aa r X i v : . [ m a t h - ph ] M a y ON CLASSICAL SOLUTIONS OF THE KDV EQUATION
SERGEI GRUDSKY AND ALEXEI RYBKIN
Abstract.
We show that if the initial profile q ( x ) for the Korteweg-de Vries(KdV) equation is essentially semibounded from below and R ∞ x / | q ( x ) | dx < ∞ , (no decay at −∞ is required) then the KdV has a unique global classicalsolution given by a determinant formula. This result is best known to date. We dedicate this paper to the memory of Jean Bourgain.1.
Introduction
We are concerned with the Cauchy problem for the Korteweg-de Vries (KdV)equation ( ∂ t u − u∂ x u + ∂ x u = 0 , x ∈ R , t ≥ u ( x,
0) = q ( x ) . (1.1)As is well-known, (1.1) is the first nonlinear evolution PDE solved in the seminal1967 Gardner-Greene-Kruskal-Miura paper [11] by the method which is now re-ferred to as the inverse scattering transform (IST). Much of the original work wasdone under generous assumptions on initial data q (typically from the Schwartzclass) for which the well-posedness of (1.1) was not an issue even in the classicalsense . But well-posedness in less nice function classes becomes a problem. Themain (but of course not the only) difficulty is related to slower decay of q at infinitywhich negatively affects regularity of the solutions. This issue drew much of atten-tion once (1.1) became in the spot light. For the earlier literature account we referthe reader to the substantial 1987 paper [2] by Cohen-Kappeler. The main resultof [2] says that if Z ∞−∞ (1 + | x | ) | q ( x ) | dx < ∞ , (1.2) Z ∞ (1 + | x | ) N | q ( x ) | dx < ∞ , N ≥ / H − ( a, ∞ ) for any real a . The uniqueness was not proven in [2] and infact it was stated as an open problem. The best known uniqueness result backthen was available for H / ( R ) which of course assumes some smoothness whereasthe conditions (1.2)-(1.3) do not. Since any function subject to (1.2)-(1.3) can Date : May, 2019.1991
Mathematics Subject Classification.
Key words and phrases.
KdV equation, Hankel operators.SG is supported by CONACYT grant 238630. AR is supported in part by the NSF grantDMS-1716975. I.e., at least three times continuously differentiable in x and once in t . R ∞ | f ( x ) | dx < ∞ means that R ∞ a | f ( x ) | dx < ∞ for all finite a . be properly included in H s ( R ) with any negative s , a well-posedness statement in H s ( R ) , s <
0, would turn the Cohen-Kappeler existence result into a classical well-posedness. The s = 0 bar was reached in 1993 in the seminal papers by Bourgain[4] where, among others, he proved that (1.1) is well-posed in L ( R ). Moreoverhis trademark harmonic analysis techniques could be pushed below s = 0. Werefer the interested reader to the influential [3] for the extensive literature prior to2003. Until very recently, the best well-posedness Sobolev space for (1.1) remained[15] H − / ( R ). Note that harmonic analysis methods break down while crossing s = − / s = − q = v + w ′ where v, w ∈ L ( R ). For s < − H s ( R ) scale(see [15] for relevant discussions and the literature cited therein).However all these spectacular achievements do not answer the natural questionabout the optimal rate of decay of initial data guaranteeing the existence of aclassical solution to (1.1) free of a priori smoothness of q ? Surprisingly enough,this important question seems to have been in the shadow and to the best of ourknowledge the Cohen-Kappeler conditions (1.2)-(1.3) have not been fully improved.The current paper is devoted to this question. In particular, we prove Theorem 1.1 (Main Theorem) . Suppose that a real locally integrable initial profile q in (1.1) satisfies: Sup | I | =1 Z I max ( − q ( x ) , dx < ∞ , (essential boundedness from below) ; (1.4) Z ∞ (1 + | x | ) N | q ( x ) | dx < ∞ , N ≥ / rate of decay at + ∞ ) , (1.5) then the KdV equation has a unique classical solution u ( x, t ) such that uniformlyon compacts in R × R + u ( x, t ) = lim b →−∞ u b ( x, t ) , (1.6) where u b ( x, t ) is the classical solution with the data q b = q | ( b, ∞ ) . We now discuss how Theorem 1.1 is related to previously known results andoutline the ideas behind our arguments.Compare first conditions (1.2) and (1.4). Note that (1.2) is the natural conditionfor solubility of the classical inverse scattering problem (the Marchenko characteri-zation of scattering data [16]), which is the backbone of the IST. Since the Cohen-Kappeler approach is based upon the Marchenko integral equation, the condition(1.2) cannot be relaxed within their framework. It is well-known however that theKdV equation is strongly unidirectional (solitons run to the right) which has to bereflected somehow in the conditions on initial data. As opposed to Cohen-Kappelerour approach is based on ”one-sided” scattering (from the right) for the full lineSchr¨odinger operator L q = − ∂ x + q ( x ), which requires the decay (1.2) only at + ∞ .The direct scattering problem can be solved then as long as q is in the so-calledlimit point case at −∞ , which is readily provided by our (1.4). But of course theIST requires by definition a suitable inverse scattering. We however do not analyzethe inverse scattering problem which could in fact be a difficult endeavour. Instead, In fact only L decay is needed for the direct scattering problem. DV EQUATION 3 we bypass it by considering first truncated data q b = q | ( b, ∞ ) covered by the classicalFaddeev-Marchenko inverse scattering theory. Since q b ∈ H − ( R ) for any b , theproblem (1.1) is well-posed in H − ( R ) (in fact in H s ( R ) for any s < u b ( x, t ) as b → −∞ and it is how our notion of well-posednesscomes about in Theorem 1.1. Justifications of our limiting procedures rely on somesubtle facts from the theory of Hankel operators. As the reader will see in Sections4-6 the Hankel operator plays an indispensable role in proving our results. We onlymention here that our Hankel operator is nothing but a different representation ofthe classical Marchenko operator. But of course it makes all the difference. Ob-serve that condition (1.4) doesn’t assume any pattern of behavior at −∞ and is, ina certain sense, optimal (see Section 7). We noticed this phenomenon first in [23]under additional technical assumptions. We eventually weeded them all out in [14]when the full power of the theory of Hankel operators was unleashed. In this sensethe condition (1.4) is not new but we present here a better proof.Our condition (1.5) is new. It apparently improves N in (1.3) by 1 /
4. Wecan actually show that N = 11 / / N = 11 /
4. This is done in the current paper by finding a new representationof the reflection coefficient, Proposition 3.1. Thus Proposition 3.1 combined withTheorem 4.1 taken from our [12] leads to the condition (1.5).What we find remarkable is that Theorem 1.1 comes with an explicit determinantformula for our solution (an extension of the Dyson formula). We postpone itsdiscussion till Section 6 when we have all necessary terminology.Theorem 1.1 immediately implies
Theorem 1.2.
Suppose that q in (1.1) is real, ∞ X n = −∞ (cid:18)Z n +1 n | q ( x ) | dx (cid:19) < ∞ , (1.7) and Z ∞ (1 + | x | ) N | q ( x ) | dx < ∞ , N ≥ / , then the problem (1.1) has a unique classical solution u ( x, t ) such that lim t → +0 u ( x, t ) = q ( x ) in H − ( R ) . (1.8)Indeed, since the condition (1.7) clearly implies (1.4) and hence Theorem 1.1applies, we have a classical solution u ( x, t ). On the other hand, (1.7) also meansthat q ∈ H − ( R ) and hence, due to the well-posedness in H − (see [15]), (1.8)holds. The convergence (1.6) is then superfluous as it merely follows from thewell-posedness.In fact, (1.7) can be replaced with q ∈ H − ( R ). The arguments follow our [13]where we treat H − ( R ) initial data supported on a left half line. We leave the fullproof out. SERGEI GRUDSKY AND ALEXEI RYBKIN
Note that Theorem 1.1 does not require specifying in what sense the initialcondition is understood. In fact, we do not rule out the existence of a differentsolution to (1.1) but such a solution will not be physical as the natural requirement(1.6) is clearly lost. In [22], under some additional condition we show that (1.8)holds in L ( a, ∞ ) for any a > −∞ . We believe our Hankel operator approach offerssome optimal statements about initial condition. We plan to address it elsewhere.Note that our theorems demonstrate a strong smoothing effect of the KdV flow(see section 7).The paper is organized as follows. The short Section 2 is devoted to our agree-ment on notation. In Section 3 we present some background on scattering theoryand establish some properties of the reflection coefficient crucially important forwhat follows. In Section 4 we give brief background information on Hankel op-erators and prepare some statements for the following sections. In Section 5 weintroduce what we maned separation of infinities principle which makes the proofof Theorem 1.1 much more structured and easier to follow. Section 6 is devoted tothe proof of Theorem 1.1 and the final section 7 is reserved for relevant discussions.2. Notations
We follow standard notation accepted in Analysis. For number sets: N = { , , , ... } , R is the real line, R ± = (0 , ±∞ ), C is the complex plane, C ± = { z ∈ C : ± Im z > } . z is the complex conjugate of z. Besides number sets, black board bold letters will also be used for (linear) oper-ators. As always, ∂ nx := ∂ n /∂x n . As usual, L p ( S ) , < p ≤ ∞ , is the Lebesgue space on a set S . If S = R thenwe abbreviate L p ( R ) = L p . We will also deal with the weighted L spaces L N ( S ) = (cid:26) f | Z S (cid:16) | x | N (cid:17) | f ( x ) | dx < ∞ (cid:27) , N > . This function class is basic for scattering theory for 1D Schr¨odinger operators.3.
The structure of the reflection coefficient
Through this section we assume that q is short-range, i.e. q ∈ L . Associatewith q the full line Schr¨odinger operator L q = − ∂ x + q ( x ). As is well-known, L q is self-adjoint on L and its spectrum consists of a finite number of simplenegative eigenvalues {− κ n } , called bound states, and two fold absolutely continuouscomponent filling R + . There is no singular continuous spectrum. Two linearlyindependent (generalized) eigenfunctions of the a.c. spectrum ψ ± ( x, k ) , k ∈ R , canbe chosen to satisfy ψ ± ( x, k ) = e ± ikx + o (1) , ∂ x ψ ± ( x, k ) ∓ ikψ ± ( x, k ) = o (1) , x → ±∞ . (3.1)The functions ψ ± are referred to as Jost solutions of the Schr¨odinger equation L q ψ = k ψ. (3.2)Since q is real, ψ ± also solves (3.2) and one can easily see that the pairs { ψ + , ψ + } and { ψ − , ψ − } form fundamental sets for (3.2). Hence ψ ∓ is a linear combination DV EQUATION 5 of { ψ ± , ψ ± } . We write this fact as follows ( k ∈ R ) T ( k ) ψ − ( x, k ) = ψ + ( x, k ) + R ( k ) ψ + ( x, k ) , (3.3) T ( k ) ψ + ( x, k ) = ψ − ( x, k ) + L ( k ) ψ − ( x, k ) , (3.4)where T, R, and L are called transmission, right, and left reflection coefficientsrespectively. The identities (3.3)-(3.4) are totally elementary but serve as a basisfor inverse scattering theory and for this reason they are commonly referred to asbasic scattering relations. As is well-known (see, e.g. [16]), the triple { R, ( κ n , c n ) } ,where c n = k ψ + ( · , iκ n ) k − , determines q uniquely and is called the scattering datafor L q . We will need Proposition 3.1 (Structure of the classical reflection coefficient) . Suppose q isreal and in L and q ± = q | R ± is the restriction of q to R ± . Let { R, ( κ n , c n ) } , { R + , ( κ + n , c + n ) } be the scattering data for L q , L q + respectively. Then R = G + R + . (3.5) The function G admits the representation G = T R − − L + R − , (3.6) where T + , L + are the transmission and the left reflection coefficients from q + and R − is the right reflection coefficient from q − . The function G is bounded on R andmeromorphic on C + with simple poles at ( iκ n ) and ( iκ + n ) with residues Res k = iκ n G ( k ) = ic n , Res k = iκ + n G ( k ) = ic + n , (3.7) Furthermore, R + ( k ) = T + ( k ) ( ik Z ∞ e − ikx q ( x ) dx + 1(2 ik ) Z ∞ e − ikx Q ′ ( x ) dx ) , (3.8) where Q is an absolutely continuous function subject to | Q ′ ( x ) | ≤ C | q ( x ) | + C Z ∞ x | q | , x ≥ , (3.9) with some (finite) constants C , C dependent on k q + k L and k q + k L only.Proof. From (3.3) we have R ( k ) = T ( k ) ψ − (0 ,k ) ψ + (0 ,k ) − ψ + (0 ,k ) ψ + (0 ,k ) R + ( k ) = T + ( k ) ψ + (0 ,k ) − ψ + (0 ,k ) ψ + (0 ,k ) . Subtracting these equations yields R ( k ) = R + ( k ) + G ( k ) , where G ( k ) := T ( k ) ψ − (0 , k ) ψ + (0 , k ) − T + ( k ) ψ + (0 , k ) (3.10)We refer to our [22] for the details of derivation of (3.6). The function G , initiallydefined and bounded on the real line, can be analytically continued into C + (since T is meromorphic in C + and ψ ± are analytic there). Its singularities (includingremovable) come apparently from the poles of T, T + and the zeros of ψ + (0 , k ). It SERGEI GRUDSKY AND ALEXEI RYBKIN is well-known from the classical 1D scattering theory (see, e.g. [5]) that the polesof
T, T + occur at ( iκ n ), ( iκ + n ), where (cid:0) − κ n (cid:1) , (cid:16) − ( κ + n ) (cid:17) are the (negative) boundstates of L q and L q + respectively and moreover,Res k = iκ n T ( k ) ψ − (0 , k ) ψ + (0 , k ) = ic n , Res k = iκ + n T + ( k ) ψ + (0 , k ) = ic + n . This combined with (3.10) implies (3.7). We now show that zeros of ψ + (0 , k ) areremovable singularities of G . It follows from (3.3) that T = 2 ikW ( ψ − , ψ + ) , T + ( k ) = 2 ikW ( ψ , − , ψ , + ) , (3.11)where ψ , ± are the Jost solutions corresponding to q + and W ( f, g ) = f g ′ − f ′ g stands for the Wronskian. For G we then have G ( k ) = 2 ikψ + (0 , k ) (cid:26) ψ − (0 , k ) W ( ψ − , ψ + ) − W ( ψ , − , ψ , + ) (cid:27) . Since ψ , − ( x, k ) = e − ikx , x ≤ , and ψ , + ( x, k ) = ψ + ( x, k ) , x ≥ , one concludesthat ( W is independent of x ) W ( ψ , − , ψ , + ) = ψ , − (0 , k ) ∂ x ψ , + (0 , k ) − ∂ x ψ , − (0 , k ) ψ , + (0 , k )= ∂ x ψ + (0 , k ) + ikψ + (0 , k ) , (3.12)and we arrive at G ( k ) = 2 ikW ( ψ − , ψ + ) ∂ x ψ − (0 , k ) + ikψ − (0 , k ) ∂ x ψ + (0 , k ) + ikψ + (0 , k ) . It now follows from (3.12) that a zero of ψ + (0 , k ) cannot be a zero of ∂ x ψ + (0 , k )(otherwise ψ , − and ψ , + were linearly dependant) and thus a zero of ψ + (0 , k ) isnot a pole of G ( k ).Turn now to (3.8). To this end we use the following representation from [5]2 ik R + ( k ) T + ( k ) = Z ∞−∞ g ( y ) e − iky dy, (3.13)where g is defined as follows. Let y ± ( x, k ) = e ∓ ikx ψ ± ( x, k ) . As is shown in [5], y ± ( x, k ) − ∈ H for every x , y ± ( x, k ) = 1 ± Z ±∞ B ± ( x, y ) e ± iky dy, (i.e. the Fourier representation of y ± ( x, k ) −
1) and g ( y ) = − ∂ x B + (0 , y ) + ∂ x B − (0 , y ) + Z ∂ x B − (0 , z ) B + (0 , y − z ) dz − Z ∂ x B + (0 , z ) B − (0 , x − z ) dz. In our case y − ( x, k ) = 1 for x ≤ B − ( x, y ) = 0. Therefore the previousequation simplifies to g ( y ) = − ∂ x B + (0 , y ) . (3.14) DV EQUATION 7 B + ( x, y ), in turn, solves the integral equation [5] B + ( x, y ) − Z y (cid:18)Z ∞ x + y − z q + ( t ) B + ( t, z ) dt (cid:19) dz = Z ∞ x + y q + ( t ) dt, y ≥ . Differentiating this equation in x and setting x = 0 yields g ( y ) = q + ( y ) + Z y q + ( y − z ) B + ( y − z, z ) dz = q + ( y ) + Z y q + ( z ) B + ( z, y − z ) dz =: q ( y ) + Q ( y ) . (3.15)Let us now study Q . It is clearly supported on (0 , ∞ ) and one has Q ′ ( y ) = q ( y ) B + ( y,
0) + Z y q ( z ) ∂ y B + ( z, y − z ) dz := g ( y ) + g ( y ) . (3.16)To obtain the desired estimate (3.9) we make use of two crucially important esti-mates from [5]: for q ∈ L | B + ( x, y ) | ≤ η ( x + y ) e γ ( x ) , (3.17)and | ∂ y B + ( x, y ) + q ( x + y ) | ≤ η ( x + y ) η ( x ) e γ ( x ) , (3.18)where γ ( x ) = Z ∞ x ( t − x ) | q ( t ) | dt, η ( x ) = Z ∞ x | q ( t ) | dt. Since for x ≥ γ ( x ) ≤ Z ∞ x t | q ( t ) | dt ≤ Z ∞ t | q ( t ) | dt = γ (0) , it follows from (3.17)-(3.18) that (recalling that y ≥ | g ( y ) | ≤ | q ( y ) | η ( y ) e γ ( y ) (3.19) ≤ η (0) e γ (0) | q ( y ) | and | g ( y ) | ≤ | q ( y ) | Z y | q | + 2 η ( y ) Z y | q ( z ) | η ( z ) e γ ( z ) dz (3.20) ≤ | q ( y ) | Z y | q | + 2 η (0) e γ (0) η ( y ) . Combining now (3.16) and (3.19)-(3.20) yields (3.9).It remains to show (3.8). Substituting (3.15) into (3.13) we have2 ik R + ( k ) T + ( k ) = Z ∞ q ( y ) e − iky dy + Z ∞ Q ( y ) e − iky dy. Evaluating the last integral by parts yields Z ∞ Q ( y ) e − iky dy = − Q ( y ) e − iλy iλ (cid:12)(cid:12)(cid:12)(cid:12) ∞ + 12 iλ Z ∞ Q ′ ( y ) e − iλy dy. It follows from (3.15) and (3.17) that the integrated term vanishes and (3.8) isproven. (cid:3)
SERGEI GRUDSKY AND ALEXEI RYBKIN
The split (3.5) implies that the right reflection coefficient R can be representedas an analytic function plus the right reflection coefficient R + which need not admitanalytic continuation from the real line. Moreover, R + is completely determined by q on (0 , ∞ ) (by simple shifting arguments, any interval ( a, ∞ ) can be considered).Some parts of Proposition 3.1 appeared in our [22] and [14]) but (3.8) is new. For q supported on the full line, it was proven in [5] that R ( k ) = T ( k )2 ik Z ∞−∞ e − ikx g ( x ) dx, where g satisfies | g ( x ) | ≤ | q ( x ) | + const (cid:26) R ∞ x | q | , x ≥ R x −∞ | q | , x < , (3.21)and nothing better can be said about g in general. In the case of q supported on(0 , ∞ ) this statement can be improved. Indeed, (3.8) implies that g ( x ) = q ( x ) + Q ( x )with some absolutely continuous on (0 , ∞ ) function which derivative Q ′ satisfies(3.21). 4. Hankel operators with oscillatory symbols
We refer the reader to [17] and [18] for background reading on Hankel operators.We recall that a function f analytic in C ± is in the Hardy space H ± ifsup y> Z ∞−∞ | f ( x ± iy ) | dx < ∞ . We will also need H ∞± , the algebra of analytic functions uniformly bounded in C ± .It is particularly important that H ± is a Hilbert space with the inner productinduced from L : h f, g i H ± = h f, g i L = h f, g i = Z ∞−∞ f ( x ) ¯ g ( x ) dx. It is well-known that L = H ⊕ H − , the orthogonal (Riesz) projection P ± onto H ± being given by ( P ± f )( x ) = ± πi Z ∞−∞ f ( s ) dss − ( x ± i . (4.1)Let ( J f )( x ) = f ( − x ) be the operator of reflection. Given ϕ ∈ L ∞ the operator H ( ϕ ) : H → H defined by the formula H ( ϕ ) f = JP − ϕf, f ∈ H , (4.2)is called the Hankel operator with symbol ϕ .It directly follows from the definition (4.2) that the Hankel operator H ( ϕ ) isbounded if its symbol ϕ is bounded and H ( ϕ + h ) = H ( ϕ ) for any h ∈ H ∞ + . Thelatter means that only part of ϕ analytic in C − (called co-analytic) matters. Morespecifically, H ( ϕ ) = H ( e P − ϕ ) , DV EQUATION 9 where ( e P − ϕ )( x ) = − πi Z ∞−∞ (cid:18) λ − ( x − i − λ + i (cid:19) ϕ ( λ ) dλ (4.3)= ( x + i ) (cid:18) P − · + i ϕ (cid:19) ( x ) , ϕ ∈ L ∞ . We note that in general e P − ϕ / ∈ H ∞− if ϕ ∈ L ∞ but the Hankel operator H ( ϕ ) isstill well-defined by (4.2) and bounded. If ϕ ∈ L then e P − ϕ differs from P − ϕ by aconstant and thus P − ϕ can be take as the co-analytic part.In the context of the KdV equation symbols of the following form ϕ ( x ) = G ( x ) ξ α,β ( x ) , naturally arise. Here G ∈ L ∞ , and ξ α,β ( x ) = exp i (cid:0) αx + βx (cid:1) , where α, β are real parameters, and β > . The main feature of ξ α,β is a rapiddecay along any line R + ih in the upper half plane and as a result the quality of H ( Gξ α,β ) may actually be better than H ( G ). E.g., if G ∈ L ∞ and is analytic in C + then (4.3) takes form ( h > e P − ϕ )( x ) = − πi Z R + ih (cid:18) λ − x − λ + i (cid:19) G ( λ ) ξ α,β ( λ ) dλ, (4.4)which is an entire function as long as this integral is absolutely convergent. Thismeans that H ( Gξ α,β ) is in any Shatten-von Neumann ideal S p (0 < p ≤ ∞ ) while H ( G ) need not be even compact. Better yet, H ( Gξ α,β ) can be differentiated in any S p norm with respect to α, β infinitely many time. Indeed, since for all m, n∂ mα ∂ nβ ( e P − ϕ )( x ) = − πi Z R + ih (cid:18) λ − x − λ + i (cid:19) G ( λ ) ∂ mα ∂ nβ ξ α,β ( λ ) dλ are entire functions the operators defined by ∂ mα ∂ nβ H ( Gξ α,β ) = H (cid:16) ∂ mα ∂ nβ e P − Gξ α,β (cid:17) (4.5)are all in S p . Note that if we formally set ∂ mα ∂ nβ H ( Gξ α,β ) = H (cid:0) ∂ mα ∂ nβ Gξ α,β (cid:1) , then we would have the Hankel operator with an unbounded symbol ( ix ) m +3 n G ( x ) ξ α,β ( x ).Thus, (4.5) can be viewed as a way to regularize Hankel operators with certain un-bounded oscillatory symbols.We have to work a bit harder if G doesn’t extend analytically into C + but hassome smoothness. We can no longer apply the Cauchy theorem to evaluate e P − ϕ but the Cauchy-Green formula will do. This is the case when G ( x ) = Z ∞ e − ixs g ( s ) ds with some g ∈ L N ( R + ), N ≥
1. Apparently for any integer n ≤ NG ( n ) ∈ H ∞ (cid:0) C − (cid:1) ∩ C ( R ) (4.6) but G doesn’t in general extend analytically into C + and we can no longer deformthe contour into the upper half plane. Let us now consider instead its pseudoan-alytic extension into C + . Following [7] we call F ( x, y ) a pseudoanalytic extensionof f ( x ) into C if F ( x,
0) = f ( x ) and ∂F ( x, y ) → , y → , where ∂ := (1 /
2) ( ∂ x + i∂ y ). Note that due to (4.6) for n ≤ N the Taylor formula G ( z, z ) = n − X m =0 G ( m ) ( z ) m ! ( z − z ) m , z ∈ C + , (4.7)defines such continuation as G ( z, z ) clearly agrees with G on the real line and for λ ∈ C + ∂G ( z, z ) = G ( n ) ( z )( n − z − z ) n − , n ≤ N. (4.8)By the Cauchy-Green formula applied, say, to the strip 0 ≤ Im z ≤ λ = u + iv ) e P − Gξ α,β ( x ) (4.9)= x + i πi Z ∞−∞ ξ α,β ( λ ) G ( λ ) λ + i dλλ − ( x − i x + i πi Z R + i ξ α,β ( λ ) G (cid:0) λ, λ (cid:1) λ + i dλλ − x + x + iπ Z ≤ Im λ ≤ ξ α,β ( λ ) ∂G (cid:0) λ, λ (cid:1) λ + i dudvλ − x . The first integral on the right hand side of (4.9) is identical to (4.4) and thus weonly need to study φ α,β ( x ) := x + iπ Z ≤ Im λ ≤ ξ α,β ( λ ) ∂G (cid:0) λ, λ (cid:1) λ + i dudvλ − x , where ∂G (cid:0) λ, λ (cid:1) = G ( n ) (cid:0) λ (cid:1) ( n − (cid:0) λ − λ (cid:1) n − = (cid:26)Z ∞ (2 s ) n e − iλs g ( s ) ds (cid:27) v n − i ( n − . We have φ α,β ( x )= x + i πi Z ≤ Im λ ≤ ξ α,β ( λ ) λ + i (cid:26)Z ∞ (2 s ) n e − iλs g ( s ) ds (cid:27) v n − ( n − dudvλ − x = x + i πi Z ∞ (cid:26)Z dve − vs v n − Z R + iv ξ α,β − s ( λ ) λ + i dλλ − x (cid:27) (2 s ) n g ( s ) ds. The integral with respect to dλ is clearly independent of contour and hence φ α,β ( x ) = Z ∞ I ( s, α, β ) γ n ( s ) (2 s ) n g ( s ) ds, (4.10) DV EQUATION 11 where I ( x, α − s, β ) := x + i πi Z R + i ξ α − s,β ( λ ) λ + i dλλ − x and γ n ( s ) := Z dve − vs v n − ( n − . Differentiating φ α,β ( x ) formally in β we have ∂ jα φ α,β ( x ) = Z ∞ ∂ jα I ( x, α − s, β ) (2 s ) n γ n ( s ) g ( s ) ds. (4.11)Apparently, this formal differentiation is valid as long as the integral is absolutelyconvergent. But ∂ jα I ( x, α − s, β ) = x + i πi Z R + i ( iλ ) j ξ α − s,β ( λ ) λ + i dλλ − x =: I j ( x, α − s, β )is clearly absolutely convergent and(2 s ) n γ n ( s ) ≤ (2 s ) n Z ∞ dve − vs v n − ( n − . Note that the integral defining I j ( x, α − s, β ) is independent of contour. The cur-rent one, R + i , is not suitable for getting required bounds on its growth in s → ∞ and we will later deform it as needed (see (4.13)). It follows from (4.11) that (cid:12)(cid:12) ∂ jα φ α,β ( x ) (cid:12)(cid:12) ≤ Z ∞ | I j ( x, α − s, β ) | | g ( s ) | ds and thus ∂ jα H ( φ α,β ) is well-defined by ∂ jα H ( φ α,β ) = H (cid:0) ∂ jα φ α,β (cid:1) as a boundedoperator if for each α and β > Z ∞− β | I j ( x, − s, β ) | | g ( s + β ) | ds ∈ L ∞ . We will however need conditions on the decay of g which guarantee the membershipof ∂ jα H ( φ α,β ) in trace class S for a specified number j . We studied this questionin [12] where we proved Theorem 4.1.
Let real g ∈ L N ( R + ) and φ α,β be given by (4.10) then the Hankeloperator H ( φ α,β ) is ⌊ N ⌋ − times continuously differentiable in α in trace normfor every real α and β > . Note that since ∂ jβ ξ α,β = ∂ jα ξ α,β , Theorem 4.1 can be restated for β accordingly.We refer to [12] for the complete proof. We only mention that our arguments relyon a deep characterization of trace class Hankel operators by Peller [18] which saysthat, given ϕ ∈ L ∞ ( R ), the Hankel operator H ( ϕ ) is trace class iff (cid:16)e P − ϕ (cid:17) ′′ ∈ L ( C − ) and sup Im z ≤− (cid:12)(cid:12)(cid:12)e P − ϕ ( z ) (cid:12)(cid:12)(cid:12) < ∞ . In our case the problem boils down to thefollowing question. Given integer n , find the least possible N such that g ∈ L N ( R + ) = ⇒ Z ∞ (Z R + i λ n ξ α − s,β ( λ )( λ − z ) dλ ) g ( s ) ds ∈ L (cid:0) C − (cid:1) . (4.12) Proving (4.12) reduces essentially to analyzing Z ∞ dss n/ | g ( s + α ) | Z C + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ e is / f ( λ ) λ n dλ ( λ − x + iy ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdy, (4.13)where f ( λ ) = λ / − λ is the phase function and Γ is a contour passing through itsstationary points λ = ±
1. The hardest part is treating the neighborhood of points x − iy close to λ = ±
1. One needs to use the steepest decent approximation withcoalescent stationary points and poles (see [24]). The payoff is however an optimalestimate for (4.13), which in turn means that, in a sense, Theorem 4.1 is optimal.5.
The separation of infinities principle
Through this section we assume that our initial data q is short-range. Let { R, ( κ n , c n ) } be the scattering data for L q . Consider the Hankel operator H ( ϕ )with the symbol ϕ ( k ) = X n c n ξ x,t ( iκ n ) ik + κ n + ξ x,t ( k ) R ( k ) , (5.1)were ξ x,t ( k ) = e i ( k t +2 kx ) . Theorem 5.1 (separation of infinities principle) . Under conditions and in notationof Proposition 3.1 H ( ϕ ) = H ( ϕ + ) + H (Φ) , where ϕ + ( k ) = X n c + n ξ x,t ( iκ + n ) ik + κ + n + ξ x,t ( k ) R + ( k ) (5.2) and Φ ( k ) = − πi Z R + ih ξ x,t ( λ ) G ( λ ) λ − k dλ, h > max ( κ n ) . Proof.
Set φ ( k ) := X n c n ξ x,t ( iκ n ) ik + κ n , φ + ( k ) := X n c + n ξ x,t ( iκ + n ) ik + κ + n , which are rational function with simple poles at ( iκ n ) , ( iκ + n ) , respectively. Considerthe co-analytic part of ξ x,t G (as is well-known, R ∈ L ) :( P − ξ x,t G ) ( k ) = − πi Z R ξ x,t ( λ ) G ( λ ) λ − ( k − i dλ. By Proposition 3.1, ξ x,t G is meromorphic in C + and by the residue theorem wethen have ( h > max ( κ n )) − πi Z R ξ x,t ( λ ) G ( λ ) λ − ( k − i dλ + 12 πi Z R + ih ξ x,t ( λ ) G ( λ ) λ − k dλ = − X n ic n ξ x,t ( iκ n ) iκ n − k + X n ic + n ξ x,t ( iκ + n ) iκ + n − k = − φ + φ + . It follows that P − ξ x,t G = Φ − φ + φ + . DV EQUATION 13
By Proposition 3.1 then H ( ϕ ) = H ( φ ) + H ( ξ x,t R )= H ( φ ) + H ( ξ x,t R + ) + H ( ξ x,t G )= H ( φ ) + H ( ξ x,t R + ) + H ( P − ξ x,t G )= H ( φ ) + H ( ξ x,t R + ) + H (Φ − φ + φ + )= H ( ξ x,t R + ) + H (Φ) + H ( φ + )= H ( ϕ + ) + H (Φ)and the theorem is proven. (cid:3) Theorem 5.1 can be interpreted as follows. Given scattering data for L q , theHankel operator H ( ϕ ) associated with these data is different from the one corre-sponding to the data for L q + by the Hankel operator with an analytic symbol. Thus H ( ϕ + ) is completely determined by q on (0 , ∞ ). The part H (Φ) depends on q onthe whole line but has some nice properties (see below).Our application of Theorem 5.1 to the KdV equation is based on what we callthe Dyson formula (aka Bargmann or log-determinant formula). It says that a L potential q ( x ) can be recovered from the scattering data { R, ( κ n , c n ) } by theformula q ( x ) = − ∂ x log det { H ( ϕ x ) } , ϕ x ( k ) := X n c n e − κ n x ik + κ n + e ikx R ( k ) . (5.3)where the determinant is understood in the classical Fredholm sense.The formula (5.3) has a long history. If R = 0 (reflectionless q ) the Marchenkointegral equation turns into a (finite) linear system and (5.3) follows immediatelyfrom the Cramer rule. This idea is extended to the general L case in Faddeev’ssurvey [9], where it naturally appears as nothing but a different (equivalent) wayof writing the solution to the Marchenko integral equation. We first learned about(5.3) from [9] but Dyson in his influential [6] refers to Faddeev’s [10] available firstin Russian in 1959. Dyson links (5.3) to Fredholm determinants arising in randommatrix theory and it is likely why (5.3) is frequently associated with him. In thecontext of integrable systems, (5.3) is revisited in 1984 by Poppe in [19] where it isrelated to the famous Hirota tau function. We have also seen (5.3) used in the KdVcontext with references to Bargmann and Moser (i.e. it was already known back inthe early 1950s). We refer the interested reader to [1] for many other applicationsof Fredholm determinants and associated numerics.Since the Marchenko integral operator is unitarily equivalent to H ( ϕ x ), our ver-sion (5.3) immediately follows from that of [9].As was discussed in Introduction, the KdV equation with data q ∈ L is well-posed at least in H − s with s > u ( x, t ) can be obtained fromsolving the Marchenko integral equation and written as u ( x, t ) = − ∂ x log det { H ( x, t ) } , H ( x, t ) := H ( ϕ ) , where ϕ is defined by (5.1). As we proved in [12], H ( x, t ) is trace class and hencedet { H ( x, t ) } is well-defined in the classical Fredholm sense. To prove the nec-essary smoothness we show that the condition (1.5) provides five continuous x derivatives of H ( x, t ) (and one in t ). This will be done in the next section. Inciden-tally, differentiability of the Fredholm determinant is also discussed in [19] underadditional smoothness assumptions on the initial data.Theorem 5.1 and the well-known formuladet (cid:18) A A A A (cid:19) = det A det (cid:0) A − A A − A (cid:1) , readily imply Theorem 5.2 (separation of infinities principle for KdV) . The solution to theCauchy problem for the KdV equation (1.1) with q ∈ L can be written in thefollowing forms u ( x, t ) = − ∂ x log det { H + ( x, t ) + H (Φ) } = u + ( x, t ) − ∂ x log det n H + ( x, t )] − H (Φ) o = − ∂ x log det H + ( x, t ) i ( H (Φ)) / i ( H (Φ)) / ! = − ∂ x log det (cid:18) H + ( x, t ) − H (Φ)1 1 (cid:19) = − ∂ x log det (cid:18) H + ( x, t ) 1 − H (Φ) 1 (cid:19) , where u + ( x, t ) is the solution to (1.1) with data q + and H + ( x, t ) = H ( ϕ + ) . This theorem is a manifestation of the unidirectional nature of the KdV equa-tion. The effect of the part of initial data supported on ( −∞ ,
0) is encoded in theHankel operator H (Φ) with an analytic symbol, while the part H + ( x, t ) is solelydetermined by the data on (0 , ∞ ). Theorem 5.2 provides a convenient startingpoint to extending the IST formalism to initial data q beyond the realm of theshort range scattering. Since, in general, there is no inverse scattering procedureavailable outside of the short range setting we have to rely on suitable limitingarguments. 6. Proof of the Main Theorem
With most of ingredients prepared in the previous sections very little is left toprove Theorem 1.1. Take b < q b = q | ( b, ∞ ) . By Theorem 5.2 for its solution we have u b ( x, t ) = − ∂ x log det { H + ( x, t ) + H (Φ b ) } , (6.1)where Φ b ( k ) = − k + i πi Z R + ih ξ x,t ( λ ) G b ( λ )( λ + i ) ( λ − k ) dλ, h > max (cid:0) κ bn (cid:1) ,G b := T R b − L + R b , and is the right reflection coefficient from q | ( b, . As is well-known (see, e.g. [5]), R b is a meromorphic function on the entire plane, and [14] uniformly on compacts DV EQUATION 15 in C + R b ( λ ) → iλ − m − (cid:0) λ (cid:1) iλ + m − ( λ ) := R ( λ ) , b → −∞ , (6.2)where m − (cid:0) k (cid:1) is the Titchmarsh-Weyl m-function of L Dq − , the Schr¨odinger operatoron L ( −∞ ,
0) with a Dirichlet boundary condition at 0. As is well-known, m − ( λ )is analytic on C away from the spectrum of L Dq − which due to the condition 1.4 isbounded from below (see e.g. [8]). Consequently, R is analytic in C + away frompurely imaginary points λ such that λ is in the negative spectrum of L Dq − . Thuslim b →−∞ G b = T R − L + R =: G (6.3)is an analytic function on C + away from a bounded set on the imaginary line. Inturn this means that Φ = lim b →−∞ Φ b is an entire function and H (Φ b ) → H (Φ) intrace norm. Following same arguments as in Section 4 (see also [20] for more details)we see that for every n, m k ∂ nx ∂ mt [ H (Φ b ) − H (Φ)] k S → , b → −∞ . (6.4)Turn now to H + ( x, t ) = H ( ϕ + ) = H ( φ + ) + H ( ξ x,t R + ) .ϕ x ( k ) = X n c n e − κ n x ik + κ n + e ikx R ( k ) . Since φ + is a rational function, H ( φ + ) is smooth in ( x, t ) in trace norm.By (3.8) we have H ( ξ x,t R + ) = H ( ϕ ) + H ( ϕ ) , (6.5)where ϕ ( k ) := T + ( k )2 ik ξ x,t ( k ) Z ∞ e − ikx q ( x ) dx,ϕ ( k ) := T + ( k )(2 ik ) ξ x,t ( k ) Z ∞ e − ikx Q ′ ( x ) dx. We remind that ϕ , ϕ are both bounded at k = 0 as T + ( k ) vanishes at k = 0 toorder 1 . Apparently, ∂ x H ( ϕ ) = H ( T + ξ x,t G ) , ∂ x H ( ϕ ) = H ( T + ξ x,t G ) . (6.6)where G ( k ) := Z ∞ e − ikx q ( x ) dx, G ( k ) := Z ∞ e − ikx Q ′ ( x ) dx. The symbols f ξ x,t G , in (6.6) are different from the ones studied in Section 4 bya factor T + of the form T + = h/B , where h ∈ H ∞ + and B is the finite Blaschkeproduct with simple zeros at ( iκ + n ). This is however a purely technical circumstancein the way of applying Theorem 4.1. The easiest way to circumvent it is to alterour original q by performing the Darboux transform on q + removing all (negative) R can be interpreted as the (right) reflection coefficient from q − (see [14], [20] for details). In fact, it happens generically. For the so-called exceptional potentials T (0) = 0 but anarbitrarily small perturbation turns such a potential into generic. In our case it can be achievedby merely shifting the data q (the KdV is translation invariant). bound states of L q + . Then T + = h ∈ H ∞ + . But if h ∈ H ∞ + and ϕ ∈ L ∞ one easilysees that H ( hϕ ) = T (cid:0) h (cid:1) H ( ϕ ) , where T (cid:0) h (cid:1) = P + h is the Toeplitz operator with symbol h . The letter is a boundedoperator independent of ( x, t ) and smoothness in trace norm of H ( T + ξ x,t G , )with respect of ( x, t ) is the same as H ( ξ x,t G , ). As is well-known, adding backthe previously removed bound states results in adding solitons corresponding to − ( κ + n ) (which are of Schwartz class).Recalling from Proposition 3.1 that | Q ′ ( x ) | ≤ C | q ( x ) | + C Z ∞ x | q | , one concludes that if q ∈ L N then Q ′ ∈ L N − . By Theorem 4.1 if N = 5 / H ( ξ x,t G ) and H ( ξ x,t G ) are differentiable in x in S four and three timesrespectively. By (6.5) and (6.6) H ( ξ x,t R + ) is differentiable in x in S five timesand hence so is H + ( x, t ). Thus, since ∂ t ξ x,t = ∂ x ξ x,t , the formula (6.1) defines aclassical solution u b ( x, t ) with initial data q b and it remains to let b → −∞ . But itfollows from (6.4) thatlim b →−∞ u b ( x, t ) = − ∂ x log det { H + ( x, t ) + H (Φ) } := u ( x, t ) (6.7)and u ( x, t ) is a classical solution to (1.1). Theorem 1.1 is proven.In fact, we have proven a stronger statement Theorem 6.1.
If in Theorem 1.1 q ∈ L N then u ( x, t ) is continuously differentiable ⌊ N ⌋ − times in x and ⌊ (2 N − / ⌋ times in t . We conclude this section with yet another solution formula, which can be viewedas a generalized Dyson formula.
Theorem 6.2.
Under conditions of Theorem 1.1, the solution to (1.1) can berepresented by u ( x, t ) = − ∂ x log det (1 + H ( ϕ x,t )) , with ϕ x,t ( k ) = Z h ξ x,t ( is ) s + ik dρ ( s ) + ξ x,t ( k ) R ( k ) , (6.8) where R is the right reflection coefficient of q and dρ is a positive finite measure. Note that the pair (
R, ρ ) can be viewed as scattering data associated with L q and only (6.8) needs proving. It is proven in our [14] where a complete treatmentof ( R, ρ ) is also given. 7.
Conclusions
Theorem 6.1 says that, loosely speaking, the KdV flow instantaneously smoothensany (integrable) singularities of q ( x ) as long q ( x ) = o (cid:0) x − (cid:1) , x → + ∞ . Such aneffect is commonly referred to as dispersive smoothing. This smoothing propertybecomes stronger as the rate of decay at + ∞ increases, the behavior at −∞ playingno role. In [21] we show that if ( C, δ > q ( x ) = O (cid:0) exp (cid:0) − Cx δ (cid:1)(cid:1) , x → + ∞ , (7.1) Note that h ( − k ) = h ( k ). DV EQUATION 17 then (1) if δ > / u ( x, t ) is meromorphic with respect to x on the wholecomplex plane (with no real poles) for any t >
0; (2) if δ = 1 / u ( x, t ) ismeromorphic in a strip around the x − axis widening proportionally to √ t ; (3) for0 < δ < / q is locally integrable can be lifted. By employingthe arguments from our [13] we may easily extend all our results to include H − type singularities (like Dirac δ − functions, Coulomb potentials, etc.) on any interval( −∞ , a ).The condition (1.4) is optimal. Indeed, what we actually need is semibounded-ness of L q from below, which is guaranteed by (1.4). If q is negative then (1.4)becomes also necessary [8].The absence of decay at −∞ ruins any hope that classical conservations lawswould take place. We do not however rule out existence of some regularized con-servation laws or at least some energy estimates. It would of course be importantto find such estimates. References [1] Bornemann, Folkmar
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Departamento de Matematicas, CINVESTAV del I.P.N. Aportado Postal 14-740, 07000Mexico, D.F., Mexico.
E-mail address : [email protected]. Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box756660, Fairbanks, AK 99775
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