On classical solutions of the KdV equation

TL;DR

This paper proves the existence and uniqueness of global classical solutions for the KdV equation under specific initial conditions, using a determinant formula, extending previous results in the field.

Contribution

It establishes the most general conditions known to date for the existence of classical solutions to the KdV equation, including non-decaying initial profiles.

Findings

Unique global classical solutions exist under broad initial conditions.

Solutions can be explicitly constructed using a determinant formula.

The results extend previous known conditions for well-posedness.

Abstract

\begin{abstract} We show that if the initial profile $q\left( x\right) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $\int^{\infty }x^{5/2}\left\vert q\left( x\right) \right\vert dx<\infty,$ (no decay at $-\infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. \end{abstract}

Equations

$$\{\begin{array}{l}{\partial }_{t}u-6u{\partial }_{x}u+{\partial }_x^{3}u=0,x\in \mathbb{R},t\ge 0\\ u(x,0)=q(x).\end{array}$$

$${\int }_{-\infty }^{\infty }\left(1+\left|x\right|\right)\left|q\left(x\right)\right|dx$$

$${\int }^{\infty }{\left(1+\left|x\right|\right)}^N\left|q\left(x\right)\right|dx$$

$$\underset{|I|=1}{\mathrm{Sup}}{\int }_I\max \left(-q\left(x\right),0\right)dx<\infty ,\text{(essential boundedness from below)};$$

$${\int }^{\infty }{\left(1+\left|x\right|\right)}^N\left|q\left(x\right)\right|dx<\infty ,N\ge 5/2(\text{rate of decay at}+\infty ),$$

$$u\left(x,t\right)=\underset{b\to -\infty }{\lim }{u}_b(x,t),$$

$$\sum _{n=-\infty }^{\infty }{\left({\int }_n^{n+1}\left|q\left(x\right)\right|dx\right)}^2<\infty ,$$

$${\int }^{\infty }{\left(1+\left|x\right|\right)}^N\left|q\left(x\right)\right|dx<\infty ,N\ge 5/2,$$

$$\underset{t\to +0}{\lim }u(x,t)=q\left(x\right)\text{ in }{H}^{-1}\left(\mathbb{R}\right).$$

$$L_N^1\left(S\right)=\left\{f|{\int }_S\left(1+{\left|x\right|}^N\right)\left|f\left(x\right)\right|dx<\infty \right\},N>0.$$

$${\psi }_{\pm }(x,k)=e^{\pm ikx}+o(1),{\partial }_x{\psi }_{\pm }(x,k)\mp ik{\psi }_{\pm }(x,k)=o(1),x\to \pm \infty .$$

$${\mathbb{L}}_q\psi =k^2\psi .$$

$$T(k){\psi }_{-}(x,k)$$

$$T(k){\psi }_{+}(x,k)$$

$$R=G+R_{+}.$$

$$G=\frac{T_{+}^2{R}_{-}}{1-L_{+}{R}_{-}},$$

$$\underset{k=i{\kappa }_n}{\mathrm{Res}}G(k)=i{c}_n,\underset{k=i{\kappa }_n^{+}}{\mathrm{Res}}G(k)=i{c}_n^{+},$$

$$R_{+}\left(k\right)=T_{+}\left(k\right)\left\{\frac{1}{2ik}{\int }_0^{\infty }{e}^{-2ikx}q\left(x\right)dx+\frac{1}{{\left(2ik\right)}^2}{\int }_0^{\infty }{e}^{-2ikx}{Q}^{\prime }\left(x\right)dx\right\},$$

$$\left|Q^{\prime }\left(x\right)\right|\le C_1\left|q\left(x\right)\right|+C_2{\int }_x^{\infty }\left|q\right|,x\ge 0,$$

$$\begin{array}[c]{ccc}R(k)&=&T(k)\frac{\psi_{-}\left(0,k\right)}{\psi_{+}\left(0,k\right)}-\frac{\overline{\psi_{+}\left(0,k\right)}}{\psi_{+}\left(0,k\right)}\\ R_{+}(k)&=&\frac{T_{+}(k)}{\psi_{+}\left(0,k\right)}-\frac{\overline{\psi_{+}\left(0,k\right)}}{\psi_{+}\left(0,k\right)}\end{array}.$$

$$R(k)=R_{+}(k)+G\left(k\right),$$

$$G\left(k\right):=T(k)\frac{{\psi }_{-}\left(0,k\right)}{{\psi }_{+}\left(0,k\right)}-\frac{T_{+}(k)}{{\psi }_{+}\left(0,k\right)}$$

$$\underset{k=i{\kappa }_n}{\mathrm{Res}}T(k)\frac{{\psi }_{-}(0,k)}{{\psi }_{+}(0,k)}=i{c}_n,\underset{k=i{\kappa }_n^{+}}{\mathrm{Res}}\frac{T_{+}(k)}{{\psi }_{+}(0,k)}=i{c}_n^{+}.$$

$$T=\frac{2ik}{W(\psi_{-},\psi_{+})},\ \ T_{+}\left(k\right)=\frac{2ik}{W(\psi_{0,-},\psi_{0,+})},\$$

$$G\left(k\right)=\frac{2ik}{{\psi }_{+}\left(0,k\right)}\left\{\frac{{\psi }_{-}\left(0,k\right)}{W({\psi }_{-},{\psi }_{+})}-\frac{1}{W({\psi }_{0,-},{\psi }_{0,+})}\right\}.$$

$$W({\psi }_{0,-},{\psi }_{0,+})$$

$$={\partial }_x{\psi }_{+}\left(0,k\right)+ik{\psi }_{+}\left(0,k\right),$$

$$G\left(k\right)=\frac{2ik}{W({\psi }_{-},{\psi }_{+})}\frac{{\partial }_x{\psi }_{-}\left(0,k\right)+ik{\psi }_{-}\left(0,k\right)}{{\partial }_x{\psi }_{+}\left(0,k\right)+ik{\psi }_{+}\left(0,k\right)}.$$

$$2ik\frac{R_{+}(k)}{T_{+}(k)}={\int }_{-\infty }^{\infty }g(y)e^{-2iky}dy,$$

$$y_{\pm }\left(x,k\right)=e^{\mp ikx}{\psi }_{\pm }(x,k).$$

$$y_{\pm }(x,k)=1\pm {\int }_0^{\pm \infty }{B}_{\pm }(x,y)e^{\pm 2iky}dy,$$

$$g(y)$$

Full text

SG is supported by CONACYT grant 238630. AR is supported in part by the NSF grant DMS-1716975.

1. Introduction

We are concerned with the Cauchy problem for the Korteweg-de Vries (KdV) equation

[TABLE]

As is well-known, (1.1) is the first nonlinear evolution PDE solved in the seminal 1967 Gardner-Greene-Kruskal-Miura paper [11] by the method which is now referred to as the inverse scattering transform (IST). Much of the original work was done under generous assumptions on initial data $q$ (typically from the Schwartz class) for which the well-posedness of (1.1) was not an issue even in the classical sense111I.e., at least three times continuously differentiable in $x$ and once in $t$.. But well-posedness in less nice function classes becomes a problem. The main (but of course not the only) difficulty is related to slower decay of $q$ at infinity which negatively affects regularity of the solutions. This issue drew much of attention once (1.1) became in the spot light. For the earlier literature account we refer the reader to the substantial 1987 paper [2] by Cohen-Kappeler. The main result of [2] says that if222$\int^{\infty}\left|f\left(x\right)\right|dx<\infty$ means that $\int_{a}^{\infty}\left|f\left(x\right)\right|dx<\infty$ for all finite $a$.

[TABLE]

then (1.1) has a classical solution, the initial condition being satisfied in the Sobolev space $H^{-1}\left(a,\infty\right)$ for any real $a$. The uniqueness was not proven in [2] and in fact it was stated as an open problem. The best known uniqueness result back then was available for $H^{3/2}\left(\mathbb{R}\right)$ which of course assumes some smoothness whereas the conditions (1.2)-(1.3) do not. Since any function subject to (1.2)-(1.3) can be properly included in $H^{s}\left(\mathbb{R}\right)$ with any negative $s$, a well-posedness statement in $H^{s}\left(\mathbb{R}\right),s<0$, 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 $L^{2}\left(\mathbb{R}\right)$. Moreover his trademark harmonic analysis techniques could be pushed below $s=0$. We refer the interested reader to the influential [3] for the extensive literature prior to 2003. Until very recently, the best well-posedness Sobolev space for (1.1) remained [15] $H^{-3/4}\left(\mathbb{R}\right)$. Note that harmonic analysis methods break down while crossing $s=-3/4$ in an irreparable way. Further improvements required utilizing complete integrability of the KdV. The breakthrough has just occurred in Killip-Visan [15] where $s=-1$ was reached. That is, (1.1) is well-posed for initial data of the form $q=v+w^{\prime}$ where $v,w\in L^{2}\left(\mathbb{R}\right)$. For $s<-1$ the KdV is ill-posed in $H^{s}\left(\mathbb{R}\right)$ scale (see [15] for relevant discussions and the literature cited therein).

However all these spectacular achievements do not answer the natural question about the optimal rate of decay of initial data guaranteeing the existence of a classical 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 our knowledge 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:

[TABLE]

[TABLE]

then the KdV equation has a unique classical solution $u\left(x,t\right)$ such that uniformly on compacts in $\mathbb{R\times R}_{+}$

[TABLE]

where $u_{b}(x,t)$ is the classical solution with the data $q_{b}=\left.q\right|_{\left(b,\infty\right)}$.

We now discuss how Theorem 1.1 is related to previously known results and outline the ideas behind our arguments.

Compare first conditions (1.2) and (1.4). Note that (1.2) is the natural condition for solubility of the classical inverse scattering problem (the Marchenko characterization 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 the KdV equation is strongly unidirectional (solitons run to the right) which has to be reflected somehow in the conditions on initial data. As opposed to Cohen-Kappeler our approach is based on ”one-sided” scattering (from the right) for the full line Schrödinger operator $\mathbb{L}_{q}=-\partial_{x}^{2}+q\left(x\right)$, which requires the decay333In fact only $L^{1}$ decay is needed for the direct scattering problem. (1.2) only at $+\infty$. The direct scattering problem can be solved then as long as $q$ is in the so-called limit point case at $-\infty$, which is readily provided by our (1.4). But of course the IST requires by definition a suitable inverse scattering. We however do not analyze the inverse scattering problem which could in fact be a difficult endeavour. Instead, we bypass it by considering first truncated data $q_{b}=\left.q\right|_{\left(b,\infty\right)}$ covered by the classical Faddeev-Marchenko inverse scattering theory. Since $q_{b}\in H^{-1}\left(\mathbb{R}\right)$ for any $b$, the problem (1.1) is well-posed in $H^{-1}\left(\mathbb{R}\right)$ (in fact in $H^{s}\left(\mathbb{R}\right)$ for any $s<0$). We then study its solution $u_{b}(x,t)$ as $b\rightarrow-\infty$ and it is how our notion of well-posedness comes about in Theorem 1.1. Justifications of our limiting procedures rely on some subtle facts from the theory of Hankel operators. As the reader will see in Sections 4-6 the Hankel operator plays an indispensable role in proving our results. We only mention here that our Hankel operator is nothing but a different representation of the classical Marchenko operator. But of course it makes all the difference. Observe that condition (1.4) doesn’t assume any pattern of behavior at $-\infty$ and is, in a 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 sense the 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 $1/4$. We can actually show that $N=11/4$ cannot be improved within the Cohen-Kappeler approach. We save extra $1/4$ by representing the symbol of our Hankel operator (the Marchenko operator in disguise) in a suitable form. This representation is very natural and common in the theory of Hankel operators but is obscured in the Marchenko form. It then invites the famous characterization of trace class Hankel operators due to Peller [18]. We first noticed the relevance of Peller’s theorem in [22] but were able to overcome numerous technical difficulties only recently in [20], [12]. We could not however achieve the condition (1.5) and in fact could not even beat $N=11/4$. This is done in the current paper by finding a new representation of the reflection coefficient, Proposition 3.1. Thus Proposition 3.1 combined with Theorem 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 determinant formula for our solution (an extension of the Dyson formula). We postpone its discussion 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,

[TABLE]

and

[TABLE]

then the problem (1.1) has a unique classical solution $u\left(x,t\right)$ such that

[TABLE]

Indeed, since the condition (1.7) clearly implies (1.4) and hence Theorem 1.1 applies, we have a classical solution $u\left(x,t\right)$. On the other hand, (1.7) also means that $q\in H^{-1}\left(\mathbb{R}\right)$ and hence, due to the well-posedness in $H^{-1}$ (see [15]), (1.8) holds. The convergence (1.6) is then superfluous as it merely follows from the well-posedness.

In fact, (1.7) can be replaced with $q\in H^{-1}\left(\mathbb{R}\right)$. The arguments follow our [13] where we treat $H_{\operatorname*{loc}}^{-1}\left(\mathbb{R}\right)$ initial data supported on a left half line. We leave the full proof out.

Note that Theorem 1.1 does not require specifying in what sense the initial condition is understood. In fact, we do not rule out the existence of a different solution 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 $L^{2}\left(a,\infty\right)$ for any $a>-\infty$. We believe our Hankel operator approach offers some 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 agreement on notation. In Section 3 we present some background on scattering theory and establish some properties of the reflection coefficient crucially important for what follows. In Section 4 we give brief background information on Hankel operators and prepare some statements for the following sections. In Section 5 we introduce what we maned separation of infinities principle which makes the proof of Theorem 1.1 much more structured and easier to follow. Section 6 is devoted to the 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: $\mathbb{N}_{0}=\left\{0,1,2,...\right\}$, $\mathbb{R}$ is the real line, $\mathbb{R}_{\pm}=(0,\pm\infty)$, $\mathbb{C}$ is the complex plane, $\mathbb{C}^{\pm}=\left\{z\in\mathbb{C}:\pm\operatorname{Im}z>0\right\}$. $\overline{z}$ is the complex conjugate of $z.$

Besides number sets, black board bold letters will also be used for (linear) operators. As always, $\partial_{x}^{n}:=\partial^{n}/\partial x^{n}.$

As usual, $L^{p}\left(S\right),\ 0<p\leq\infty$, is the Lebesgue space on a set $S$. If $S=\mathbb{R}$ then we abbreviate $\ L^{p}\left(\mathbb{R}\right)=L^{p}$. We will also deal with the weighted $L^{1}$ spaces

[TABLE]

This function class is basic for scattering theory for 1D Schrödinger operators.

3. The structure of the reflection coefficient

Through this section we assume that $q$ is short-range, i.e. $q\in L_{1}^{1}$. Associate with $q$ the full line Schrödinger operator $\mathbb{L}_{q}=-\partial_{x}^{2}+q(x)$. As is well-known, $\mathbb{L}_{q}$ is self-adjoint on $L^{2}$ and its spectrum consists of a finite number of simple negative eigenvalues $\{-\kappa_{n}^{2}\}$, called bound states, and two fold absolutely continuous component filling $\mathbb{R}_{+}$. There is no singular continuous spectrum. Two linearly independent (generalized) eigenfunctions of the a.c. spectrum $\psi_{\pm}(x,k),\;k\in\mathbb{R}$, can be chosen to satisfy

[TABLE]

The functions $\psi_{\pm}$ are referred to as Jost solutions of the Schrödinger equation

[TABLE]

Since $q$ is real, $\overline{\psi_{\pm}}$ also solves (3.2) and one can easily see that the pairs $\{\psi_{+},\overline{\psi_{+}}\}$ and $\{\psi_{-},\overline{\psi_{-}}\}$ form fundamental sets for (3.2). Hence $\psi_{\mp}$ is a linear combination of $\{\psi_{\pm},\overline{\psi_{\pm}}\}$. We write this fact as follows ($k\in\mathbb{R}$)

[TABLE]

where $T,R,$ and $L$ are called transmission, right, and left reflection coefficients respectively. The identities (3.3)-(3.4) are totally elementary but serve as a basis for inverse scattering theory and for this reason they are commonly referred to as basic scattering relations. As is well-known (see, e.g. [16]), the triple $\{R,(\kappa_{n},c_{n})\}$, where $c_{n}=\left\|\psi_{+}(\cdot,i\kappa_{n})\right\|^{-1}$, determines $q$ uniquely and is called the scattering data for $\mathbb{L}_{q}$. We will need

Proposition 3.1** (Structure of the classical reflection coefficient).**

Suppose $q$ is real and in $L_{1}^{1}$ and $q_{\pm}=\left.q\right|_{\mathbb{R}_{\pm}}$ is the restriction of $q$ to $\mathbb{R}_{\pm}$. Let $\{R,(\kappa_{n},c_{n})\},$ $\{R_{+},(\kappa_{n}^{+},c_{n}^{+})\}$ be the scattering data for $\mathbb{L}_{q},\mathbb{L}_{q_{+}}$ respectively. Then

[TABLE]

The function $G$ admits the representation

[TABLE]

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 $\mathbb{R}$ and meromorphic on $\mathbb{C}^{+}$ with simple poles at $\left(i\kappa_{n}\right)$ and $\left(i\kappa_{n}^{+}\right)$ with residues

[TABLE]

Furthermore,

[TABLE]

where $Q$ is an absolutely continuous function subject to

[TABLE]

with some (finite) constants $C_{1},C_{2}$ dependent on $\left\|q_{+}\right\|_{L^{1}}$ and $\left\|q_{+}\right\|_{L_{1}^{1}}$ only.

Proof.

From (3.3) we have

[TABLE]

Subtracting these equations yields

[TABLE]

where

[TABLE]

We refer to our [22] for the details of derivation of (3.6). The function $G$, initially defined and bounded on the real line, can be analytically continued into $\mathbb{C}^{+}$ (since $T$ is meromorphic in $\mathbb{C}^{+}$ and $\psi_{\pm}$ are analytic there). Its singularities (including removable) come apparently from the poles of $T,T_{+}$ and the zeros of $\psi_{+}\left(0,k\right)$. It is well-known from the classical 1D scattering theory (see, e.g. [5]) that the poles of $T,T_{+}$ occur at $\left(i\kappa_{n}\right)$, $\left(i\kappa_{n}^{+}\right)$, where $\left(-\kappa_{n}^{2}\right)$, $\left(-\left(\kappa_{n}^{+}\right)^{2}\right)$ are the (negative) bound states of $\mathbb{L}_{q}$ and $\mathbb{L}_{q_{+}}$ respectively and moreover,

[TABLE]

This combined with (3.10) implies (3.7). We now show that zeros of $\psi_{+}\left(0,k\right)$ are removable singularities of $G$. It follows from (3.3) that

[TABLE]

where $\psi_{0,\pm}$ are the Jost solutions corresponding to $q_{+}$ and $W\left(f,g\right)=fg^{\prime}-f^{\prime}g$ stands for the Wronskian. For $G$ we then have

[TABLE]

Since $\psi_{0,-}\left(x,k\right)=e^{-ikx},\ x\leq 0,$ and $\psi_{0,+}\left(x,k\right)=\psi_{+}\left(x,k\right),\ x\geq 0,$ one concludes that ($W$ is independent of $x$)

[TABLE]

and we arrive at

[TABLE]

It now follows from (3.12) that a zero of $\psi_{+}\left(0,k\right)$ cannot be a zero of $\partial_{x}\psi_{+}\left(0,k\right)$ (otherwise $\psi_{0,-}$ and $\psi_{0,+}$ were linearly dependant) and thus a zero of $\psi_{+}\left(0,k\right)$ is not a pole of $G\left(k\right)$.

Turn now to (3.8). To this end we use the following representation from [5]

[TABLE]

where $g$ is defined as follows. Let

[TABLE]

As is shown in [5], $y_{\pm}\left(x,k\right)-1\in H^{2}$ for every $x$,

[TABLE]

(i.e. the Fourier representation of $y_{\pm}\left(x,k\right)-1$) and

[TABLE]

In our case $y_{-}\left(x,k\right)=1$ for $x\leq 0$ and hence $B_{-}(x,y)=0$. Therefore the previous equation simplifies to

[TABLE]

$B_{+}(x,y)$, in turn, solves the integral equation [5]

[TABLE]

Differentiating this equation in $x$ and setting $x=0$ yields

[TABLE]

Let us now study $Q$. It is clearly supported on $\left(0,\infty\right)$ and one has

[TABLE]

To obtain the desired estimate (3.9) we make use of two crucially important estimates from [5]: for $q\in L_{1}^{1}$

[TABLE]

and

[TABLE]

where

[TABLE]

Since for $x\geq 0$

[TABLE]

it follows from (3.17)-(3.18) that (recalling that $y\geq 0$)

[TABLE]

and

[TABLE]

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 have

[TABLE]

Evaluating the last integral by parts yields

[TABLE]

It follows from (3.15) and (3.17) that the integrated term vanishes and (3.8) is proven. ∎

The split (3.5) implies that the right reflection coefficient $R$ can be represented as an analytic function plus the right reflection coefficient $R_{+}$ which need not admit analytic continuation from the real line. Moreover, $R_{+}$ is completely determined by $q$ on $\left(0,\infty\right)$ (by simple shifting arguments, any interval $\left(a,\infty\right)$ 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

[TABLE]

where $g$ satisfies

[TABLE]

and nothing better can be said about $g$ in general. In the case of $q$ supported on $\left(0,\infty\right)$ this statement can be improved. Indeed, (3.8) implies that

[TABLE]

with some absolutely continuous on $\left(0,\infty\right)$ function which derivative $Q^{\prime}$ 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 $\mathbb{C}^{\pm}$ is in the Hardy space $H_{\pm}^{2}$ if

[TABLE]

We will also need $H_{\pm}^{\infty}$, the algebra of analytic functions uniformly bounded in $\mathbb{C}^{\pm}$. It is particularly important that $H_{\pm}^{2}$ is a Hilbert space with the inner product induced from $L^{2}$:

[TABLE]

It is well-known that $L^{2}=H_{+}^{2}\oplus H_{-}^{2},$ the orthogonal (Riesz) projection $\mathbb{P}_{\pm}$ onto $H_{\pm}^{2}$ being given by

[TABLE]

Let $(\mathbb{J}f)(x)=f(-x)$ be the operator of reflection. Given $\varphi\in L^{\infty}$ the operator $\mathbb{H}(\varphi):H_{+}^{2}\rightarrow H_{+}^{2}$ defined by the formula

[TABLE]

is called the Hankel* *operator with symbol $\varphi$.

It directly follows from the definition (4.2) that the Hankel operator $\mathbb{H}(\varphi)$ is bounded if its symbol $\varphi$ is bounded and $\mathbb{H}(\varphi+h)=\mathbb{H}(\varphi)$ for any $h\in H_{+}^{\infty}$. The latter means that only part of $\varphi$ analytic in $\mathbb{C}^{-}$ (called co-analytic) matters. More specifically,

[TABLE]

where

[TABLE]

We note that in general $\widetilde{\mathbb{P}}_{-}\varphi\notin H_{-}^{\infty}$ if $\varphi\in L^{\infty}$ but the Hankel operator $\mathbb{H}(\varphi)$ is still well-defined by (4.2) and bounded. If $\varphi\in L^{2}$ then $\widetilde{\mathbb{P}}_{-}\varphi$ differs from $\mathbb{P}_{-}\varphi$ by a constant and thus $\mathbb{P}_{-}\varphi$ can be take as the co-analytic part.

In the context of the KdV equation symbols of the following form

[TABLE]

naturally arise. Here $G\in L^{\infty}$, and

[TABLE]

where $\alpha,\beta$ are real parameters, and $\beta>0.$ The main feature of $\xi_{\alpha,\beta}$ is a rapid decay along any line $\mathbb{R}+ih$ in the upper half plane and as a result the quality of $\mathbb{H}\left(G\xi_{\alpha,\beta}\right)$ may actually be better than $\mathbb{H}\left(G\right)$. E.g., if $G\in L^{\infty}$ and is analytic in $\mathbb{C}^{+}$ then (4.3) takes form ($h>0$)

[TABLE]

which is an entire function as long as this integral is absolutely convergent. This means that $\mathbb{H}\left(G\xi_{\alpha,\beta}\right)$ is in any Shatten-von Neumann ideal $\mathfrak{S}_{p}$ ($0<p\leq\infty$) while $\mathbb{H}\left(G\right)$ need not be even compact. Better yet, $\mathbb{H}\left(G\xi_{\alpha,\beta}\right)$ can be differentiated in any $\mathfrak{S}_{p}$ norm with respect to $\alpha,\beta$ infinitely many time. Indeed, since for all $m,n$

[TABLE]

are entire functions the operators defined by

[TABLE]

are all in $\mathfrak{S}_{p}$. Note that if we formally set

[TABLE]

then we would have the Hankel operator with an unbounded symbol $\left(ix\right)^{m+3n}G\left(x\right)\xi_{\alpha,\beta}\left(x\right)$. Thus, (4.5) can be viewed as a way to regularize Hankel operators with certain unbounded oscillatory symbols.

We have to work a bit harder if $G$ doesn’t extend analytically into $\mathbb{C}^{+}$ but has some smoothness. We can no longer apply the Cauchy theorem to evaluate $\widetilde{\mathbb{P}}_{-}\varphi$ but the Cauchy-Green formula will do. This is the case when

[TABLE]

with some $g\in L_{N}^{1}\left(\mathbb{R}_{+}\right)$, $N\geq 1$. Apparently for any integer $n\leq N$

[TABLE]

but $G$ doesn’t in general extend analytically into $\mathbb{C}^{+}$ and we can no longer deform the contour into the upper half plane. Let us now consider instead its pseudoanalytic extension into $\mathbb{C}^{+}$. Following [7] we call $F\left(x,y\right)$ a pseudoanalytic extension of $f\left(x\right)$ into $\mathbb{C}$ if

[TABLE]

where $\overline{\partial}:=\left(1/2\right)\left(\partial_{x}+i\partial_{y}\right)$. Note that due to (4.6) for $n\leq N$ the Taylor formula

[TABLE]

defines such continuation as $G\left(z,\overline{z}\right)$ clearly agrees with $G$ on the real line and for $\lambda\in\mathbb{C}^{+}$

[TABLE]

By the Cauchy-Green formula applied, say, to the strip $0\leq\operatorname{Im}z\leq 1$ we have ($\lambda=u+iv$)

[TABLE]

The first integral on the right hand side of (4.9) is identical to (4.4) and thus we only need to study

[TABLE]

where

[TABLE]

We have

[TABLE]

The integral with respect to $d\lambda$ is clearly independent of contour and hence

[TABLE]

where

[TABLE]

and

[TABLE]

Differentiating $\phi_{\alpha,\beta}\left(x\right)$ formally in $\beta$ we have

[TABLE]

Apparently, this formal differentiation is valid as long as the integral is absolutely convergent. But

[TABLE]

is clearly absolutely convergent and

[TABLE]

Note that the integral defining $I_{j}\left(x,\alpha-s,\beta\right)$ is independent of contour. The current one, $\mathbb{R}+i$, is not suitable for getting required bounds on its growth in $s\rightarrow\infty$ and we will later deform it as needed (see (4.13)). It follows from (4.11) that

[TABLE]

and thus $\partial_{\alpha}^{j}\mathbb{H}\left(\phi_{\alpha,\beta}\right)$ is well-defined by $\partial_{\alpha}^{j}\mathbb{H}\left(\phi_{\alpha,\beta}\right)=\mathbb{H}\left(\partial_{\alpha}^{j}\phi_{\alpha,\beta}\right)$ as a bounded operator if for each $\alpha$ and $\beta>0$

[TABLE]

We will however need conditions on the decay of $g$ which guarantee the membership of $\partial_{\alpha}^{j}\mathbb{H}\left(\phi_{\alpha,\beta}\right)$ in trace class $\mathfrak{S}_{1}$ for a specified number $j$. We studied this question in [12] where we proved

Theorem 4.1**.**

Let real $g\in L_{N}^{1}\left(\mathbb{R}_{+}\right)$ and $\phi_{\alpha,\beta}$ be given by (4.10) then the Hankel operator $\mathbb{H}\left(\phi_{\alpha,\beta}\right)$ is $\left\lfloor 2N\right\rfloor-1$ times continuously differentiable in $\alpha$ in trace norm for every real $\alpha$ and $\beta>0$.

Note that since $\partial_{\beta}^{j}\xi_{\alpha,\beta}=\partial_{\alpha}^{3j}\xi_{\alpha,\beta}$, Theorem 4.1 can be restated for $\beta$ accordingly. We refer to [12] for the complete proof. We only mention that our arguments rely on a deep characterization of trace class Hankel operators by Peller [18] which says that, given $\varphi\in L^{\infty}\left(\mathbb{R}\right)$, the Hankel operator $\mathbb{H}(\varphi)$ is trace class iff $\left(\widetilde{\mathbb{P}}_{-}\varphi\right)^{\prime\prime}\in L^{1}\left(\mathbb{C}^{-}\right)$ and $\sup_{\operatorname{Im}z\leq-1}\left|\widetilde{\mathbb{P}}_{-}\varphi\left(z\right)\right|<\infty$. In our case the problem boils down to the following question. Given integer $n$, find the least possible $N$ such that

[TABLE]

Proving (4.12) reduces essentially to analyzing

[TABLE]

where $f\left(\lambda\right)=\lambda^{3}/3-\lambda$ is the phase function and $\Gamma$ is a contour passing through its stationary points $\lambda=\pm 1$. The hardest part is treating the neighborhood of points $x-iy$ close to $\lambda=\pm 1$. One needs to use the steepest decent approximation with coalescent stationary points and poles (see [24]). The payoff is however an optimal estimate 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 $\{R,(\kappa_{n},c_{n})\}$ be the scattering data for $\mathbb{L}_{q}$. Consider the Hankel operator $\mathbb{H}(\varphi)$ with the symbol

[TABLE]

were

[TABLE]

Theorem 5.1** (separation of infinities principle).**

Under conditions and in notation of Proposition 3.1

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Set

[TABLE]

which are rational function with simple poles at $\left(i\kappa_{n}\right),\left(i\kappa_{n}^{+}\right),$ respectively. Consider the co-analytic part of $\xi_{x,t}G$ (as is well-known, $R\in L^{2}$) :

[TABLE]

By Proposition 3.1, $\xi_{x,t}G$ is meromorphic in $\mathbb{C}^{+}$ and by the residue theorem we then have ($h>\max\left(\kappa_{n}\right)$)

[TABLE]

It follows that

[TABLE]

By Proposition 3.1 then

[TABLE]

and the theorem is proven. ∎

Theorem 5.1 can be interpreted as follows. Given scattering data for $\mathbb{L}_{q}$, the Hankel operator $\mathbb{H}\left(\varphi\right)$ associated with these data is different from the one corresponding to the data for $\mathbb{L}_{q_{+}}$ by the Hankel operator with an analytic symbol. Thus $\mathbb{H}\left(\varphi_{+}\right)$ is completely determined by $q$ on $\left(0,\infty\right)$. The part $\mathbb{H}(\Phi)$ depends on $q$ on the whole line but has some nice properties (see below).

Our application of Theorem 5.1 to the KdV equation is based on what we call the Dyson formula (aka Bargmann or log-determinant formula). It says that a $L_{1}^{1}$ potential $q\left(x\right)$ can be recovered from the scattering data $\{R,(\kappa_{n},c_{n})\}$ by the formula

[TABLE]

where the determinant is understood in the classical Fredholm sense.

The formula (5.3) has a long history. If $R=0$ (reflectionless $q$) the Marchenko integral equation turns into a (finite) linear system and (5.3) follows immediately from the Cramer rule. This idea is extended to the general $L_{1}^{1}$ case in Faddeev’s survey [9], where it naturally appears as nothing but a different (equivalent) way of 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 first in Russian in 1959. Dyson links (5.3) to Fredholm determinants arising in random matrix theory and it is likely why (5.3) is frequently associated with him. In the context of integrable systems, (5.3) is revisited in 1984 by Poppe in [19] where it is related to the famous Hirota tau function. We have also seen (5.3) used in the KdV context with references to Bargmann and Moser (i.e. it was already known back in the early 1950s). We refer the interested reader to [1] for many other applications of Fredholm determinants and associated numerics.

Since the Marchenko integral operator is unitarily equivalent to $\mathbb{H}(\varphi_{x})$, our version (5.3) immediately follows from that of [9].

As was discussed in Introduction, the KdV equation with data $q\in L_{1}^{1}$ is well-posed at least in $H^{-s}$ with $s>0$ and its solution $u(x,t)$ can be obtained from solving the Marchenko integral equation and written as

[TABLE]

where $\varphi$ is defined by (5.1). As we proved in [12], $\mathbb{H}(x,t)$ is trace class and hence $\det\left\{1+\mathbb{H}(x,t)\right\}$ is well-defined in the classical Fredholm sense. To prove the necessary smoothness we show that the condition (1.5) provides five continuous $x$ derivatives of $\mathbb{H}(x,t)$ (and one in $t$). This will be done in the next section. Incidentally, differentiability of the Fredholm determinant is also discussed in [19] under additional smoothness assumptions on the initial data.

Theorem 5.1 and the well-known formula

[TABLE]

readily imply

Theorem 5.2** (separation of infinities principle for KdV).**

The solution to the Cauchy problem for the KdV equation (1.1) with $q\in L_{1}^{1}$ can be written in the following forms

[TABLE]

where $u_{+}(x,t)$ is the solution to (1.1) with data $q_{+}$ and $\mathbb{H}_{+}(x,t)=\mathbb{H}\left(\varphi_{+}\right)$.

This theorem is a manifestation of the unidirectional nature of the KdV equation. The effect of the part of initial data supported on $\left(-\infty,0\right)$ is encoded in the Hankel operator $\mathbb{H}\left(\Phi\right)$ with an analytic symbol, while the part $\mathbb{H}_{+}(x,t)$ is solely determined by the data on $\left(0,\infty\right)$. Theorem 5.2 provides a convenient starting point to extending the IST formalism to initial data $q$ beyond the realm of the short range scattering. Since, in general, there is no inverse scattering procedure available outside of the short range setting we have to rely on suitable limiting arguments.

6. Proof of the Main Theorem

With most of ingredients prepared in the previous sections very little is left to prove Theorem 1.1. Take $b<0$ and consider the problem (1.1) with initial data $q_{b}=\left.q\right|_{\left(b,\infty\right)}$. By Theorem 5.2 for its solution we have

[TABLE]

where

[TABLE]

and is the right reflection coefficient from $\left.q\right|_{\left(b,0\right)}$. As is well-known (see, e.g. [5]), $R_{b}$ is a meromorphic function on the entire plane, and [14] uniformly on compacts in $\mathbb{C}^{+}$

[TABLE]

where $m_{-}\left(k^{2}\right)$ is the Titchmarsh-Weyl m-function of $\mathbb{L}_{q_{-}}^{D}$, the Schrödinger operator on $L^{2}\left(-\infty,0\right)$ with a Dirichlet boundary condition at [math]. As is well-known, $m_{-}\left(\lambda\right)$ is analytic on $\mathbb{C}$ away from the spectrum of $\mathbb{L}_{q_{-}}^{D}$ which due to the condition 1.4 is bounded from below (see e.g. [8]). Consequently, $R$ is444$R$ can be interpreted as the (right) reflection coefficient from $q_{-}$ (see [14], [20] for details). analytic in $\mathbb{C}^{+}$ away from purely imaginary points $\lambda$ such that $\lambda^{2}$ is in the negative spectrum of $\mathbb{L}_{q_{-}}^{D}$. Thus

[TABLE]

is an analytic function on $\mathbb{C}^{+}$ away from a bounded set on the imaginary line. In turn this means that $\Phi=\lim_{b\rightarrow-\infty}\Phi_{b}$ is an entire function and $\mathbb{H}\left(\Phi_{b}\right)\rightarrow\mathbb{H}\left(\Phi\right)\mathbb{\ }$in trace norm. Following same arguments as in Section 4 (see also [20] for more details) we see that for every $n,m$

[TABLE]

Turn now to

[TABLE]

[TABLE]

Since $\phi_{+}$ is a rational function, $\mathbb{H}\left(\phi_{+}\right)$ is smooth in $\left(x,t\right)$ in trace norm.

By (3.8) we have

[TABLE]

where

[TABLE]

We remind that $\varphi_{0},\varphi_{1}$ are both bounded at $k=0$ as $T_{+}\left(k\right)$ vanishes at $k=0$ to order 1555In fact, it happens generically. For the so-called exceptional potentials $T\left(0\right)\neq 0$ but an arbitrarily small perturbation turns such a potential into generic. In our case it can be achieved by merely shifting the data $q$ (the KdV is translation invariant).. Apparently,

[TABLE]

where

[TABLE]

The symbols $f\xi_{x,t}G_{0,1}$ in (6.6) are different from the ones studied in Section 4 by a factor $T_{+}$ of the form $T_{+}=h/B$, where $h\in H_{+}^{\infty}$ and $B$ is the finite Blaschke product with simple zeros at $\left(i\kappa_{n}^{+}\right)$. This is however a purely technical circumstance in the way of applying Theorem 4.1. The easiest way to circumvent it is to alter our original $q$ by performing the Darboux transform on $q_{+}$ removing all (negative) bound states of $\mathbb{L}_{q_{+}}$. Then $T_{+}=h\in H_{+}^{\infty}$. But if $h\in H_{+}^{\infty}$ and $\varphi\in L^{\infty}$ one easily sees that666Note that $h\left(-k\right)=\overline{h\left(k\right)}$.

[TABLE]

where $\mathbb{T}\left(\overline{h}\right)=\mathbb{P}_{+}\overline{h}$ is the Toeplitz operator with symbol $\overline{h}$. The letter is a bounded operator independent of $\left(x,t\right)$ and smoothness in trace norm of $\mathbb{H}\left(T_{+}\xi_{x,t}G_{0,1}\right)$ with respect of $\left(x,t\right)$ is the same as $\mathbb{H}\left(\xi_{x,t}G_{0,1}\right)$. As is well-known, adding back the previously removed bound states results in adding solitons corresponding to $-\left(\kappa_{n}^{+}\right)^{2}$ (which are of Schwartz class).

Recalling from Proposition 3.1 that

[TABLE]

one concludes that if $q\in L_{N}^{1}$ then $Q^{\prime}\in L_{N-1}^{1}$. By Theorem 4.1 if $N=5/2$ then $\mathbb{H}\left(\xi_{x,t}G_{0}\right)$ and $\mathbb{H}\left(\xi_{x,t}G_{1}\right)$ are differentiable in $x$ in $\mathfrak{S}_{1}$ four and three times respectively. By (6.5) and (6.6) $\mathbb{H}\left(\xi_{x,t}R_{+}\right)$ is differentiable in $x$ in $\mathfrak{S}_{1}$ five times and hence so is $\mathbb{H}_{+}(x,t)$. Thus, since $\partial_{t}\xi_{x,t}=\partial_{x}^{3}\xi_{x,t}$, the formula (6.1) defines a classical solution $u_{b}\left(x,t\right)$ with initial data $q_{b}$ and it remains to let $b\rightarrow-\infty$. But it follows from (6.4) that

[TABLE]

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 $q\in L_{N}^{1}$ then $u\left(x,t\right)$ is continuously differentiable $\left\lfloor 2N\right\rfloor-2$ times in $x$ and $\left\lfloor\left(2N-2\right)/3\right\rfloor$ times in $t$.

We conclude this section with yet another solution formula, which can be viewed as a generalized Dyson formula.

Theorem 6.2**.**

Under conditions of Theorem 1.1, the solution to (1.1) can be represented by

[TABLE]

with

[TABLE]

where $R$ is the right reflection coefficient of $q$ and $d\rho$ is a positive finite measure.

Note that the pair $\left(R,\rho\right)$ can be viewed as scattering data associated with $\mathbb{L}_{q}$ and only (6.8) needs proving. It is proven in our [14] where a complete treatment of $\left(R,\rho\right)$ is also given.

7. Conclusions

Theorem 6.1 says that, loosely speaking, the KdV flow* *instantaneously smoothens any (integrable) singularities of $q\left(x\right)$ as long $q\left(x\right)=o\left(x^{-2}\right)$, $x\rightarrow+\infty$. Such an effect is commonly referred to as dispersive smoothing. This smoothing property becomes stronger as the rate of decay at $+\infty$ increases, the behavior at $-\infty$ playing no role. In [21] we show that if ($C,\delta>0$)

[TABLE]

then (1) if $\delta>1/2$ then $u(x,t)$ is meromorphic with respect to $x$ on the whole complex plane (with no real poles) for any $t>0$; (2) if $\delta=1/2$ then $u(x,t)$ is meromorphic in a strip around the $x-$axis widening proportionally to $\sqrt{t}$; (3) for $0<\delta<1/2$ the solution need not be analytic but is at least Gevrey smooth.

Actually, the requirement that $q$ is locally integrable can be lifted. By employing the arguments from our [13] we may easily extend all our results to include $H^{-1}$ type singularities (like Dirac $\delta-$functions, Coulomb potentials, etc.) on any interval $\left(-\infty,a\right)$.

The condition (1.4) is optimal. Indeed, what we actually need is semiboundedness of $\mathbb{L}_{q}$ from below, which is guaranteed by (1.4). If $q$ is negative then (1.4) becomes also necessary [8].

The absence of decay at $-\infty$ ruins any hope that classical conservations laws would take place. We do not however rule out existence of some regularized conservation laws or at least some energy estimates. It would of course be important to find such estimates.