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Non-linear WKB Analysis of the String Equation

arXiv:hep-th/9112074v1 30 Dec 199191/26July 29, 2021Non-linear WKB Analysis of the String EquationF. Fucito⋆Dipartimento di Fisica, Universit`a di Roma II Tor Vergata and INFN,sezione di Roma Tor Vergata, Via Carnevale, 00173 Roma, ItalyA.

GambaDottorato in Matematica, Universit`a diMilano, via Saldini 50, 20133 Milano, ItalyM. Martellini†INFN, sezione di Roma I, piazzale Aldo Moro 5, Roma, ItalyO.

Ragnisco‡Dipartimento di Fisica, Universit`a di Roma I La Sapienza andINFN, sezione di Roma I, piazzale Aldo Moro 5, Roma, ItalyABSTRACTWe apply non-linear WKB analysis to the study of the string equation. Eventhough the solutions obtained with this method are not exact, they approximateextremely well the true solutions, as we explicitly show using numerical simula-tions.

“Physical” solutions are seen to be separatrices corresponding to degenerateRiemann surfaces. We obtain an analytic approximation in excellent agreementwith the numerical solution found by Parisi et al.

for the k = 3 case.To appear in the proceedings of the Research Conference on Advanced QuantumField Theory and Critical Phenomena, held in Como (Italy), June 17-21, 1991 –World Scientific⋆email addresses: vaxtov::fucito and fucito@roma2.infn.it†permanent address: Dipartimento di Fisica, Universit`a di Milano, 20133 Milano, Italyand INFN, sezione di Pavia, 27100 Pavia, Italy; email addresses: vaxmi::martellini andmartellini@milano.infn.it‡email addresses: vaxrom::ragnisco and ragnisco@roma1.infn.it

1. IntroductionDuring the last two years, the partition function of two-dimensional gravity hasbeen set in correspondence with the τ-function of the KdV hierarchy, subjectedto the constraint of the so-called string equation[1].

In the case of pure gravity,described by a matrix model with criticality index k = 2, the string equationreduces to the well known Painlev´e type I equation for the specific heat of thetheory. The string equation can be seen as a perturbation of a stationary KdVequation, and thus solved in a semiclassical approximation[2] around the stationaryKdV solution.

Novikov and Krichever[3] have conjectured that such an approachlead to exact solutions already at 0-th order in perturbation theory.We haveresorted to accurate numerical simulations in order to check this conjecture. Onthe other side, we show that “physical” solutions to the k-th multicritical modelare obtained as separatrices corresponding to degenerate Riemann surfaces.

Weexplicitly compare the k = 3 solution with the one found numerically by Parisiet al.[4]. Purely asymptotic analysis seems to give in this case the same degree ofprecision of the semiclassical approximation in the non-degenerate case.1.1.

The String EquationThe string equation−x +NXj=0(j + 1) tj+1Rj[u(x)] = 0(1.1)(where Rj are the Gel’fand-Dikii differential polynomials) for finite N gives a con-straint which is compatible with the first N flows of KdV. For N →∞this givesa constraint compatible with all of the KdV flows.

As a matter of fact, write thej-th KdV flow as ∂u∂tj = Kj[u(x)] ≡∂∂xRj[u(x)], differentiate (1.1) with respect tox and compute∂∂ts1 +NXj=0(j + 1) tj+1Kj=s Ks−1 +NXj=0(j + 1) tj+1K′j[Ks] =s Ks−1 + K′s[NXj=0(j + 1) tj+1Kj] =s Ks−1 + K′s[−1] = 0(1.2)where K′j[φ](u) ≡∂∂ǫKj(u + ǫφ)|ǫ=0 and we used known properties of the KdV1

flows, in particular (i) commutativity, i.e. K′j[Ks] −K′s[Kj] ≡[Kj, Ks] = 0; (ii)K′s[1] = [Ks, 1] = s Ks−1, expressing the fact that 1 = τ−1 is the first mastersymmetry[5] of the KdV hierarchy.

We note by the way that (1.2) can be writtenin a more satisfactory invariant form[6] and easily generalized to arbitrary master-symmetries of the KdV equation.1.2. The Whitham methodThe Whitham method is the application to non-linear equations of the semi-classical approximation known in the realm of linear equations as WKB method.Let us consider e.g.

the wave equationϕxx −1c2ϕtt = 0(1.3)and the plane wave solutions ϕ = Aei(kx + ωt). If we are studying propagation oflight we have a “small” scale, i.e.

the characteristic length of oscillation, which isabout ǫ = 10−6 times smaller than the natural unit length used by the observer:so we can consider that solutions be locally given by plane waves, while on the“observational” scale the parameters A, k are “slowly” varying:ϕ(x, t) = A(ǫx, ǫt) eiS(ǫx, ǫt)ǫ(1.4)where ∂S∂X = k(X, T), ∂S∂T = ω(X, T), X = ǫx, T = ǫt,∂∂x = ǫ ∂∂X ,∂∂t = ǫ ∂∂T . At0-th order, we find the eikonal equation of geometrical optics: ( ∂S∂X )2 =1c2( ∂S∂T )2.The evolution of k =∂S∂X gives the paths of the “rays”; the eikonal equation isequivalent tok2 = ω2c2∂k∂T = ∂ω∂X(1.5)In other words, we started with a class of exact solutions ϕ(x, t; A, k, ω) = Aei(kx + ωt)of (1.3) and passed from the precise description of the oscillation process to theapproximate description of the “slow” variation of parameters k, ω given by (1.5);as a matter of fact we averaged over the rapid variation of the function ϕ and choseto observe only “secular” variations.

The same method could have been readilyapplied to non-linear equations if we had2

i) a family of exact solutions depending on an adequate number of parametersE1, . .

. , E2g+1;ii) a way of “averaging out” fast oscillations, in order to obtain an analog of thesecond equation in (1.5), which we will call “Whitham equation”.This is the case for the KdV equation[7] and more generally for equations thatcan be written in the form⋆∂L∂t −∂A∂y + [L, A] = 0,(1.6)that is, as the compatibility condition for the existence of a solution ψ of the linearsystemLψ = ∂ψ∂yAψ = ∂ψ∂t(1.7)where L = Pnj=0 uj(x, y, t) ∂j∂xj , A = Pmk=0 vk(x, y, t) ∂k∂xk are differential operatorswith scalar or matrix coefficients: as is known, these equations admit large sets ofexact solutions (the so-called g-zone solutions) expressed in term of the Riemannθ function.In particular, the KdV equation4ut = uxxx −6uux(1.8)can be written as Lt = [L, A] withL = −∂2 + u(x, t)A = ∂3 −32u(x, t)∂−34ux(x, t),∂≡∂∂x(1.9)and admits the “cnoidal wave” solutionu(x, t) = ℘(x −vt; g2, g3)(1.10)where ℘is the Weierstrass elliptic function and g2, g3 are arbitrary constants.Roughly speaking, the Whitham method consists in slowly varying the constants⋆[L, A] = LA −AL is the usual commutator of differential operators, while∂∂y and∂∂t aresupposed to act on the coefficients uj, vk.3

g2, g3, that is in finding the correct dependence g2 = g2(X, T), g3 = g3(X, T),in order to approximate either (i) new solutions of (1.8) (corresponding e.g. tonon-periodic initial data!) or (ii) solutions to the perturbed equation4ut = uxxx −6uux + ǫK(x).

(1.11)The cnoidal wave solution is a so-called 1-zone (g = 1) solution; general g-zonesolutions are given byu(x) = −2 ∂2∂x2 log θ(Ux|B) + C(1.12)where the exact form of U = U(Ej), B = B(Ej), C = C(Ej) is given in theAppendix.2. The Whitham methodLet’s start with perturbation theory for a non-linear equation (e.g.

the sta-tionary KdV equation):uxxx −12 uux = ǫK(2.1)and look for solutions in the formu = u0 + ǫu1 + ǫ2u2 + · · ·(2.2)withuk = uk(t|X) = uk(S(X)ǫ|Ej(X)),X = ǫx,k = 0, 1, 2, . .

. (2.3)where 1ǫS1(X), .

. .

, 1ǫSg(X) are rapidly oscillating functions that will be determinedin the following, and uk depends on some parameters Ej = Ej(X) which on theirturn are slowly varying with x, and will be determined in the sequel. Substitute(2.2) and (2.3) in (2.1).Note that the form of the functions uk implies∂∂x =4

∂S∂X · ∂∂t + ǫ ∂∂X . Now take the various order in ǫ: e.g.

at order O(1) we get( ∂S∂X · ∂∂t)3u0(t|X) −12 u0(t|X)( ∂S∂X · ∂∂t)u0(t|X) = 0(2.4)Let L = ∂S∂X · ∂∂t = Pgj=1∂Sj∂X∂∂tj , and go on writing equations at all orders in thecompact form:O(1) :L3u0 −12 u0 Lu0 = 0O(ǫ) :L3u1 −12 L(u0u1) = F1 + KO(ǫk) :L3uk −12 L(u0uk) = Fk,k = 2, 3, 4, . .

. (2.5)Note that only the first equation is non-linear, and that equations for the orderO(ǫk), k = 1, 2, 3, .

. ., differ only in the non-homogenous term Fk.This termwould not be present in na¨ıve perturbation theory, and comes from differentiationwith respect to slow variables: for instance u0x = ( ∂S∂X · ∂∂t + ǫ ∂∂X )u0 producesthe term ∂u0∂X of order ǫ giving contribution to F1, together with derivatives like∂∂xSj coming from u0xx, etc.

(the explicit form of the term F1 is given in theAppendix). The idea is that at order O(ǫ) the term F1 should compensate forthe perturbation term K in order that the “correction” u1 be bounded, and thisgives equations for the correct dependence of the parameters Ej = Ej(X) on theslow variables; otherwise, we can think of averaging the O(ǫ) equation in (2.5)over the fast variables t1, .

. .

, tg, thus remaining with the only variable X. A thirdpoint of view is that the O(ǫ) equation has the form Lu1 = F1 + K, where L is alinear operator: this means that periodic solutions exist iffF1 +K is orthogonal toKer(L†).

The three points of view are all equivalent and give the same Whithamequations for Ej = Ej(X).Equation (2.4) has been written in explicit form in order to make clear an im-portant point: at any order the variable X appears as a parameter, while equationsare only in the differential variables t1, . .

. , tg.⋆Moreover, putting∂S∂X = U(Ej(X)),t = Ux(2.6)we see that (2.4) becomes equivalent to the unperturbed u0xxx −12u0u0x = 0, so⋆This explains why the variables x and X are usually treated as independent variables in thetwo-scale method, and makes rigorous the usual argument of “freezing” the slow variable.5

we immediately get the form of u0†:u0(t) = −2(U · ∂∂t)2 log θ(t|Ej) + C(Ej)(2.7)that is, in the genus g = 1 case, u0(t1) = ℘(2ω · t1; g2, g3).Functions of the form (2.7) are periodic in t1, . .

. , tg; moreover, we have seenthat the variables t and X must be regarded as independent: so we can averageboth sides of the O(ǫ) equation in (2.5) over t1, .

. .

, tg, thus being left with the onlyvariable X. The equations for E1(X), .

. .

, E2g+1(X) thus obtained in the case ofthe stationary KdV equation have the form∂∂X√(E−E1(X))(E−E2(X))(E−E3(X)) dE = −6E+r(E1(X),E2(X),E3(X))√(E−E1(X))(E−E2(X))(E−E3(X))dE(2.8)where r(E1, E2, E3) =R E2E3EdE√(E−E1)(E−E2)(E−E3)/R E2E3dE√(E−E1)(E−E2)(E−E3) andE is a dummy variable.In general, the Whitham equations for KdV are differential conditions on thefunctions p = p(E, Ej(X, T)), Ω= Ω(E, Ej(X, T)) (quasi-momentum and quasi-energy) appearing in the E →∞leading term ψ ≃epx + Ωt of the solutions tothe associated linear system (1.7). For KdV they take the form∂Ω∂X −∂p∂T = ⟨ψ†Kψ⟩⟨ψ†ψ⟩∂p∂E(2.9)Equation (2.8) can equivalently be written as ∂Ω∂X = −6 ∂p∂E.

A complete derivationand explanation of these formulas is given in the Appendix.† Actually, this is the complex form of u0. A rigorous treatment needs the use of real variablest1, .

. .

, t2g. In the g = 1 case we would get u0(t1, t2) = ℘(2ω · t1 + 2ω′ · t2; g2, g3).6

3. The Painlev´e type I equationThe Painlev´e equationu′′ = 6(u2 −x)(3.1)(which is obtained from (1.1) when we leave only t3 ̸= 0) can be differentiated oncewith respect to x and seen as a perturbation of the (integrable) stationary KdVequation:u′′′ = 12uu′ + ǫK(3.2)with ǫ = 1, K = −6.

An asymptotic relation between the two equations is obtainedthrough the change of variables u(x) = √ξ v(ξ)ξ(x) = 45x5/4. (3.3)It comes out(2.9) ↑[8] that solutions of (3.1) are asymptotic to functions of theform u(x) = ℘(45x5/4; 12, g3); nothing is said about the parameter g3.

We will nowresort to a finer analysis of the problem.We are here concerned(2.9) ↑[9] with the case of the torus of equation (2w)2 =4E3 −g2E −g3 = 4(E −E1)(E −E2)(E −E3); we distinguish two cases; for∆= g32 −27g23 > 0 we have real roots E1, E2, E3 and two periods 2ω and 2ω′ whichare respectively real and pure imaginary(2.9) ↑[10]:2ω = 2E2ZE3dE√4E3−g2E−g3 =2K(m)√E1−E3,2ω′ = 2E1ZE2dE√4E3−g2E−g3 =2iK(1−m)√E1−E3 ,(3.4)whereK(m) =π2Z0dφp1 −m sin2 φ,E(m) =π2Z0q1 −m sin2 φ dφ,m = E2 −E3E1 −E3(3.5)are standard elliptic integrals of the 1st and 2nd kind, respectively, and m is calledthe Jacobi modulus. We remind that the solutions of (3.2) with ǫ = 0 can be7

expressed in term of the Weierstrass elliptic function (see Ref. 10)℘(x, g2, g3) = −∂2∂x2 log θ1( x2ω) −ηω.

(3.6)The Whitham equations for (3.2) take the form∂w∂X dE = −6dp(3.7)or, explicitly,∂∂Xp4E3 −g2E −g3 dE ≡−∂g2∂X E −∂g3∂X2p4E3 −g2E −g3dE = −6E + r(X)p4E3 −g2E −g3dE(3.8)giving∂g2∂X = 12,∂g3∂X = 12 r(X). (3.9)The first equation is readily integrated, giving g2 = 12 · X + const; moreover,the second too is integrable by quadratures, because ∂w∂X dE already has the samebehaviour as −6dp for E →∞, so we need only to impose the normalizationcondition (compare with (5.4))ImIw dE= Re1iE1ZE2q4E3 −12XE −g3(X) dE= const = h(3.10)in order to get ImH ∂w∂X dE =∂∂X ImHwdE = 0 and∂∂X wdE = −6dp.

But (3.10)can be solved explicitly for g3, yielding the correct dependence g3 = g3(X).In order to do this it is convenient to introduce new parameters (λ, m) in placeof (g2, g3):λ = (E1 −E3)29,m = E2 −E3E1 −E3,g2 = 12λ(1 −m + m2),g3 = 4λ3/2(2 −3m −3m2 + 2m3),E1 =√λ(2 −m),E2 =√λ(−1 + 2m),E3 =√λ(−1 −m),E1ZE2p4E3 −g2E −g3 dE ≡iλ5/45√3Φ(m);(3.11)8

giving (for ∆> 0) the solutionλ(m) = 5√3Φ(m)! 45.

(3.12)In Fig. 1 we show the form of the resulting function x = x(m), both for the cases∆> 0 and ∆< 0 (for graphic convenience we plotted x versus1m instead of m).The pole in the ∆< 0 region gives rise to two distinct curves in the space ofparameters (g2, g3), which we plotted in Fig.

2. Knowing the properties(2.9) ↑[11]of the Weierstrass elliptic function ℘we see that for x = g212 > 0 the Whithammethod gives us either small oscillations (curve in the ∆> 0 region) or a sequenceof poles (lower part of the curve, lying in the ∆< 0 region).

If we start withoscillatory behaviour at +∞and go in the direction of decreasing x, we ultimatelyreach ∆= 0 where a transition to polar behaviour occurs. Note that for x < 0 wecan have only polar behaviour.

The two parts of the curve correspond to solutionshaving or not having poles at +∞. The “physical” solutions will be seen to beseparatrices lying between these two kinds of solutions.The function Φ(m) is computed reducing the integral in (3.10) to standardelliptic integrals(2.9) ↑[12].

We getR E2E3dE2w =K(m)√3λ1/4,R E1E2dE2w = K(1 −m)√3λ1/4,R E2E3EdE2w= λ1/4√3 K(m) −√3λ1/4E(m),R E1E2EdE2w= λ1/4(−1−m)√3iK(1−m) +√3λ1/4iE(1−m),and finallyE2ZE3p4E3 −g2E −g3 = 25λ5/4√3 (18(2 −3m + m2)K(m) −36(1 −m + m2)E(m)),E1ZE2p4E3 −g2E −g3 = 25iλ5/4√3(−18(m + m2)K(1 −m) + 36(1 −m + m2)E(1 −m)),Φ(m) = −18(m + m2)K(1 −m) + 36(1 −m + m2)E(1 −m). (3.13)This procedure gives us also the “actions” S1, S2:S1 = −13E1ZE2p4E3 −g2E −g3 dE,S2 = −13E2ZE3p4E3 −g2E −g3 dE, (3.14)9

because from ∂w∂X dE = −6dp it follows∂S1∂X = U1 = 2E1ZE2dp,∂S2∂X = U2 = −2E2ZE3dp. (3.15)Take S = S1 + τS2 as prescribed by (2.6), (5.15) and (5.16), where τ = iK(1−m)K(m)is the 1-dimensional analog of the period matrix B, and find from (3.13) and theLegendre relation EK′ + E′K −KK′ = π2 (see Ref.

10):S = i√35 λ5/4 · 12(1 −m + m2)K(m)= 45i x2ω. (3.16)The Whitham solution has the formu0(x) = −℘(45ix + ω; 12x, 4I(x)x3/2) = −√x℘(45ix5/4 + ω; 12, 4I(x));(3.17)(We have used here the homogeneity property ℘(tx; g2t4 , g3t6 ) =1t2℘(x; g2, g3), t =x−1/4; the ω-shift is needed in order to get non-singular solutions).

We thus recoverthe “Boutroux” asymptotic form (3.3), but with a more preciseI(x) = I(m(x)) = 2 −3m −3m2 + 2m3(1 −m + m2)3/2. (3.18)Asymptotic analysis(2.9) ↑[13] shows that the amplitude of the oscillatory solu-tions of (3.1) decreases at +∞as1x1/8.

This feature, not recovered by the simpleBoutroux-type solution, is obtained from the fine tuning realized by the term I(x).Just use the expansionK(m) = π2 (1 + m4 + 964m2 + · · ·),E(m) = π2(1 −m4 −364m2 + · · ·), (3.19)for m →0 (see Ref. 10).

From (3.11), in the limit m →1, we get x = g212 ≃λ(m),Φ(m) ≃135π8 (1 −m)2, λ(m) ≃(8√3h27π )4/5(1 −m)−8/5, −E1, −E2 ≃√x, E1 −E2 ≃q8h√3πx−1/8 (Notice that the function ℘(ix + ω) oscillates between the extremalvalues −E1 and −E2, see Ref. 11).10

If we want to examine the case x < 0, corresponding to ∆< 0, formulas (3.11)are no longer convenient, and we resort to the following real parameterizationm∗= 1m = 12 + iσ,λ∗= m2λ = −ρ2,iλ5/4Φ(m) = −iλ5/4∗Φ(m∗)(3.20)The solution corresponding to (3.12) in the complex case becomesλ∗(m∗) = − 5√3hRe(√iΦ(m∗))!4/5. (3.21)A numerical computation shows that the function Φ(m∗) ≡Φ(12 + iσ) ≡Φ(σ) hasa zero for σ = σ0 = −0.231026398427 .

. ..In the case ∆< 0 real and pure imaginary combinations of periods are givenby ω = K( ˜m)√H , ω′2 = iK(1−˜m)√H, where H2 = 3E21 −g24 = |3−4σ21+4σ2|, ˜m = 12 +σ√1+4σ2(see Ref.

10).With this parameterization we recover the numerical result ofRef. 13, asserting that the distance of poles goes asymptotically as˜cx1/4, with ˜c =7.276726 .

. .

; as a matter of fact we find c = 2ω′2i=2√3K(12 −σ0√1+4σ20 )(3−4σ201+4σ20 )1/4 =2.970711275212 . .

., which exactly coincides with the result of Ref. 13 after therescaling ˜c =√6 c (due to our factor 6 in (3.1)).

An analogous reasoning for x > 0gives the period of the oscillatory solutions going ascx1/4, with c = 2ω′i=π√3.In Fig. 3 we show the approximate solution (with h = 1), together with anexact numerical solution obtained with the Runge-Kutta method (the Painlev´eequation is satisfied with an error of 10−14).

Fig. 4 is a magnification of the regionaround the zero where the approximation seems to be less effective.

We see thatthe solution we are considering seems to be out of phase with respect to the exactnumerical one. This is no surprise because we really have neglected a phase: theWhitham method made variating the “constants” g2 and g3, but in ℘(x + c; g2, g3)we have also a third integration constant c, that we assumed to be zero.

We dobelieve that an equation for this phase can be deduced(2.9) ↑[14], and its behaviourwill be discussed in a forthcoming paper. Here we have just made a fit of such aphase on the side of the positive x, getting a correction δφ(x) ≃.175(x−.0098).538 .

Wehave also done the same on the negative x side but we got a really tiny correctionthat we chose to neglect. The fit of the “experimental” data is shown in Fig.

5,while Fig. 6 shows the effect of putting in the correction by hand (for x ≃0 we letδφ(x) die smoothly).11

4. Degenerate solutionsThe solutions u(x) of Painlev´e-like equations represent specific heats of the ran-dom matrix models, which in the planar limit must satisfy the boundary conditionu(x) ≃x1/k (x →∞) for scaling arguments of the partition function: Z ∼x−γ+2,where γ is the string susceptibility.

Thus u(x) ≡∂2F∂x2 ∼∂∂xx−γ+1 ∼x−γ, and theresult comes from the fact that γ = 1k in the proximity of the critical point. Thesituation is common to other non-linear physical models in the critical r´egime.

Itis well-known that the problems of mathematical physics must be complementedby boundary conditions, and that the boundary conditions contain in some sensethe physics of the problem. In our particular case we come to the request that thesolutions to the string equation for the k-th multicritical model (which is obtainedfrom (1.1) putting all tj = 0, except tk+1) must satisfy the physical constraintu(x) ≃x1/k for x →+∞.

However, the Whitham method gave us either smalloscillations modulating over −√x (see Fig. 3), or solutions with poles.

The onlypossibility to get non-periodic solutions is to considerate degenerate Riemann sur-faces, where the length of the bands is sent to 0. These solutions are degeneratecases of the periodic solutions and are themselves unstable separatrices, lying be-tween the two sets of solutions with poles and without poles for x →+∞.All we have to do is to compute the spectral curve corresponding to the givenk-th stationary KdV equation and imposing the coincidence of the pairs of branchpoints; g = k −1 conditions are found by requesting that dw = dEg+ 12 + O(1); onemore condition comes from fixing the periods of the solutions at x = ±∞; the lastg conditions come from the request that the branch points coalesce in pairs.For the k = 2 case we require dw ≃dE3/2 + O(1), givingE1 + E2 + E3 = 0E1E2 + E2E3 + E3E1 = c.(4.1)The asymptotic condition fixes c = 3, and we ask for E2 = E3.

This gives E2 =E3 = −E12 = 1. The θ function degenerate to a combination of hyperbolic functionsand we finally getu0(x) =( √x (1 −3(cosh(√3· 45x54 ))2),x ≥0,√−x (℘(45(−x)54; −12, −I(12 + iσ0)),x < 0.

(4.2)(In the x ≤0 case the surface does not degenerate, but I(x) →I(12 +iσ0) = const,where σ0 is the constant introduced in the previous section). The function u(x) isplotted in Fig.

7. In x = 0 we get a cusp as we are trying to connect at finite xtwo asymptotic solutions: a smoother curve would probably require a phase-typecorrection as suggested in the previous section.12

In the k = 3 case the condition dw = dE5/2 + O(1) givesP5i=1 Ei = 0Pi

(4.3)where a = (30)14 cos ϑ2, b = (30)14 sin ϑ2, ϑ = arctan 1√5. (We checked that this so-lution satisfies the higher order stationary KdV equation).Fig.8 shows thatu0(x) = 3√x · v0(67x76) approximate very well the form of the solution found nu-merically in Ref.

4, except that in the proximity of x = 0. (In the graph we haveshifted z1 7→z1 + 14, z2 7→z2 −14 for x < 0 to match the phase of Ref 4.

Theresulting function still satisfies the KdV equation).Acknowledgements:One of us (A. G.) wants to thank I. Krichever for having clarified to him severalparts of his work, and F. Magri for useful discussions.5. AppendixIn sections 1, 2, 3 we review some facts about algebraic geometry and KdVequations, mainly for notational convenience.

In section 4 we report the proof ofKrichever’s theorem, following Ref. 2.13

5.1. Complex curvesThe algebraic equationw2 = E2g+1 + a1E2g−1 + a2E2g−2 + .

. .

+ a2g−1E + a2g ≡p2g+1(E)(5.1)defines a curve Γ in the complex plane of the variables (E, w). The curve is com-pactified at the ∞and is known to be topologically equivalent to a compact surfacewith g holes.

If the polynomial p2g+1(E) has 2g+1 real roots E1, . .

. , E2g+1, we candraw them on the complex plane and use solid lines for the segments (E2k−1, E2k),where the square root w = ±pp2g+1(E) = ±p(E −E1)(E −E2) · · ·(E −E2g+1)takes real values.

In spectral theory these are the forbidden zones of the spectrum.Coordinates on Γ are given byu = Ealmost everywhere,u = pE −Ejin the neighborhood of Ej,u =1√Ein the neighborhood of ∞. (5.2)Consider integrals of the formRΩk =REkdE2√(E−E1)...(E−E2g+1), k = 0, 1, 2, .

. .

Us-ing (5.2) it is easy to see that Ω0, . .

. , Ωg−1 are everywhere non-singular, whileΩg, Ωg+1, .

. .

have poles at the infinity of order 2, 4, . .

., etc.To fix a basis of differentials we chose first a canonical(2.9) ↑[17] basis ofpaths a1, . .

. , ag and b1, .

. .

, bg on Γ and take ω1, . .

. , ωg as linear combinationsof Ω0, .

. .

, Ωg−1 satisfying the normalization conditionIakωj ≡2E2kZE2k−1ωk = δjk,j, k = 1, . .

. , g(5.3)We are left with the b-periods, forming a g × g matrix Bjk = Bkj =Hbk ωj ≡2R E2g+1E2kωj.

Differentials with poles of order 2j will be indicated by ω(2j−1) forfuture commodity, and can be fixed by requiring that they go at the infinity asω(2j−1) ≃dEj−12 +O(1). the arbitrariness on the holomorphic tail can be eliminatedby imposing the 2g real conditionsImIakω(2j−1) = 0,ImIbkω(2j−1) = 0,k = 1, .

. .

, g.(5.4)(Another standard choice of the normalization is to impose instead of (5.4) the g14

complex conditionsHak ω(2j−1) = 0, k = 1, . .

. , g).

We will also use the notationdp = ω(1) ≃d√E,dΩ= ω(3) ≃dE3/2,E →∞;(5.5)these are the differentials of the quasi-momentum p(E) and quasi-energy Ω(E),fundamental in the theory of the KdV equation (see 5.11); p(E) =R E∞dp andΩ(E) =R E∞dΩare multivalued functions with periods Uj =Hbj dp, Wj =Hbj dΩ,j = 1, . .

. , g. Note that p(E) and Ω(E) are uniquely determined by the asymptoticbehaviour and the normalization conditions.5.2.

Functions on the surface ΓThe Abel map P 7→A(P) ≡t(R P∞ω1, . .

. ,R P∞ωg) maps any point P on Γ onthe g-dimensional torus Cg/{period lattice}.

The Abel map can be inverted bymeans of the Fourier seriesθ(z|B) =Xn∈ζgeπin·B·n+2πin·z(5.6)defining (for positive definite ImB) the Riemann θ function, which has the period-icity properties θ(z + ej) = θ(z), θ(z + Bej) = e−πiBjj−2πizjθ(z) (see Ref. 17).

Asa matter of fact, a theorem of Jacobi asserts that the functionf(P; P1, . .

. , Pg) = θ(PZ∞ωk −gXj=1PjZ∞ωk + Kk)(5.7)(where K = (Kk) is a certain constant vector) has exactly g zeroes P1, .

. .

, Pg.This theorem gives an analog of the development of a rational function in simplefractions.5.3. The KdV equationThe KdV equation 4ut = uxxx −6uux admits the Lax representation Lt =15

[L, A], with L, A, given by (1.9), and has exact solutions of the formu(x, t) = −2 ∂2∂x2 log θ(Ux + Wt + z0|B) + C(B)(5.8)where (see (5.5))Uj =Ibjdp,Wj =IbjdΩ,j = 1, . .

. , g.(5.9)This can be seen as follows(2.9) ↑[18].

The equation Lt = [L, A] is the compatibilitycondition for the system of linear equations( Lψ = EψAψ = ∂ψ∂t . (5.10)For E ≃∞we get L ≃d2, A ≃∂3, so the asymptotic form of the commoneigenvectors ψ will beψ(x, t; E) = ep(E)x + Ω(E)t·φ(x, t; E),withp(E) ≃√E,Ω(E) ≃E3/2asE 7→∞.

(5.11)The exact form of φ(x, t) is (compare with (5.7); see Ref. 9)φ(x, t; E) =θ(R E∞ωk −Pgj=1R E(Pj)∞ωk + Ukx + Vkt + Kk)θ(R E∞ωk −Pgj=1R E(Pj)∞ωk + Kk)(5.12)for given Γ and P1, .

. .

, Pg on Γ. It is easy to verify that ψ = epx + Ωt · φ is aone-valued function of E. The function ψ is uniquely determined by the behaviourat infinity (ψ ≃eE1/2x+E3/2t) and the position of the g poles P1, .

. .

, Pg, as can beeasily seen with the help of the Riemann-Roch theorem. For E ≡k2 →∞we getψ(x, t; E) = cekx + k3t · (1 + ξ1(x, t)k+ ξ2(x, t)k2+ · · ·)(5.13)If ψ satisfies Lψ = Eψ, Aψ = ∂ψ∂t , then we can collect terms of the same order in 1kand ξ1, ξ2, .

. .

should satisfy some equation at any order. It is easily seen that thefirst of these equation gives u(x, t) = 2∂ξ1∂x (x, t), so for this choice of the potentialu we get (5.10) verified at order O( 1k).

But note that (L −E)ψ, (A −∂∂t)ψ againhave the behaviour ekx + k3t and poles at P1, . .

. , Pg, so they must again have theform (5.13) with no O(1) term: this means that they are identically zero; so (5.11)and (5.12) give an exact solution to (5.10).

Developing (5.12) at 1st order in 1k weget precisely formula (5.8) for u = 2 ∂∂xξ1(x, t).16

5.4. Krichever’s theoremWe will here report the form found by Krichever for the Whitham equationsof systems of KP type, following Ref.

2.It is convenient to consider directlythe general case of equations of the form (1.6), as for instance the KP equation3uyy +∂∂x(4ut −6uux + uxxx) = 0. (1.6) is the compatibility condition for theexistence of a solution ψ of the linear system Lψ = ∂ψ∂y , Aψ = ∂ψ∂t .

The commoneigenvector ψ will be given here using a particular real normalization, necessaryfor the successive averaging procedure:ψ(x, y, t; P) = epx + Ey + Ωt + s · t · φ(Ux + Vy + Wt + t, P)(5.14)here the spectral curve Γ is no more hyperelliptic, and consequently the spectralparameter E becomes itself a multi-valued function E(P) of the point P on thesurface; the functions p(P), E(P), Ω(P) are normalized by requiring that they havepure imaginary periods along all cycles a1, . .

. , ag, b1, .

. .

, bg; U, V, W are the real2g-dimensional vectors of periods of the multi-valued functions p(P), E(P), Ω(P):U = t(Ib1dp, . .

. ,Ibgdp, −Ia1dp, .

. .

, −Iagdp),etc. ;(5.15)t1, .

. .

, t2g are auxiliary “times” needed for further procedure of averaging; s1, . .

. , s2gare the corresponding “momenta”, not needed in what follows; φ is a periodic func-tion with period 1 with respect to all of the 2g variables x1, .

. .

, xg, y1, . .

. , yg (Ais the Abel map):φ(xy; P) = c e2πiA(P) · y · θ(A(P) + x + By + z0)θ(z0)θ(A(P) + z0)θ(x + By + z0)(5.16)We will also need the solutions to the adjoint system ψ†L = −∂ψ†∂y , ψ†A = −∂ψ†∂twhere differential operators written on the left should be intended according toψ†(u ∂j∂xj ) ≡(−∂∂x)j(ψ†u).

(formal integration by parts). The left and right actiondiffer only for a complete derivative:(ψ†L)ψ = ψ†Lψ + ∂∂x(ψ†L(1)ψ) + ∂2∂x2 (ψ†L(2)ψ) + · · ·(5.17)as can readily be seen by repeated applications of the Leibnitz rule.

Here L(r) ≡(−1)rr!drd(∂)r L (formal derivation with respect to the symbol ∂: for instance, A(1) =−3∂2 + 32u).17

Solutions to the adjoint system can be written in the form ψ†(x, y, t; P) =e−px−Ey−Ωt−s·t · φ†(−Ux −Vy −Wt −t, P)The application of the Whitham method to equations of the KP type is allowedby the possibility of averaging identities by means of some “ergodic theorem”: theaverage of periodic functions φ(t1, . .

. , t2g; Ik) with period 1 with respect to allarguments is given by ⟨φ⟩≡Rφ(t)d2gt, and for a generic vector U it coincides withthe limit ⟨φ⟩x ≡limx0→+∞12x0R x0−x0 φ(Ux)dx, because the line Ux winds denselyon a 2g-dimensional torus.

Note that the derivative of φ along any direction haszero average value:⟨∂∂xφ(Ux)⟩= U · ⟨∂φ∂t ⟩=2gXj=1Uj1Z0∂φ∂tjdt1 . .

. dt2g = 0(5.18)Note also that the functions ∂φ∂Ik are again periodic, because in our real normaliza-tion the periods have the fixed value 1 (not depending on the Ik).Solutions of the KP equation have the formu(x, y, t) = −2 ∂2∂x2 log θ(Ux + Vy + Wt + z0|B) + C(B)(5.19)Thus, we look for a semiclassical approximation in the formu0(x, y, t) = −2 ∂2∂x2 log θ(S(X, Y, T)ǫ|I(X, Y, T)) + C(X, Y, T),where ∂S∂X = U, ∂S∂Y = V, ∂S∂T = W, ψ0 = e1ǫ s·Sφ(Sǫ ).

The operators L0, A0 obtainedsubstituting u 7→u0 are taken as first terms of the asymptotic seriesA = A0 + ǫA1 + · · · ,L = L0 + ǫL1 + · · ·,(5.20)Let us introduce the notationˆ∂∂τ ≡∂I∂τ · ∂∂I ≡Xj∂Ij∂τ∂∂Ij;(5.21)then the substitution u 7→u0 implies∂∂x 7→∂∂x + ǫ ˆ∂∂X ; taking terms of order O(ǫ)18

in (1.6) we get the linearized equation∂L1∂t −∂A1∂y + [L0, A1] + [L1, A0] = K −F,(5.22)where F is the term due to derivation with respect to slow variables: this termmust be adjusted in order to compensate for K. We readily find for F the formF =ˆ∂L∂T −ˆ∂A∂Y + (L(1) ˆ∂A∂X −A(1) ˆ∂L∂X ). (5.23)Now use ∂ψ∂y = Lψ, ∂ψ†∂y = −ψ†L, ∂ψ∂t = Aψ, ∂ψ†∂t = −ψ†A, and (5.17):∂∂t(ψ†L1ψ)−∂∂y(ψ†A1ψ) = ψ†∂L1∂t −∂A1∂y +[L0, A1]+[L1, A0]ψ+ ∂∂x(.

. .) (5.24)Thus, the average of the left hand side of (5.22) comes out to be zero (being theaverage of a total derivative) and we obtain the Whitham equations in the implicitform⟨ψ†Kψ⟩= ⟨ψ†Fψ⟩(5.25)Explicit computing of the Whitham term ⟨ψ†Fψ⟩will give us the final form.

Takerespectively(i) a curve I = I(τ) in the space of parameters (P = const, t = const);(ii) a curve P = P(τ) moving the point P on the surface Γ (I = const, t =const);(iii) a curve t = t(τ) moving only the “times” ti (I = const, P = const).Correspondingly, we get L(τ), A(τ), etc., andψ(τ) = ep(τ)x + E(τ)y + Ω(τ)t + s · t · φ(U(τ)x + V(τ)y + W(τ)t + t)ψ† = e−px −Ey −Ωt −s · t · φ†(−Ux −Vy −Wt −t)(5.26)Now compute∂∂ττ=0 ψ†ψ(τ) in all the three cases and use a point to denote deriva-19

tion with respect to τ:(i)∂∂τ0ψ†ψ(τ) = ( ˙px + ˙Ey + ˙Ωt) ψ†ψ + ( ˙Ux + ˙Vy + ˙Wt) · ψ†∂ψ∂t + ˙I · ψ†∂ψ∂I ,(ii)∂∂τ0ψ†ψ(τ) = (dp x + dE y + dΩt) φ†φ,(iii)∂∂τ0ψ†ψ(τ) = s · ˙t φ†φ + φ†∂φ∂t · ˙t;(5.27)Using ∂ψ∂t = Aψ, ∂ψ†∂= −ψ†A and (5.17), see that∂∂t(ψ†ψ(τ)) = ψ†(A(τ) −A)ψ(τ) −∂∂x(ψ†A(1)ψ(τ)) + ∂2∂x2(. .

. )(5.28)Deriving the left-hand side with respect to τ and using (i) we get for instance∂∂τ0∂∂t(ψ†ψ(τ)) = ˙Ωψ†ψ + ˙W · ψ†∂ψ∂t+ {( ˙px + ˙Ey + ˙Ωt) ∂∂t(ψ†ψ)+ ( ˙Ux + ˙Vy + ˙Wt) · ∂∂t(φ†∂φ∂t ) + ˙I · ∂∂t(φ†∂φ∂I )}(5.29)Now fix x, y, t and average upon d2gt: the terms in braces are linear combinationsof total derivatives with constant coefficients and thus vanish, giving⟨∂∂τ0∂∂t(ψ†ψ(τ))⟩= ˙Ω⟨ψ†ψ⟩+ ˙W · ⟨ψ†∂ψ∂t ⟩;(5.30)this passage contains the essence of the method of averaging.

We can now go onderiving both sides of (5.28) with respect to τ, applying (i), (ii), (iii) and findingafter averaging the following identities:(i) :˙Ω⟨ψ†ψ⟩+ ˙W · ⟨ψ†∂ψ∂t ⟩= ⟨ψ†∂A∂τ ψ⟩−˙p⟨ψ†A(1)ψ⟩−U · ⟨φ† ˆA(1)∂φ∂t ⟩(ii) :dΩ⟨ψ†ψ⟩= −dp⟨ψ†A(1)ψ⟩(iii) :0 = ⟨ψ†∂A∂tj⟩(5.31)Note that ψ†ψ = φ†φ, that we posed ˆA(1) = e(−px −· · ·)A(1)e(px + · · ·), and that20

(iii) implies⟨ψ†∂A∂τ ψ⟩≡⟨ψ†(ˆ∂A∂τ A + ( ˙Ux + ˙Vy + ˙Wt) · ∂A∂t )ψ⟩= ⟨ψ† ˆ∂A∂τ ψ⟩(5.32)We get analogous identities for E, V, L if we derive instead of (5.28) the identity∂∂y(ψ†ψ(τ)) = ψ†(L(τ) −L)ψ(τ) −∂∂x(ψ†L(1)ψ(τ)) + ∂2∂x2 (. .

. )(5.33)Now letting τ = Y, T (remember that Y, T are independent of y, t), we can rewrite(5.31) and the analogous identities for L as−⟨ψ† ˆ∂A∂y ψ⟩= −∂Ω∂Y ⟨ψ†ψ⟩−∂W∂Y · ⟨φ†∂φ∂t ⟩−∂p∂Y ⟨ψ† ˆA(1)ψ⟩−∂U∂Y · ⟨φ†∂φ∂t ⟩⟨ψ† ˆ∂L∂t ψ⟩= ∂E∂T ⟨ψ†ψ⟩+ ∂V∂T · ⟨φ†∂φ∂t ⟩+ ∂p∂T ⟨ψ†ˆL(1)ψ⟩+ ∂U∂T · ⟨φ†∂φ∂t ⟩(5.34)The last term we need comes from the identity∂∂t(ψ†L(1)ψ(τ))−∂∂y (ψ†A(1)ψ(τ)) = ψ†[L(1)(A(τ)−A)−A(1)(L(τ)−L)]ψ(τ)+ ∂∂x(.

. .

)(5.35)which, after putting τ = X and averaging, gives⟨ψ†(L(1) ∂A∂X −A(1) ∂L∂X )ψ⟩= ∂Ω∂X ⟨ψ†L(1)ψ⟩−∂E∂X ⟨ψ†A(1)ψ⟩+∂W∂X ·⟨ψ†ˆL(1)∂φ∂t ⟩−∂W∂X ·⟨φ† ˆA(1)∂φ∂t ⟩(5.36)Summing up (5.34) and (5.36), and using the compatibility conditions∂U∂Y = ∂W∂X ,∂U∂T = ∂W∂X ,∂V∂T = ∂W∂Y(5.37)we get⟨ψ†Fψ⟩= (∂Ω∂Y −∂E∂T )⟨ψ†ψ⟩+( ∂Ω∂X −∂p∂T )⟨ψ†L(1)ψ⟩+( ∂p∂Y −∂E∂X )⟨ψ†A(1)ψ⟩(5.38)using (ii) from (5.31) we can rewrite the Whitham equations in the final form(∂Ω∂Y −∂E∂T )dp + ( ∂p∂T −∂Ω∂X )dE + ( ∂E∂X −∂p∂Y )dΩ= ⟨ψ†Kψ⟩⟨ψ†ψ⟩dp. (5.39)21

5.5. Whitham equations for the stationary and evolutive KdVThe KdV is a particular case of the KP equation.

The solutions of the KPequation correspond to generic (non-hyperelliptic, that is not of the form w2 =p(E)) Riemann surfaces Γ: in this case the function E(P), P ∈Γ itself is nomore one-valued and its differential dE has non-zero periods Vj =12πiHbj dE. Thesolutions have the form (5.19).

When Γ is hyperelliptic the differential dE is exact,V = 0 and the dependence on y disappears, giving solutions to the KdV equation.For KdV the Whitham equations have the form( ∂p∂T −∂Ω∂X ) = ⟨ψ†Kψ⟩⟨ψ†ψ⟩dpdE(5.40)Solutions to the stationary KdV equation, which is equivalent to the linear systemLψ = Eψ, Aψ = w(E) ψ, come out when dΩtoo is exact, and this is true forΩ(E) = 2w(E)dE =p4E3 −g2E −g3 dE; the corresponding Whitham equationgives∂w∂X dE = ⟨ψ†Kψ⟩⟨ψ†ψ⟩dp(5.41)where dp is normalized with ImHdp = 0.5.6. Integrability of the Whitham equations for K=0For KdV the Whitham equations have the form (5.40).

In this case, the param-eters Ij of the preceding section are simply the branch points E = t(E1, . .

. , E2g+1)of the spectral curve.

For K = 0 this comes out as∂p∂T = ∂Ω∂X(5.42)Krichever has shown that (5.42) has solutions E1(X, T), . .

. , E2g+1(X, T) givenimplicitly by the conditionsdΛdp (E(X, T))|E=Ej(X,T )+X+T dΩdp (E(X, T))|E=Ej(X,T ) = 0,j = 1, .

. .

, 2g+1,(5.43)where dΛ is an arbitrary differential with possibly discontinuities and singularitiesnot depending on X, T. Analogous solutions exist for the KP case (see Ref. 2).22

In order to see it, consider that if the function S(X, T) =R P∞dΛ(X, T) +Xdp(X, T) + TdΩ(X, T) is such that ∂S∂X = p, ∂S∂T = Ω, then(5.42) is automaticallysatisfied. Now, (5.43) is equivalent to dS|E=Ej(X,T ) = 0 for j = 1, .

. .

, 2g + 1;the form∂∂X dS = dp + ( ∂∂X dΛ + X ∂∂X dp + T ∂∂X dΩ) has the same normalizationas dp, the same singularity at the infinity and is holomorphic everywhere else, ex-cept, generally speaking, in the points Ej, because e.g.∂∂XdE√(E−E1)(E−E2)(E−E3) =−12(...)dE[(E−E1)(E−E2)(E−E3)]3/2 + · · · cease to be holomorphic in these points. However,thanks to dS|E=Ej = 0 we have dS = pE −EjdE + O(E −Ej)3/2 in a neigh-borhood of Ej, and∂∂X dS = −∂Ej∂XdE2√E−Ej + · · · = −∂Ej∂X2udu2u+··· comes out to benon-singular in Ej too: thus∂∂X dS coincides with dp, and similarly∂∂T dS = dΩ.5.7.

Integrability of the stationary Whitham equation for K = 1The equation∂w∂X dE = dp can be integrated in the following way (see Ref.3): take k =√E, dE = 2k dk, w =pE2g+1 + c1E2g−1 + · · · + c2g = k2g+1 +2g+12 T2g+1k2g−1+2g−12 T2g−1k2g−3+· · ·+32T3k+x21k+O( 1k2) and require ImHa1 w dE =hi = const, ImHb1 w dE = h′i = const, i = 1, . .

. , g.The first condition givesc1, .

. .

, cg as algebraic functions of T2g+1, . .

. , T3; the second condition fixes alsocg+1, .

. .

, c2g as transcendental functions of T2g+1, . .

. , T3, X.

Moreover, the firstcondition gives∂∂X w dE ≃d√E ≃dp, while the second assures that both sideshave the same normalization: this means that∂∂X w dE = dp, as desired. (Intro-ducing the functions Ωj ≃kj+O(1), E →∞, it is easy to see that∂∂Tj w dE = dΩj,thus giving also ∂Ωi∂Tj = ∂Ωj∂Ti ).23

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