The effect of a positive bound state on the KdV solution. A case study

TL;DR

This paper investigates how a positive bound state influences the solution of the KdV equation for a specific oscillatory potential, revealing a new bounded term in the solution related to the positive eigenvalue.

Contribution

It demonstrates that the KdV equation admits a classical solution for a potential with a positive eigenvalue, introducing a novel bounded term akin to a positon, using Hankel operators.

Findings

The solution includes a new bounded term due to the positive eigenvalue.

The solution can be expressed in closed form using Hankel operators.

The approach extends inverse scattering methods to non-rapidly decaying potentials.

Abstract

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schr\"odinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extend this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

Equations

$$\begin{array}[c]{cc}\partial_{t}u-6u\partial_{x}u+\partial_{x}^{3}u=0,&-\infty<x<\infty,\ t\geq 0,\\ u\left(x,0\right)=q\left(x\right).&\end{array}$$

$${\mathbb{L}}_q=-{\partial }_x^2+q(x)\text{ on }{L}^2\left(\mathbb{R}\right).$$

$$u\left(x,t\right)=-2{\partial }_x^2\log \tau \left(x,t\right),$$

$$q\left(x\right)=\left(c/x\right)\sin 2x+O\left(x^{-2}\right),x\to \pm \infty .$$

$$Q\left(x\right)=-2\frac{d^2}{d{x}^2}\log \left(1+\rho x-\frac{\rho }{2}\sin 2x\right),$$

$$\underset{h>0}{\sup }{\int }_{\mathbb{R}\pm ih}{\left|f\left(z\right)\right|}^2\left|dz\right|<\infty .$$

$$⟨f,g{⟩}_{H_{\pm }^2}=⟨f,g{⟩}_{L^2}=⟨f,g⟩=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f\left(x\right)\bar{g}\left(x\right)dx.$$

$$({\mathbb{P}}_{\pm }f)(x)$$

$$=\pm \frac{1}{2\pi i}{\int }_{-\infty }^{\infty }\frac{f(s)ds}{s-(x\pm i0)}.$$

$${\mathbb{P}}_{-}\left(\frac{1}{\cdot -\lambda }f\right)=\frac{1}{\cdot -\lambda }f(\lambda ),\lambda \in {\mathbb{C}}^{+}.$$

$$k_{\lambda }\left(z\right):=\frac{i}{z-\overline{\lambda }},\lambda \in {\mathbb{C}}^{\pm }$$

$$\Vert k_{\lambda }\Vert =\sqrt{⟨k_{\lambda },k_{\lambda }⟩}=\frac{1}{\sqrt{2\mathrm{Im}\lambda }}$$

$$f$$

$$B\left(z\right)=\prod _n{b}_n\left(z\right),b_n\left(z\right)=\frac{z-z_n}{z-\overline{z_n}}.$$

$$K_B=\mathrm{span}\left\{k_{z_n}\right\}.$$

$$K_B=H^2\ominus B{H}^2,\text{ where }B{H}^2:=\left\{Bf:f\in H^2\right\}.$$

$${\mathbb{P}}_B=B{\mathbb{P}}_{-}\overline{B},{\mathbb{P}}_B^{\perp }=B{\mathbb{P}}_{+}\overline{B}.$$

$${\left({\mathbb{P}}_{B}A{P}_B\right)}_{mn}=⟨A{k}_{z_n},k_{z_m}^{\perp }⟩,$$

$$k_{z_n}^{\perp }\left(z\right):=\frac{2\mathrm{Im}{z}_n}{B_n\left(z_n\right)}{B}_n\left(z\right)k_{z_n}\left(z\right),B_n:=B/b_n$$

$${\mathbb{P}}_{-}\overline{B}f=-\frac{1}{2\pi i}{\int }_{-\infty }^{\infty }\frac{f\left(s\right)}{B\left(s\right)}\frac{ds}{s-\left(x-i0\right)}$$

$$\left({\mathbb{P}}_{-}\overline{B}f\right)\left(x\right)$$

$$=\sum _n\frac{2i\mathrm{Im}{z}_n}{B_n\left(z_n\right)}\frac{f\left(z_n\right)}{x-z_n}$$

$$=\sum _n\frac{2i\mathrm{Im}{z}_n}{B_n\left(z_n\right)}\frac{⟨f,k_{z_n}⟩}{x-z_n}\text{ (by (refform for rep ker)).}$$

$${\mathbb{P}}_{B}f$$

$$=\sum _n⟨f,k_{z_n}⟩k_{z_n}^{\perp },$$

$$⟨k_{z_n}^{\perp },k_{z_m}⟩$$

$$=\frac{2\mathrm{Im}{z}_n}{B_n\left(z_n\right)}{B}_n\left(z_m\right)k_{z_n}\left(z_m\right).$$

$$⟨k_{z_n}^{\perp },k_{z_m}⟩=2\mathrm{Im}{z}_n{k}_{z_n}\left(z_n\right)=1.$$

$$(\mathbb{H}f)(x)={\int }_0^{\infty }h(x+y)f(y)dy,f\in L^2({\mathbb{R}}_{+}),x\ge 0$$

$$(\mathbb{J}f)(x)=f(-x)$$

$$\mathbb{H}(\phi )f={\mathbb{JP}}_{-}\phi f,f\in H^2,$$

$$\mathbb{H}(\phi +h)=\mathbb{H}(\phi )\text{ for any }h\in H^{\infty }.$$

Full text

The author is supported in part by the NSF grant DMS 1716975.

1. 1 Introduction

2. 2 Our analytic tools

3. 2.1 Riesz projections

4. 2.2 Reproducing kernels

5. 2.3 Hankel operators

6. 3 Our explicit potential and its short-range approximation

7. 3.1 An explicit WvN type potential

8. 3.2 Short-range approximation of $Q$

9. 4 Main results

1. Introduction

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

[TABLE]

As is a well-known, for smooth rapidly decaying $q$’s (1.1) was solved in closed form in the short 1967 paper [11] by Gardner-Greene-Kruskal-Miura (GGKM). This seminal paper introduces what we now call the* inverse scattering transform *(IST). Conceptually, it is similar to the Fourier transform (see e.g. the classical books [1], [29]) but based on the inverse scattering theory for the Schrödinger operator

[TABLE]

Moreover, the solution $q\left(x,t\right)$ to (1.1) for each $t>0$ can be obtained by the formula

[TABLE]

where $\tau$ is the so-called Hirota tau-function introduced in [17] which admits an explicit representation in terms of the scattering data of the pair $\left(\mathbb{L}_{q},\mathbb{L}_{0}\right)$. The solution has a relatively simple and by now well understood wave structure of running (finitely many) solitons accompanied by radiation of decaying waves (see e.g. Grunert-Teschl [14] for a streamlined modern exposition). In about 1973, the IST was extended to $q$’s rapidly approaching different constants $q_{\pm}$ as $x\rightarrow\pm\infty$ (step initial profile). It appeared first in the physical literature [15] and was rigorously treated in 1976 by Hruslov111Also transcripted as Khruslov. [18]. The formula (1.3) is also available in this case with an explicit representation of the  tau-function in terms of certain scattering data. We refer to our recent [13] and [32] where (1.3) is extended to essentially arbitrary $q$’s with a rapid decay only at $+\infty$. The main feature of such initial profiles is infinite sequence of solitons emitted by the initial step. Note that a complete rigorous investigation of all other asymptotic regimes and their generalizations was done only recently by Teschl with his collaborators (see e.g. [4], [9], [10]).

Another equally important and explicitly solvable case is when $q$ is periodic. The periodic IST is quite different from the GGKM one and is actually the inverse spectral transform (also abbreviated as IST) since it relies on the Floquet theory for $\mathbb{L}_{q}$ and analysis of Riemann surfaces and hence is much more complex than the rapidly decaying case. The solution $u\left(x,t\right)$ is given essentially by the same formula (1.3), frequently referred to as the Its-Matveev formula [19] (see also [8] by Dubrovin-Matveev-Novikov and the 2003 Gesztesy-Holden book [12] where a complete history is given), but $\tau$ is a multidimensional222Infinite dimensional in general. theta-function of real hyperelliptic algebraic curves explicitly computed in terms of spectral data of the associated Dirichlet problem for $\mathbb{L}_{q}$. It is therefore very different from the rapidly decaying case. The main feature of a periodic solution is its quasi-periodicity in time $t$.

We have outlined two main classes of initial data $q$ in (1.1) for which a suitable form of the IST was found during the initial boom followed by [11]. Such progress was possible due to well-developed inverse scattering/spectral theories for the underlying potentials $q$. However, while we have proven [13] that no decay at $-\infty$ is required to do the IST but slower than $x^{-2}$ decay at $+\infty$ results in serious complications. The main issue here is that the classical inverse scattering theory, the foundation for the IST, has not been extended beyond short-range potentials, i.e. $q\left(x\right)=O\left(\left|x\right|^{-2-\varepsilon}\right)$, $x\rightarrow\pm\infty$. We emphasize that during the boom in scattering theory there was a number of results on (direct) scattering/spectral theory for a variety of long-range potentials but the inverse scattering theory is a different matter. It was shown in 1982 [2] that the short-range scattering data no longer determine the potential uniquely even in the case when $q\left(x\right)=O\left(x^{-2}\right)$ and it is not merely a technical issue of adding some extra data. The problem appears to be open even for $L^{1}$ potentials (see Aktuson-Klaus [3]) for which all scattering quantities are well-defined but may exhibit an erratic behavior at zero energy which is notoriously difficult to analyze and classify. Besides, a possible infinite negative spectrum begets an infinite sequence of norming constants which can be arbitrary. Consequently, it is even unclear how to state a (well-posed) Riemann-Hilbert problem which would solve the inverse scattering problem. Once we leave $L^{1}$ then infinite embedded singular spectrum may appear leaving no hope to figure out what true scattering data might be. We note that any attempt to try the inverse spectral transform instead runs into equally difficult problems (see, e.g. our [31] and the literature cited therein) as spectral data evolve in time under the KdV flows by a simple law essentially only for the so-called finite gap potentials. In addition, it makes sense to find a suitable IST for (1.1) if (1.1) is actually well-posed. The seminal 1993 Bourgain’s paper [6] says that (1.1) is well-posed if $q$ is in $L^{2}$ and not much better result should be expected regarding the decay at $+\infty$.

In the current paper we look into a specific representative of the important class of continuous potentials asymptotically behaving like

[TABLE]

In the half line context such potentials333In fact, for 3D radially symmetric potentials. first appeared in 1929 in the famous paper [25] by Wigner-von Neumann where they explicitly constructed a potential of type (1.4) with $c=-8$ which supports bound state $+1$ embedded in the absolutely continuous spectrum. Note that in general any $q$ of type (1.4) with $\left|c\right|>2$ may support a bound state $+1$ which is extremely unstable and turns into the so-called Wigner-von Neumann resonance under a small perturbation. If $\left|c\right|>1/\sqrt{2}$ then the negative spectrum (necessarily discrete) of $\mathbb{L}_{q}$ is infinite in general [20]. While there is a very extensive literature on potentials of type (1.4) (commonly referred to as Wigner-von Neumann type potentials) but, as Matveev puts it in [7], ”The related inverse scattering problem is not yet solved and the study of the related large times evolution is a very challenging problem”. Observe that since any Wigner-von Neumann potential is clearly in $L^{2}$, the Bourgain Theorem [6] guarantees well-posedness of (1.1) and the good open problem is if we can solve it by a suitable IST. Our goal here is to investigate a specific case of (1.4) which can be done by the IST. Namely, we consider an even potential $Q\left(x\right)$ defined for $x\geq 0$ by

[TABLE]

where $\rho$ is an arbitrary positive constant. One can easily check that $Q\,$ is continuos and behaves like (1.4) with $c=-4$. The main feature of $Q$ is that $\mathbb{L}_{Q}$ admits an explicit spectral analysis and consequently the scattering problem for the pair $\left(\mathbb{L}_{Q},\mathbb{L}_{0}\right)$ can also be solved explicitly. In particular, $+1$ is a positive bound state of $\mathbb{L}_{Q}$ but its negative spectrum consists of just one bound state. We show that for (1.1) with initial data $Q$ the tau-function in (1.3) can be explicitly calculated. The formula however is expressed in the language of Hankel operators (which is not commonly used in integrable systems) and we have to postpone it till Section 4. We only mention here that, comparing to the short range case, the tau-function $\tau$ gains an extra factor responsible for the positive bound state. Unfortunately, we were unable to find the IST even in this case but we able to detour it by means of suitable limiting arguments. Our limiting arguments are based on certain short range approximations of $Q$ combined with techniques of Hankel operators developed in our [13].

The reader will see that our approach is not restricted to just one initial condition and should work for a whole class of initial data (at least [28] gives some hopes). We however do not make an attempt to be more general for two reasons. First of call, our consideration would complicate a great deal due to numerous extra technicalities. But the main reason is that the scattering theory, the backbone of our approach, is not developed well enough outside of short-range potentials. (At least not to our satisfaction). For instance, there are only some results on regularity properties of scattering data for Wigner-von Neumann type potentials (see [21]) but almost nothing is known about their small energy behavior. The latter was posed as an open question in [21] but, to the best of our knowledge, there has been no progress in this direction since then. This is a major impediment to our approach as it requires a careful control of the scattering matrix at all energy regimes.

2. Our analytic tools

To translate our problem into the language of Hankel operators some common definitions and facts are in order [26], [30].

2.1. Riesz projections

Recall, that a function $f$ analytic in the upper half plane $\mathbb{C}^{\pm}:=\left\{z|\pm\operatorname{Im}z>0\right\}$ is in the Hardy space $H_{\pm}^{2}$ of $\mathbb{C}^{\pm}$ if

[TABLE]

It is a fundamental fact of the theory of Hardy spaces that any $f\in H_{\pm}^{2}$ has non-tangential boundary values $f\left(x\pm i0\right)$ for almost every (a.e.) $x\in\mathbb{R}$ and $H_{\pm}^{2}$ are subspaces of $L^{2}:=L^{2}\left(\mathbb{R}\right)$. Thus, $H_{\pm}^{2}$ are Hilbert spaces 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]

In what follows, we set $H_{+}^{2}=H^{2}$. Notice that for any $f\in H^{2}$

[TABLE]

Besides $H_{\pm}^{2}$, we will also use $H_{\pm}^{\infty}$, the algebra of uniformly bounded in $\mathbb{C}^{\pm}$ functions.

2.2. Reproducing kernels

Recall that, a given fixed $\lambda\in\mathbb{C}^{\pm}$ the function

[TABLE]

is called the* reproducing (or Cauchy-Szego) kernel* for $H_{\pm}^{2}$. Clearly,

[TABLE]

and hence $k_{\lambda}\in H_{\pm}^{2}$ if $\lambda\in\mathbb{C}^{\pm}$. The main reason why reproducing kernels are convenient is the following

[TABLE]

[TABLE]

Introduce

[TABLE]

It is an easy but nevertheless fundamentally important fact in interpolation of analytic functions, the study of the shift operator, so-called model operators, etc. that

[TABLE]

Lemma 2.1**.**

The orthogonal projections $\mathbb{P}_{B}$ of $H^{2}$ onto $K_{B}$ and $\mathbb{P}_{B}^{\bot}=I-\mathbb{P}_{B}$ are given by

[TABLE]

Furthermore, if $A$ is a linear bounded operator in $H^{2}$ then the matrix of $\mathbb{P}_{B}A\mathbb{P}_{B}$ with respect to $\left(k_{z_{n}}\right)$ is given by

[TABLE]

where

[TABLE]

form a bi-orthogonal basis for $\left(k_{z_{n}}\right)$. I.e., $\left\langle k_{z_{n}}^{\bot},k_{z_{m}}\right\rangle=\delta_{nm}$.

Proof.

(2.7) are proven in [27]. To show (2.8) we first explicitly evaluate $\mathbb{P}_{B}$. By (2.1) for $f\in H^{2}$ we have

[TABLE]

and by residues

[TABLE]

Hence, by (2.7),

[TABLE]

where $k_{z_{n}}^{\bot}$ is given by (2.9). It remains to verifies that $\left(k_{z_{n}}^{\bot}\right)$ forms a bi-orthogonal basis for $K_{B}$. Indeed,

[TABLE]

If $n\neq m$ then $B_{n}\left(z_{m}\right)=0$. If $n=m$ then by (2.4)

[TABLE]

The formula (2.8) easily follows now. ∎

2.3. Hankel operators

A Hankel operator is an infinitely dimensional analog of a Hankel matrix, a matrix whose $(j,k)$ entry depends only on $j+k$. In the context of integral operators the Hankel operator is usually defined as an integral operator on $L^{2}(\mathbb{R}_{+})$ whose kernel depends on the sum of the arguments

[TABLE]

and it is this form that Hankel operators typically appear in the inverse scattering formalism. It is much more convenient for our purposes to consider Hankel operators on $H^{2}$ (cf. [26], [30]).

Let

[TABLE]

be the operator of reflection on $L^{2}$ and let $\varphi\in L^{\infty}$. The operators $\mathbb{H}(\varphi)$ defined by

[TABLE]

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

It is clear that $\mathbb{H}(\varphi)$ is bounded from $H^{2}$ to $H^{2}$ and

[TABLE]

It is also straightforward to verify that $\mathbb{H}(\varphi)$ is selfadjoint if $\mathbb{J}\varphi=\bar{\varphi}.$

The following elementary lemma on Hankel operators with analytic symbols will be particularly useful.

Lemma 2.2**.**

Let a function $\varphi$ be meromorphic on $\mathbb{C}$ and subject to

[TABLE]

If $\varphi$ has finitely many simple poles $\left\{z_{n}\right\}_{n=-N}^{N}$ in $\mathbb{C}^{+}$, is bounded on $\mathbb{R}$, and for any $h\geq 0$

[TABLE]

then the Hankel operator $\mathbb{H}(\varphi)$ is selfadjoint, trace class, and admits the decomposition

[TABLE]

where $\phi$ is a rational function and $\Phi$ is an entire function given respectively by

[TABLE]

[TABLE]

Moreover,

[TABLE]

[TABLE]

where $k_{\lambda}\left(z\right)=\frac{i}{z-\overline{\lambda}}$ is the reproducing kernel of $H^{2}$.

Proof.

The selfadjointness follows from (2.13). By (2.12)

[TABLE]

and hence we have to worry only about $\mathbb{P}_{-}\varphi$. By by the residue theorem ($h>\max_{k}\operatorname{Im}z_{k}$), we have

[TABLE]

and (2.15) follows. Apparently $\Phi$ is analytic (and bounded) below the line $\mathbb{R}+ih$. Since $h$ is arbitrary, $\Phi$ is then entire. Moreover, all derivatives of $\Phi$ are bounded on $\mathbb{R}$ and therefore $\mathbb{H}(\Phi)$ is at least trace class (in any Shatten-von Neumann ideal).

It follows from (2.2) that for any $z\in\mathbb{C}^{+}$

[TABLE]

and (2.17)-(2.18) follow. ∎

Corollary 2.3**.**

If $\varphi$ has no poles in $\mathbb{C}^{+}$ then $\mathbb{H}(\varphi)=\mathbb{H}(\Phi)$.

Corollary 2.4**.**

If (2.14) holds uniformly in $h\geq h_{0}>\max_{n}\operatorname{Im}z_{n}$ then $\Phi=0.$

A very important feature of analytic symbols is that $\mathbb{H}(\varphi)$ is well-defined outside of $H^{2}$. In particular, $\mathbb{H}(\varphi)k_{x+i0}$ is a smooth element of $H^{2}$ for any $x\in\mathbb{R}$ while $k_{x+i0}\not\in H^{2}$. We will need the following statement.

Corollary 2.5**.**

For every $x,s\in\mathbb{R}$

[TABLE]

Moreover, if $\varphi_{\varepsilon}\rightarrow\varphi$ uniformly on $\mathbb{R}+ih$ then for every $x,s\in\mathbb{R}$

[TABLE]

Convergence in (2.19) and (2.20) also holds in $L^{2}$.

Proof.

It follows from (2.18) that

[TABLE]

where we have used two obvious facts: (a) $k_{x+i\varepsilon}\left(z\right)\rightarrow k_{x}\left(z\right)$ uniformly on $\mathbb{R}+ih$, and (b) by the Lebesgue dominated convergence

[TABLE]

Thus (2.19) is proven. (2.20) is proven similarly. ∎

3. Our explicit potential and its short-range

approximation

In this section we explicitly construct a symmetric Wigner-von Neumann type potentials supporting one negative and one positive bound state. Our construction is base upon a classical Gelfand-Levitan example [22] of an explicit potential of a half-line Schrödinger operator which spectral measure has one positive pure point. The symmetric extension of this potential to the whole line will be our initial condition. We then find its explicit short range approximation, which will be crucial to our consideration.

3.1. An explicit WvN type potential

Consider the function

[TABLE]

where $\rho$ is some positive number. This is a Herglotz function (i.e. analytic function mapping $\mathbb{C}^{+}$ to $\mathbb{C}^{+}$) which coincides with the Titchmarsh-Weyl $m-$function555We recall that the problem $-\partial_{x}^{2}u+q(x)u=\lambda u,\ x\in\left(0,\pm\infty\right),\ u\left(\pm 0,\lambda\right)=1$ has a unique square integrable (Weyl) solution $\Psi_{\pm}(x,\lambda)$ for any $\operatorname{Im}\lambda>0$ for broad classes of $q$’s (called* limit point case*). Define then the (Titchmarsh-Weyl) m-function $m_{\pm}$ for $\left(0,\pm\infty\right)$ as follows: $m_{\pm}\left(\lambda\right)=\pm\partial_{x}\Psi_{\pm}\left(\pm 0,\lambda\right)$. of the (Dirichlet) Schrödinger operators $-d^{2}/dx^{2}+q_{0}\left(x\right)$ on $L^{2}\left(0,\infty\right)$ with a Dirichlet boundary condition at [math]. The potential $q_{0}$ has the following explicit form

[TABLE]

where

[TABLE]

Introduce

[TABLE]

i.e., $Q$ is an even extension of $q_{0}$. One can easily see that the function $Q$ is continuous and $Q\left(0\right)=0$ but not continuously differentiable. In fact, $Q\,$ is as smooth at $x=0$ as $\left|\sin x\right|$. Moreover, one has

[TABLE]

and hence $Q\in L^{2}\left(\mathbb{R}\right)$ but $\left(1+\left|x\right|\right)Q\left(x\right)$ is not in $L^{1}\left(\mathbb{R}\right)$. Thus, $Q$ is not short-range. Also note that

[TABLE]

The main feature of $Q$ is that $\mathbb{L}_{Q}$ admits an explicit spectral and scattering theory.

Theorem 3.1**.**

The Schrödinger operator $\mathbb{L}_{Q}$ on $L^{2}\left(\mathbb{R}\right)$ with $Q$ given by (3.4) has the following properties:

1. (1)

(Spectrum) The spectrum of $\mathbb{L}_{Q}$ consists of the two fold absolutely continuos part filling $\left(0,\infty\right)$, one negative bound state $-\kappa^{2}$ found from the real solution of

[TABLE]

and one positive (embedded) bound state $+1$. 2. (2)

(Scattering quantities) For the norming constant $c$ of $-\kappa^{2}$ we have

[TABLE]

and for the scattering matrix we have

[TABLE]

where $T$ and $R$ are, respectively, the transmission and reflection coefficients given by

[TABLE]

[TABLE]

Proof.

Due to symmetry $m_{-}=m_{+}=m$ it follows from the general theory [33] that the eigenvalues of the Schrödinger operator $\mathbb{L}_{Q}$ are the (necessarily simple) poles of $m$ and $1/m$. Thus, $\mathbb{L}_{Q}$ has one positive bound state $+1$ (the pole of $m\left(\lambda\right)$) and one negative bound state $-\kappa^{2}$ (the zero of $m\left(\lambda\right)$). Clearly (3.6) holds. The fact about the absolutely continuos spectrum also follows from the general theory (as well as from (2) below) and therefore (1) is proven.

Turn to (2). By a direct computation one verifies that

[TABLE]

solve the Schrödinger equation $\mathbb{L}_{Q}f=k^{2}f$ for $\pm x\geq 0$ if $k\neq\pm 1$. Since clearly

[TABLE]

we can claim that $f_{\pm}$ are Jost solution corresponding to $\pm\infty$. By the general formulas (see e.g. [16])

[TABLE]

[TABLE]

[TABLE]

It remains to demonstrate (3.7). Recall the general fact (see e.g. [3]) that for any short-range $q$

[TABLE]

where $c_{n}^{\pm}$ are right/left norming constant associated with the bound states $-\kappa_{n}^{2}$ ($n=1,2,...)$ enumerated in the increasing order. If $q$ is even then $c_{n}^{+}=c_{n}^{-}=c_{n}$ and hence in our case of a single bound state $-\kappa^{2}$ we have

[TABLE]

and the first equation in (3.7) follows. The second and third equations in (3.7) can be verified by a direct computation. ∎

Remark 3.2**.**

Same way as we did in the proof, one can find an analog of Theorem 3.1 for the truncated potentials $Q|\mathbb{R}_{\pm}$. There will be no positive bound state but the formulas (3.10a)-(3.12a) immediately yield same (3.9) where $2\rho$ is replaced with $\rho$. Indeed, for $Q|\mathbb{R}_{+}$ we have

[TABLE]

and the claim follows. Moreover, (3.7) also holds for the truncated $Q$ with the same substitution. This demonstrates clearly that the standard triple $\left(R,\kappa,c\right)$ no longer constitutes scattering data.

3.2. Short-range approximation of $Q$

The simples short range approximation is based upon a truncation but the limiting procedure will not be simple. We instead approximate the scattering data. While much more complicated than truncation, the limiting procedure becomes easier to track.

If you recall the famous characterization of the scattering matrix [23] of a short-range potential, one of the conditions is that $T\left(k\right)$ can vanish on $\overline{\mathbb{C}^{+}}$ only at $k=0$. But in our case this occurs if $P\left(k\right)=0$ which happens also for $k=\pm 1$. This prompts to replace $P\left(k\right)$ in $T\left(k\right)$ given by (3.9) with $P\left(k\right)+i\varepsilon$ with some small $\varepsilon>0$. Clearly

[TABLE]

where

[TABLE]

Thus two real zeros $\pm 1$ move to $\mathbb{C}^{-}$. Form the Blaschke product $B^{\varepsilon}$ with zeros $z_{-1}=-\mu,z_{1}=\overline{\mu},z_{0}=i\nu\in\mathbb{C}^{+}$. I.e.,

[TABLE]

It follows from (3.14) that as $\varepsilon\rightarrow 0$

[TABLE]

The Blaschke product $B^{\varepsilon}$ will be a building block in our approximation. Apparently, $B^{\varepsilon}\rightarrow 1$ as $\varepsilon\rightarrow 0$ uniformly on compacts in $\mathbb{C}^{+}$ and a.e. on $\mathbb{R}$. We are now ready to present our approximation.

Theorem 3.3**.**

Let ($\varepsilon>0$)

[TABLE]

Then

1. (1)

The matrix

[TABLE]

is the scattering matrix of a short-range potential having two bound states $-\left(\kappa_{\pm}^{\varepsilon}\right)^{2},\ \kappa_{+}^{\varepsilon}>\kappa_{-}^{\varepsilon}$, subject to

[TABLE] 2. (2)

If we choose the left and right norming constants associated with $-\left(\kappa_{\pm}^{\varepsilon}\right)^{2}$ equal to each other and to satisfy

[TABLE]

then the unique potential $Q_{\varepsilon}\left(x\right)$ corresponding to the scattering data

[TABLE]

is even and everywhere

[TABLE]

Proof.

To prove part 1 of the statement one needs to check all the conditions of the Marchenko characterization [23]. Is is straightforward but quite involved and we omit it. By the general theory, the bound states are the squares of the (simple) poles of $T_{\varepsilon}$ in $\mathbb{C}^{+}$, i.e. the solutions of two

[TABLE]

Since each equation has only one imaginary solution $i\kappa_{\pm}^{\varepsilon}$, we have exactly two bound states $-\left(\kappa_{\pm}^{\varepsilon}\right)^{2}$ which are clearly subject to (3.16). Note that $z_{0}$, the zero of $b_{0}$, is a removable singularity by our very construction of $B^{\varepsilon}$.

Turn now to part 2. Consider the reflection coefficient $R_{\varepsilon}$. Apparently, $R_{\varepsilon}$ is a rational function with five simple poles. Two imaginary poles $i\kappa_{\pm}^{\varepsilon}$ are shared with $T_{\varepsilon}$ plus $z_{n},n=0,\pm 1$, the zeros of $B^{\varepsilon}\left(k\right)$. By a direct computation, one checks

[TABLE]

We now solve the inverse scattering problem for the data

[TABLE]

basing upon our Hankel operator approach [13]. To this end, form the symbol

[TABLE]

One immediately sees that $\varphi_{x}^{\varepsilon}$ is subject to the conditions of Lemma 2.2 with three (symmetric) poles $z_{n},n=0,\pm 1$. By condition, the left and right scattering data are identical and hence $Q_{\varepsilon}$ must be even and it enough to recover it only on $\left(0,\infty\right)$. Therefore we can assume that $x>0$ in (3.20) which by Corollary 2.4 implies that the $\Phi$-part of our symbol is zero. By Lemma 2.2

[TABLE]

Thus, our Hankel operator is rank 3 and by the Dyson formula [13] we have

[TABLE]

Note that our $Q_{\varepsilon}$ has an exponential decay and can be explicitly evaluated. We however don’t really need it. We will take the limit as $\varepsilon\rightarrow 0$ in the next section. ∎

We emphasize that Part 2 of Theorem 3.3 is essential because, due to nonuniqueness, it is a priori unclear if our approximations indeed converges to the original potential.

Note also, that $Q_{\varepsilon}\left(x\right)$ all have the property that $T_{\varepsilon}\left(0\right)\neq 0$. Such potentials are called exceptional because generically $T\left(0\right)=0.$

4. Main results

Through this section

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Theorem 4.1**.**

Let $Q$ be the initial condition (3.4) in the KdV equation (1.1),

[TABLE]

and $\mathbb{H}_{x,t}:=\mathbb{H}\left(\varphi_{x,t}\right)$, the associated Hankel operator. Then (1.1) has the (unique) classical solution given by

[TABLE]

where

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and

[TABLE]

[TABLE]

Here, as before,$\ k_{\lambda}\left(s\right)=\dfrac{i}{s-\overline{\lambda}}$ is the reproducing kernel.

Proof.

Since our approximation $Q_{\varepsilon}\left(x\right)$ decays exponentially, the (classical) solution to the KdV equation can be found in closed form by Dyson’s formula

[TABLE]

where

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Note that due to the Bourgain theorem [6] the limit $\lim_{\varepsilon\rightarrow 0}Q_{\varepsilon}\left(x,t\right)$ does exist but we cannot pass to the limit in (4.3) under the determinant sign since, as we will see later, $\mathbb{H}\left(\varphi_{x,t}^{\varepsilon}\right)$ doesn’t converge in the trace norm to $\mathbb{H}\left(\varphi_{x,t}\right)$, where

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To detour this circumstance we split our determinant as follows. Consider

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and decompose $H^{2}$ into the orthogonal sum (see Subsection 2.2)

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The decomposition (4.5) induces the block representation

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where

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and

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Examine the block $\mathbb{H}_{1}$ first. It follows from (4.4) that the poles of $\varphi_{x,t}^{\varepsilon}$ coincide with zeros $\left(z_{n}\right)$ of $B^{\varepsilon}$ and therefore by Lemma 2.2 ($h>\kappa_{+}^{\varepsilon}$)

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One immediately sees that

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Since $\varphi_{x,t}^{\varepsilon}\rightarrow\varphi_{x,t}$ uniformly on $\mathbb{R}+ih$, we obviously have

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in $C^{n}\left(\mathbb{R}\right)$ for any $n$ which in turn implies [30] that $\lim_{\varepsilon\rightarrow 0}\mathbb{H}\left(\Phi_{x,t}^{\varepsilon}\right)=\mathbb{H}\left(\Phi_{x,t}\right)$ in the trace norm (in fact in all $\mathfrak{S}_{p},p>0$). Since

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we see that $i\kappa$ is a removable singularity for $\varphi_{x,t}$ and hence by Corollary 2.3

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Since $B^{\varepsilon}\rightarrow 1$ a.e., it follows from (2.7) that in the strong operator topology

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But [5], if $H_{n}\rightarrow H$ in trace norm, $A_{n}$ is self-adjoint, $\sup_{n}\left\|A_{n}\right\|<\infty$, and $A_{n}\rightarrow A$ strongly, then $A_{n}H_{n}A_{n}\rightarrow AHA$ in trace norm. Therefore, we can conclude that in trace norm

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We now make use of a well-known formula from matrix theory:

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which yields

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Our goal is to study what happens to (4.9) as $\varepsilon\rightarrow 0$. The determinants on the right hand side of (4.9) behave very differently and we treat them separately. It follows from (4.7) that

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Turn now to the second determinant in (4.9). It is clearly a $3\times 3$ determinant. We are going to show that, in fact, this determinant vanishes as $O\left(\varepsilon\right)$. To this end, we explicitly evaluate it in the basis $\left(k_{z_{n}}\right)$

[TABLE]

where $h_{mn}$ and $d_{mn}$ are the matrix entries of

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and

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respectively. By Lemma 2.1

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Incidentally, (4.16) implies $h_{1-1}=\overline{h_{-11}},\ h_{-1-1}=\overline{h_{11}}$. Recall that $z_{n}$ are chosen so that $P\left(z_{n}\right)-i\varepsilon=0$ if $n=\pm 1$ and $P\left(z_{n}\right)+i\varepsilon=0$ if $n=0$. Rewriting (3.15) as

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for the residues we then have

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One now readily verifies that

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and thus

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Inserting (4.17) into (4.16) yields

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Observe, that $h_{n,m}=O\left(\varepsilon\right)$ if $n\not=-m$ and $h_{m,-m}$ doesn’t vanish as $\varepsilon\rightarrow 0$ (which is an important fact for what follows). As we will see, only $h_{-11}$ and $h_{11}$ matter. Recalling that $R\left(1\right)=-1$ we have

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Similarly, for the matrix $\left(d_{mn}\right)$ we have

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where $D_{mn}$ will be computed later. For the determinant in (4.11) we clearly have

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Evaluate each term in the right hand side of (4.21) separately. By (4.18)-(4.19) one has

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and

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Since $\mathbb{H}\left(\Phi_{x,t}^{\varepsilon}\right)$ is a self-adjoint operator, by Corollary 2.5 we have ($m=\pm 1,n=1$)

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where

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Similarly, by (4.6), (4.7), and Corollary 2.5 we have

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Therefore, combining (4.26) and (4.27) we have

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Substituting this and (4.25) into (4.21) yields

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We have now prepared all the ingredients to find the solution to the KdV equation with the initial data $Q_{\varepsilon}$ by the Dyson formula. Indeed,

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We are now able to fill the gap left in the proof of Theorem 3.3, i.e. (3.18). To this end, set $t=0$ in (4.29) and take $x>0$. In this case $\xi_{x,0}\in H^{\infty}$ and hence $\varphi_{x,0}\in H^{\infty}$. Therefore, $\mathbb{H}\left(\varphi_{x,t}\right)=0$ and by the Lebesgue dominated convergence theorem (or by Corollary 2.5) we also have

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Eq. (4.29) simplifies now to read

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Recalling (3.2), we conclude that $Q_{\varepsilon}\left(x\right)=Q_{\varepsilon}\left(x,0\right)\rightarrow q_{0}\left(x\right)$ for $x>0$. Since $Q_{\varepsilon}\left(x\right)$ is even, (3.18) follows.

Pass now in (4.29) to the limit as $\varepsilon\rightarrow 0$. Apparently,

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By the Bourgain theorem $Q\left(x,t\right)=\lim_{\varepsilon\rightarrow 0}Q_{\varepsilon}\left(x,t\right)$ is the (unique) solution to the KdV equations with data $Q\left(x\right)$. Recalling Corollary 2.5, we see that

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This completes the proof of the theorem. ∎

Note that the first term $u_{0}\left(x,t\right)$ in the solution (4.1) is given by the same Dyson formula (4.2) as in the short-range case but of course $u_{0}\left(x,0\right)$ is not a short range potential. The second term $u_{1}\left(x,t\right)$ in (4.1) is responsible for the bound state $+1$ and if $\rho=1$ it resembles the so-called positon solution

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Such solutions seem to have appeared first in the late 70s earlier 80s but a systematic approach was developed a decade later by V. Matveev (see his 2002 survey [24]).

The formula (4.30) readily yields basic properties of one-position solutions. (1) As a function of the spatial variable $u_{\text{pos}}\left(x,t\right)$ has a double pole real singularity which oscillates in the $1/2$ neighborhood of the moving point $x=-12t-1$. (2) For a fixed $t\geq 0$

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Observe that

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which coincides on $\left(0,\infty\right)$ with our $Q\left(x\right)$ for $\rho=1$. Moreover, comparing (3.5) with (4.31) one can see that the asymptotic behaviors for $x\rightarrow-\infty$ of our $Q\left(x\right)$ with $\rho=1$ and $u_{\text{pos}}\left(x,0\right)$ differ only by $O\left(x^{-2}\right)$. But, of course, $Q\left(x\right)$ is bounded on $\left(-\infty,0\right)$ while $u_{\text{pos}}\left(x,0\right)$ is not. Note also that the positon is somewhat similar to the soliton given by

[TABLE]

As opposed to the soliton, the positon has a square singularity (not a smooth hump) moving in the opposite direction three times as fast.

We note that multi-positon as well as soliton-positon solutions have been studied in great detail (see [24] the references cited therein). In [24] Matveev also raises the equation if there is a bounded positon, i.e. a solution having all properties of a positon but is regular. We are unable to tell if our solution is a bounded positon or not.