine equations˙u = d δI ndx δu(x) , (1)admitting the Lax representation ˙L = [A n , L], where L = (d 2 /dx 2 ) + u and the A nare certain skew-symmetric operators of order 2n + 1,∫∫ ( ) uI 0 = u 2 ′2dx, I 1 =2 + u3 dx,∫ ( u′′2I 2 =2 − 5 2 u2 u ′′ + 5 )2 u4 dx,A 0 = d(dx , A 1 = 4 d3dx 3 − 3 u ddx + d )dx u ,A 2 = d5dx 5 − 5 2 u d3dx 3 − 15 d2u′4 dx 2 + 15u2 − 25u ′′ d8 dx + 15 8)(uu ′ − u′′′. (2)2We shall call these equations “higher KV equations.” Further, beginning with thepapers [9, 10], a number of other important equations were found which admitthe Lax representation ˙L = [A, L], where L is no longer a Schrödinger operator(and is not always symmetric). In the papers of L. D. Faddeev, V. E. Zakharov,and A. B. Shabat ([10, 11, 12]) the needed generalization of scattering theory wascarried out for the new operators which has made it possible to solve the direct andinverse problems and to carry through the “Kruskal” integration of the Cauchyproblem for rapidly decreasing (as x → ±∞) initial data. A considerable literaturehas recently been devoted to discovering such new equations and carrying overthe Kruskal mechanism to them. However, even for the original KV equation theperiodic problem has not moved forward. The only new result in the periodic case,which was obtained by the method of Gardner, Green, Kruskal, and Miura, is thetheorem of Faddeev and Zakharov to the effect that the eigenvalues of the operatorL are commuting integrals of the KV equation as a Hamiltonian system (we remarkthat the integrals themselves can be expressed in a one-to-one manner in terms ofthe previously known polynomial integrals I n which are thus also involutive). Thisresult has not been used in an essential way until the present work, but here itplays an important role (cf. §2).The interaction of simple waves [solutions of the type u(x − ct), usually called“solitons”] are of major interest in the theory of the KV equation. This interactionis described by means of so-called “multisoliton solutions” where b(k) ≡ 0 for realk. 1 These solutions decay into solitons for x, t → ±∞ and describe their interactionfor finite t. For this case the Gel’fand–Levitan equations are completely solvable(cf., for example [8, 9]). Another method of obtaining multisoliton solutions hasbeen developed in [13]. However, all these results refer to the case of rapidlydecreasing functions u(x). In the periodic case the solitons u(x − ct) of the KVequation are of a more complicated structure; there are many more of them and their1 I. M. Gel’fand has informed the author that such potentials u(x) were first considered byBargmann.
THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION 3interaction has not been studied at all. In the present paper we propose a method ofstudying certain analogs of the “multisoliton” solutions of the KV equation which,generally speaking, are found to be not only periodic, but also conditionally periodicfunctions u(x) describing the interaction of periodic solitons. Our work is based oncertain simple but fundamental algebraic properties of equations admitting the Laxrepresentation which are strongly degenerate in the problem with rapidly decreasingfunctions (for x → ±∞), and have therefore not been noted. Finally, it is essentialto note the nonlinear “superposition law for waves” for the KV equation whichin the periodic case has an interesting algebraic-geometric interpretation. Thesuperposition law will be discussed in the second part of the work.In conclusion, we call the attention of the reader to the following circumstance:in classical mechanics and mathematics the appearance of integrals in conservativesystems (conservation laws) is almost always related to a Lie symmetry group ofthe problem in question. Other fundamental algebraic mechanisms of integrabilitywere previously unknown. However, there were several exceptions: for example,the Jacobi case (geodesics on a triaxial ellipsoid) or the case of Kovalevskaya (theproblem of the motion of a solid body with a fixed point in a gravitational field).Other exceptional examples are now known. There is not the slightest doubt thatall these cases are the manifestation of a Kruskal-type algebraic mechanism basedon the possibility of a Lax-type representation for these dynamical systems.1. The Schrödinger (Sturm–Liouville) Equations with PeriodicCoefficients. The Monodromy MatrixWe shall first list systematically simple facts which we shall need.Let u(x) be a smooth function where u(x + T ) = u(x), and let L = (d 2 /dx 2 ) +u. We consider on the line the equation Lψ k = λψ k , where λ = k 2 is a realnumber. We consider the pair of linearly independent solutions ψ k (x, x 0 ), ¯ψ k (x, x 0 ),where ψ k (x 0 , x 0 ) = 1, ψk ′ (x 0, x 0 ) = ik, or the pair φ k , ¯φ k , where φ k (x 0 , x 0 ) = 1,¯φ k (x 0 , x 0 ) = i. (The pair ψ k , ¯ψ k is more convenient but is meaningful only fork 2 > 0.) We can define the “monodromy matrix”whereT (k, x 0 ) =( ) a b, a = a(k, x ¯b ā 0 ), b = b(k, x 0 ),ψ k (x + T, x 0 ) = aψ k (x, x 0 ) + b ¯ψ k (x, x 0 ),¯ψ k (x + T, x 0 ) = ¯bψ k (x, x 0 ) + ā ¯ψ k (x, x 0 ),or the analogous matrix in another basis. In the basis φ k , ¯φ k , for example, themonodromy matrix is an entire function of λ = k 2 . The trace of the matrix Sp T =a + ā = 2a R is real, while the determinant is equal to one, det T = |a| 2 − |b| 2 = 1for all real k, since the Wronskian determinant is conserved.The eigenvalues of the matrix T (k, x 0 ) do not depend on the point x 0 and havethe form: µ ± = a R ± √ a 2 R − 1. In particular, we have two cases: µ ± = e ±ip , |a R | ≤1; µ ± = e ±p , |a R | ≥ 1, where p is a real number (a R = cos p for |a R | ≤ 1). The Blocheigenfunctions of the operator L are those solutions of the equation Lf = k 2 f suchthat f(x+T ) = e ±ip f(x), where the number p is called the “quasi-momentum.” Wehave periodic eigenfunctions for e ip = 1 or a R = 1, and antiperiodic eigenfunctionsf(x + T ) = −f(x), where a R = −1. The permitted zones are the regions onthe axis λ = k 2 , where |a R | ≤ 1, and the forbidden zones are the regions on the
Page 1: THE PERIODIC PROBLEM FOR THE KORTEW
Page 6 and 7: 6 S. P. NOVIKOVzeros (the spectral
Page 8 and 9: 8 S. P. NOVIKOVThis proposition has
Page 10 and 11: 10 S. P. NOVIKOV1) The Case n = 1.
Page 12 and 13: 12 S. P. NOVIKOVoru = u(c, d), q =

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

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