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

THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIESEQUATIONS. P. NOVIKOVIntroductionThe Korteweg–de Vries equation (KV) arose in the nineteenth century in connectionwith the theory of waves in shallow water. It is now known (cf., for example,[15]) that this equation also describes the propagation of waves with weak dispersionin various nonlinear media. In reduced form it is writtenu t = 6uu x − u xxx .In 1967 in the well-known work of Gardner, Green, Kruskal, and Miura a remarkableprocedure for integrating the Cauchy problem for this equation was discoveredfor functions u(X) which are rapidly decreasing for x → ±∞. This procedureconsists in the following: we consider the Schrödinger (Sturm–Liouville) operatorL = −(d 2 /dx 2 ) + u; let f(x, k) and g(x, k) be such that Lf(x, k) = k 2 f, Lg(x, k) =k 2 g, whereby f(x, k) → e −ikx for x → −∞ and g(x, y)e −ikx for x → +∞. Wethen have two pairs of linearly independent solutions f, ¯f and g, ḡ and a transitionmatrixf(x, k) = a(k)g(x, k) + b(k)ḡ(x, k),¯f(x, k) = ¯b(k)g(x, k) + ā(k)ḡ(x, k).If the potential u changes in time according to the KV equation, then the coefficientsa and b vary according to the law ȧ = 0, ḃ = 8ik3 b, and for the eigenfunctionf(x, k) there is the equation f ˙ = Af + λf, where A = 4 d3dx− 3 ( u d3 dx + ddx u) ,λ = 4ik 3 . If λ n = +kn 2 = (iκ n ) 2 is a point of the discrete spectrum of the potentialu(x), then ˙κ n = 0 and ċ n = +8κ 3 c n , where c n is the natural normalization of theeigenfunction. These formulas make it possible to reduce the Cauchy problem forthe KV equation with rapidly decreasing functions to an inverse problem of scatteringtheory and to use the results of I. M. Gel’fand, B. I. Levitan, V. A. Marchenko,and L. D. Faddeev (cf. [2, 3, 4]). Subsequently, P. Lax [7] discovered that theKV equation is identical to the operator equation ˙L = [A, L], where L = − d2dx+ u, 2A = 4 d3dx−3 ( u d3 dx + ddx u) , since ˙L is the operator of multiplication by ˙u, while [A, L]is the operator of multiplication by the function 6uu ′ − u ′′′ . In particular, it followsfrom this that the spectrum of the operator L is an integral of the KV equation(this is also true for the periodic problem). This fact reveals the algebraic meaningof the procedure in [5] and is extremely useful for the application of these ideas toother problems. Further, L. D. Faddeev and V. E. Zakharov [8] have shown thatthe KV equation is a completely integrable Hamiltonian system, where the canonicalvariables are 2k π ln |a(k)|, arg b(k), ( κ2 n, ln(b n ), b n = ic da)n dk k=iκ n. In particular,the eigenvalues λ n = kn 2 are commuting integrals of the KV equation also in theperiodic problem [and not only in the case of rapidly decreasing functions u(x)]. A1

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

THE PERIODIC PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION 5In our bases the matrix Q has the form( ) 1 0Q = −ik + iu ( ) 1 −1(basis ψ0 −1 2k 1 −1k , ¯ψ k ),Q = − i ( )1 1− i ( )1 −12 −1 −1 2 (u − k2 )(basis φ1 −1k , ¯φ k ).Thus, the monodromy matrix T can be sought as a periodic solution of Eq. (3)which satisfies the condition det T = |a| 2 − |b| 2 = 1. At the nondegenerate pointsof both spectra |a R | = 1, where b(k n ) ≠ 0 and |b| = ±a I the following equation isobtained from Eq. (3) by direct substitution:(4)φ ′ ± u k sin φ = −2k + u k , k = k n, (3 ′ )where φ = arg b(k, x) for k = k n , a I = ∓|b| ≠ 0. In principle, the nondegeneratepoints of the spectrum k 2 n are determined from the requirement that the latterEq. (3 ′ ) should have a periodic solution. In general, this condition may also include“extraneous roots” k n ; however, for the zero potential u ≡ 0 we see that ∆φ =2πn = 2kT gives the points of the spectrum exactly (analogously for the caseu = const). Formally this equation is applicable only in the nondegenerate case,but in view of the stability of the properties of this equation under small (smooth)variations of the potential u, use can be made of the fact that all the levels becomenondegenerate after almost any small perturbation. From this there follows:Proposition 1.1. 1) For potentials close to zero (or to a constant) the condition ofthe existence of periodic solutions of Eq. (3’) includes all the nondegenerate pointsof the spectrum k n and is not solvable for k n which are not spectral points;2) If a) the function φ(k, x) is known as a function of u for a given potentialu for all k, x, where it is meaningful (i.e., on the entire line with the exceptionof degenerate points of both spectra a I = b = 0), and b) the function (u/k) sin φextends as a smooth function of the variables x, k to all x, k including the degeneratepoints of the spectrum k = k n , then each spectral points satisfies the transcendentalequation∫ x0 +T (2πn = 2k − ux 0k ± u )k sin φ dx. (5)Assertion 1) of Proposition 1.1 was proved above, since for a constant potentialthe condition on the spectral points is exact. For the proof of assertion 2) we remarkthat the function sin φ at all points x, k where it is well defined (i.e., b ≠ 0) dependssmoothly on the potential u(x). If after an arbitrarily small perturbation δu of thepotential u which makes all levels nondegenerate Eq. (5) is not satisfied for given k,then this point k is not a spectral point for the potential u. This implies assertion2) of Proposition 1.1.2. Potentials with a Finite Number of Zones and MultisolitonSolutions of the KV EquationWe recall that for an infinite period T = ∞ where u(x) → 0 for x → ±∞;the monodromy matrix is defined from the transition from −∞ to +∞. The multisolitonsolutions u(x, t) of the KV equation are determined from the conditionb ≡ 0 for real k for these potentials u at any t. In this case a is defined by its

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 ).

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.

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

12 S. P. NOVIKOVoru = u(c, d), q = q(c, d), E = V (c, d), I = D(c, d).In parameter space this is a two-dimensional surface E = E(c, d), I = I(c, d).In general, under these conditions we obtain in phase space compact separatedlevel surfaces E = const, I = const of torus type with one degenerate cycle. Thetrajectories of the KV equation on these surfaces describe the interaction of a periodicsimple wave with a rapidly decreasing soliton. The interaction of two rapidlydecreasing solitons (a two-soliton solution of the problem with rapidly decreasinginitial data) is obtained [up to an additive constant U(c, d)] if still another conditionon the parameters is satisfied under which both cycles on the torus degenerate toa point. This relation has the form of a condition on the parameters under whichthe polynomial det Λ has three distinct roots, two of which are double roots (thetwo zones contract to points).With the exception of this special case, the evolution in time of two-zone potentialsaccording to the Korteweg–de Vries equation is characterized by two constants∆ 1 , ∆ 2 such thatu(x + ∆ 1 , t + ∆ 2 ) = u(x, t).Calculation of these constants ∆ 1 , ∆ 2 in terms of the zone boundaries will begiven in the second part of the work.In the second part of the work we shall study n-soliton solutions in more detail.Does there exist a superposition law synthesizing them from single-solitonsolutions — an algebraic function of pairs of solitons (elliptic functions) whichcontains doubly valued points (roots) and therefore, in general, leads beyond thefield of elliptic functions? In terms of the characteristic polynomials this appearsas follows: there are two solitons u 1 (x, c 1 , d 1 , E 1 ), u 2 (x, c 2 , d 2 , E 2 ) with matricesdet Λ (1) = (k 2 − λ 1 )(k 2 − λ 2 )(k 2 − λ 3 ), det Λ (2) = (k 2 − µ 1 )(k 2 − µ 2 )(k 2 − µ 3 ).Suppose that the roots λ j , µ j are nonmultiple and λ 1 = µ 1 (a condition for thepossibility of composition); the remaining 4 roots λ 2 , λ 3 , µ 2 , µ 3 are all distinct. Thesuperposition law of solitons is such that the 2-soliton potential v = F (u 1 , u 2 ) hasa characteristic polynomial det Λ in the form of the least common multiple of theinitial polynomials det Λ = (k 2 − λ 2 )(k 2 − λ 3 )(k 2 − µ 2 )(k 2 − µ 3 )(k 2 − λ 1 ), whereλ 1 = µ 1 . The correct analog of the amplitude a k for rapidly decreasing functionsis here the quantity det Λ. However, this definition of superposition is ineffective.Completely different representations of the superposition law are possible; we shalldiscuss these in Part II.Remark 5. V. B. Matveev and L. D. Faddeev have informed the author that in1961 N. I. Akhiezer essentially formulated and began the solution of the problemof constructing examples of finite-zone potentials starting from the results of [2, 3]on the inverse scattering problem on the half line. In this work [1] N. I. Akhiezerdeveloped an interesting approach to the construction of finite-zone potentials usingfacts from the theory of hyperelliptic Riemann surfaces. His construction, however,gives for a prescribed zone structure only a finite number of potentials which satisfyspecific parity conditions in x. Theorem 2.1 of the present work gives many moreperiodic and almost-periodic n-zone potentials — they depend on n continuousparameters for a prescribed zone structure. Since the KV equation { ˙u = 6uu ′ −u ′′ }is not invariant under the transformation x → −x, it follows that during evolutionin time other potentials will be obtained from those of Akhiezer which are notcontained in his construction of n-zone potentials. Judging from the work [1],

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