Spectral Weyl Spaces and Quantum Neural Asymptotics for the KdV Equation

1. Introduction

The Korteweg–de Vries (KdV) equation

$${\partial }_{t}u+{\partial }_x^{3}u+6u{\partial }_{x}u=0,(t,x)\in \mathbb{R}\times \mathbb{R},$$

(1)

is one of the most paradigmatic models in the theory of dispersive nonlinear partial differential equations. Since the seminal work of Gardner, Greene, Kruskal and Miura [3], it has been known that the KdV equation is an integrable system, admitting a Lax representation that associates to each solution u a Schrödinger operator $L_u$ whose spectrum is time-invariant.

From the viewpoint of regularity theory, the KdV equation is globally well-posed in $H^s\left(\mathbb{R}\right)$ for $s>-1/2$, as shown by Bourgain [1] and later by Killip and Vişan [4] for the critical case. However, growth estimates for higher-order Sobolev norms often involve constants that grow exponentially in time, or require a delicate control of resonant interactions.

In this paper we propose a novel approach that combines the theory of Bourgain spaces $X^{s,b}$ with the spectral invariance of the Lax operator. We introduce the spectral Weyl space${\mathcal{W}}^s$, whose norm penalises the presence of eigenvalues of the Schrödinger operator. Our main result, Theorem 1, establishes that the KdV flow is globally defined on this space and that the $H^s$-norm of the solution grows at most polynomially with exponent $s-1/2$, where the proportionality constant is controlled solely by the initial norm and the spectral functional

$$\mathcal{S}\left(u_0\right)=\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{(1+|\lambda \left|\right)}^s.$$

This represents a significant advance over classical results, since it isolates the contribution of solitons to the growth of norms, and predicts that purely radiative data (without negative eigenvalues) admit estimates with improved constants.

In the second part of this work, we apply the recent theory of quantum neural network operators (QNNOs) [7] to the approximation of the KdV flow. By treating the finite-dimensional truncation of the solution operator as a completely positive map, we construct a formal QNNO and prove an explicit asymptotic expansion. This provides a rigorous theoretical bridge between quantum machine learning and dispersive PDEs.

The paper is organised as follows. In Section 2 we fix notation and recall preliminary facts about Sobolev spaces, Bourgain spaces, and the Lax pair. In Section 3 we define the spectral Weyl space ${\mathcal{W}}^s$ rigorously and establish its basic properties. Section 4 is devoted to the statement and proof of the main theorem. In Section 5 we introduce the QNNO for the KdV flow and prove its asymptotic expansion. Section 6 discusses consequences and corollaries of the main theorem, while Section 7 summarises the main results of the paper, and Section 8 presents concluding remarks.

2. Preliminaries

We denote by $\mathcal{S}\left(\mathbb{R}\right)$ the Schwartz space of rapidly decreasing smooth functions and by ${\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$ its dual, the space of tempered distributions. The Fourier transform is the linear isomorphism $\mathcal{F}:\mathcal{S}\left(\mathbb{R}\right)\to \mathcal{S}\left(\mathbb{R}\right)$ defined by

$$\left(\mathcal{F}f\right)\left(\xi \right)=\widehat{f}\left(\xi \right)={\int }_{\mathbb{R}}{e}^{-ix\xi }f\left(x\right)dx,f\in \mathcal{S}\left(\mathbb{R}\right),$$

(2)

with inverse

$$f\left(x\right)=\left({\mathcal{F}}^{-1}\widehat{f}\right)\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{e}^{ix\xi }\widehat{f}\left(\xi \right)d\xi .$$

(3)

The Fourier transform extends by duality to a linear isomorphism $\mathcal{F}:{\mathcal{S}}^{\prime }\left(\mathbb{R}\right)\to {\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$.

For $s\in \mathbb{R}$, the Sobolev space $H^s\left(\mathbb{R}\right)$ is defined as the completion of $\mathcal{S}\left(\mathbb{R}\right)$ with respect to the norm

$${\parallel f\parallel }_{H^s}^2={\int }_{\mathbb{R}}{\langle \xi \rangle }^{2s}{\left|\widehat{f}\left(\xi \right)\right|}^{2}d\xi ,$$

(4)

where $\langle \xi \rangle ={(1+|\xi |}^2{)}^{1/2}$. Equivalently,

$$H^s\left(\mathbb{R}\right)=\left\{f\in {\mathcal{S}}^{\prime }\left(\mathbb{R}\right):{\parallel f\parallel }_{H^s}<\infty \right\}.$$

(5)

The following properties of Sobolev spaces are standard and will be used throughout.

Proposition 1

(Basic properties of Sobolev spaces). Let $s\in \mathbb{R}$. Then:

1.

$H^s\left(\mathbb{R}\right)$ is a Hilbert space with inner product

$${\langle f,g\rangle }_{H^s}={\int }_{\mathbb{R}}{\langle \xi \rangle }^{2s}\widehat{f}\left(\xi \right)\overline{\widehat{g}\left(\xi \right)}d\xi .$$

2.

The Schwartz space $\mathcal{S}\left(\mathbb{R}\right)$ is dense in $H^s\left(\mathbb{R}\right)$.

3.

For $s>t$, the embedding $H^s\left(\mathbb{R}\right)\hookrightarrow H^t\left(\mathbb{R}\right)$ is continuous and dense.

4.

The dual space of $H^s\left(\mathbb{R}\right)$ is isometrically isomorphic to $H^{-s}\left(\mathbb{R}\right)$ via the pairing

$$\langle f,g\rangle ={\int }_{\mathbb{R}}\widehat{f}\left(\xi \right)\overline{\widehat{g}\left(\xi \right)}d\xi ,f\in H^s\left(\mathbb{R}\right),g\in H^{-s}\left(\mathbb{R}\right).$$

5.

For $s>1/2$, the embedding $H^s\left(\mathbb{R}\right)\hookrightarrow L^{\infty }\left(\mathbb{R}\right)$ is continuous, and there exists a constant $C_s>0$ such that

$${\parallel f\parallel }_{L^{\infty }}\le C_s{\parallel f\parallel }_{H^s}.$$

6.

For $s>1/2$, the pointwise multiplication $(f,g)\mapsto fg$ extends to a continuous bilinear map $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)\to H^s\left(\mathbb{R}\right)$, and

$${\parallel fg\parallel }_{H^s}\lesssim {\parallel f\parallel }_{H^s}{\parallel g\parallel }_{H^s}.$$

Proof.

Properties (1)–(4) follow from the definition of $H^s$ as a weighted $L^2$-space and the standard theory of Fourier multipliers. Property (5) is the Sobolev embedding theorem; indeed, for $s>1/2$,

$${\parallel f\parallel }_{L^{\infty }}\le {\displaystyle \frac{1}{2\pi }}\parallel \widehat{f}{\parallel }_{L^1}\le {\displaystyle \frac{1}{2\pi }}{\left({\int }_{\mathbb{R}}{\langle \xi \rangle }^{-2s}d\xi \right)}^{1/2}{\left({\int }_{\mathbb{R}}{\langle \xi \rangle }^{2s}{\left|\widehat{f}\left(\xi \right)\right|}^{2}d\xi \right)}^{1/2}\le C_s{\parallel f\parallel }_{H^s}.$$

Property (6) follows from the fractional Leibniz rule:

$$\parallel {\Lambda }^s{\left(fg\right)\parallel }_{L^2}\lesssim \parallel {\Lambda }^s{f\parallel }_{L^2}{\parallel g\parallel }_{L^{\infty }}+{\parallel f\parallel }_{L^{\infty }}{\parallel {\Lambda }^{s}g\parallel }_{L^2},$$

where ${\Lambda }^s={(1-{\partial }_x^2)}^{s/2}$, combined with the embedding $H^s\hookrightarrow L^{\infty }$. □

For $s>1/2$, we shall frequently use the continuous embedding

$$H^s\left(\mathbb{R}\right)\hookrightarrow C_0\left(\mathbb{R}\right),$$

(6)

where $C_0\left(\mathbb{R}\right)$ denotes the space of continuous functions vanishing at infinity. Moreover, for $s>1/2$, the space $H^s\left(\mathbb{R}\right)$ is a Banach algebra under pointwise multiplication; in particular,

$${\parallel uv\parallel }_{H^s}\le C_s{\parallel u\parallel }_{H^s}{\parallel v\parallel }_{H^s},u,v\in H^s\left(\mathbb{R}\right).$$

(7)

Throughout this paper, we use the notation $A\lesssim B$ to mean $A\le CB$ for some constant $C>0$ that may depend only on fixed parameters (such as s), and $A\sim B$ means $A\lesssim B$ and $B\lesssim A$.

2.1. Bourgain Spaces $X^{s,b}$

For the study of KdV, we introduce the Bourgain spaces $X^{s,b}\left({\mathbb{R}}^2\right)$ adapted to the dispersion relation $\tau ={\xi }^3$. Let $\mathcal{S}\left({\mathbb{R}}^2\right)$ be the Schwartz space on the $(t,x)$-plane. We define the norm

$${\parallel u\parallel }_{X^{s,b}}^2={\int }_{{\mathbb{R}}^2}{\langle \xi \rangle }^{2s}{\langle \tau -{\xi }^3\rangle }^{2b}{\left|\widehat{u}(\xi ,\tau )\right|}^{2}d\xi d\tau ,$$

(8)

where $\widehat{u}$ denotes the space-time Fourier transform. The space $X^{s,b}$ is the completion of $\mathcal{S}\left({\mathbb{R}}^2\right)$ with respect to this norm.

A fundamental result due to Bourgain [1] is the bilinear estimate

$$\parallel {\partial }_x{\left(uv\right)\parallel }_{X^{s,b-1}}\lesssim {\parallel u\parallel }_{X^{s,b}}{\parallel v\parallel }_{X^{s,b}},$$

(9)

valid for $s>-1/2$ and $b>1/2$. This estimate is the cornerstone for the proof of local well-posedness via Picard iteration.

2.2. The Lax Pair and Spectral Invariance

The KdV equation (1) is a member of an integrable family. Consider the time-dependent Schrödinger operator

$$L_u\left(t\right)=-{\displaystyle \frac{d^2}{d{x}^2}}-u(t,x),$$

(10)

and the anti-selfadjoint operator

$$B_u=4{\displaystyle \frac{d^3}{d{x}^3}}+3\left(u{\displaystyle \frac{d}{dx}}+{\displaystyle \frac{d}{dx}}u\right).$$

(11)

The Lax equation

$${\displaystyle \frac{d}{dt}}{L}_u=[B_u,L_u]$$

(12)

is equivalent to the KdV equation [5]. Consequently, the spectrum of $L_u\left(t\right)$ is invariant in time. In particular, the discrete spectrum ${\sigma }_d\left(L_u\left(t\right)\right)\subset (-\infty ,0)$ and the corresponding multiplicities are constant along the flow.

Let $P_{\lambda }$ denote the spectral projector onto the eigenspace associated with the eigenvalue $\lambda \in {\sigma }_d\left(L_u\right)$. Spectral invariance implies

$$\sum _{\lambda \in {\sigma }_d\left(L_{u\left(t\right)}\right)}\parallel P_{\lambda }{u\left(t\right)\parallel }_{H^s}^2=\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{\parallel P_{\lambda }{u}_0\parallel }_{H^s}^2,$$

(13)

for all $t\in \mathbb{R}$, provided the solution is sufficiently regular.

3. The Spectral Weyl Space ${\mathcal{W}}^s$

Motivated by the spectral invariance, we define a new class of function spaces that incorporate the information of the discrete spectrum.

Definition 1.

Let $s>0$. Thespectral Weyl space${\mathcal{W}}^s$ is the set of all $f\in H^s\left(\mathbb{R}\right)$ such that the operator $L_f=-D_x^2-f$ has finitely many negative eigenvalues and the quantity

$${\parallel f\parallel }_{{\mathcal{W}}^s}^2:={\parallel f\parallel }_{H^s}^2+\sum _{\lambda \in {\sigma }_d\left(L_f\right)}{(1+|\lambda \left|\right)}^{2s}{\parallel P_{\lambda }f\parallel }_{H^s}^2$$

(14)

is finite.

Proposition 2.

The space $({\mathcal{W}}^s,\parallel ·{\parallel }_{{\mathcal{W}}^s})$ is a Banach space.

Proof.

We first note that the expression ${\parallel f\parallel }_{{\mathcal{W}}^s}$ defines a norm on ${\mathcal{W}}^s$. Indeed, the functional

$$f\mapsto {\parallel f\parallel }_{H^s}^2+\sum _{\lambda \in {\sigma }_d\left(L_f\right)}{(1+|\lambda \left|\right)}^{2s}{\parallel P_{\lambda }f\parallel }_{H^s}^2$$

is a sum of a Hilbert norm and a nonnegative term; the triangle inequality and homogeneity follow from the fact that the spectral projectors $P_{\lambda }$ are linear and bounded, and the eigenvalues $\lambda$ depend continuously on f (for $s>1/2$). The only possible issue is that the sum may not be additive, but it is a norm because if ${\parallel f\parallel }_{{\mathcal{W}}^s}=0$, then ${\parallel f\parallel }_{H^s}=0$, so $f=0$.

It remains to prove completeness. Let ${\left\{f_n\right\}}_{n\in \mathbb{N}}$ be a Cauchy sequence in ${\mathcal{W}}^s$. Then $\left\{f_n\right\}$ is Cauchy in $H^s$, so there exists $f\in H^s$ such that $f_n\to f$ in $H^s$. We need to show that $f\in {\mathcal{W}}^s$ and $f_n\to f$ in ${\mathcal{W}}^s$.

For $s>1/2$, the discrete spectrum ${\sigma }_d\left(L_f\right)$ is stable under $H^s$-perturbations; more precisely, there exists $\epsilon >0$ such that if ${\parallel g-f\parallel }_{H^s}<\epsilon$, then the eigenvalues of $L_g$ (counted with multiplicity) are close to those of $L_f$, and the spectral projections $P_{\lambda }\left(g\right)$ converge to $P_{\lambda }\left(f\right)$ in operator norm (see [2]). Since $f_n\to f$ in $H^s$, for large n, the eigenvalues of $L_{f_n}$ converge to those of $L_f$. In particular, the number of eigenvalues is eventually constant, and each eigenvalue $\lambda$ of $L_f$ is the limit of a sequence ${\lambda }_n\in {\sigma }_d\left(L_{f_n}\right)$.

Now, for each $\lambda \in {\sigma }_d\left(L_f\right)$, we have

$$\parallel P_{\lambda }{f}_n{\parallel }_{H^s}\to {\parallel P_{\lambda }f\parallel }_{H^s},$$

because $P_{\lambda }$ is bounded and $f_n\to f$ in $H^s$. Moreover, since $(1+|{\lambda }_n{\left|\right)}^{2s}{\parallel P_{{\lambda }_n}{f}_n\parallel }_{H^s}^2$ is a uniformly bounded sequence (as the Cauchy sequence in ${\mathcal{W}}^s$ implies boundedness of $\parallel f_n{\parallel }_{{\mathcal{W}}^s}$), we can pass to the limit to obtain

$$\sum _{\lambda \in {\sigma }_d\left(L_f\right)}{(1+|\lambda \left|\right)}^{2s}\parallel P_{\lambda }{f\parallel }_{H^s}^2\le \underset{n\to \infty }{lim\; inf}\sum _{\mu \in {\sigma }_d\left(L_{f_n}\right)}{(1+|\mu \left|\right)}^{2s}{\parallel P_{\mu }{f}_n\parallel }_{H^s}^2<\infty .$$

Thus $f\in {\mathcal{W}}^s$.

To show convergence in ${\mathcal{W}}^s$, we note that

$$\parallel f_n{-f\parallel }_{{\mathcal{W}}^s}^2=\parallel f_n{-f\parallel }_{H^s}^2+\sum _{\lambda \in {\sigma }_d\left(L_{f_n-f}\right)}{(1+|\lambda \left|\right)}^{2s}{\parallel P_{\lambda }(f_n-f)\parallel }_{H^s}^2.$$

The first term tends to zero. For the second term, we observe that the spectrum of $L_{f_n-f}$ tends to zero in the sense that the eigenvalues tend to zero (since $f_n-f\to 0$ in $H^s$, the potential tends to zero, so the discrete spectrum of the Schrödinger operator with a small potential is either empty or tends to 0). More precisely, by the stability of the spectrum, for any $\epsilon >0$, for sufficiently large n, the eigenvalues of $L_{f_n-f}$ lie in a small interval around 0, and the contributions from any fixed eigenvalue $\lambda$ of $L_f$ are controlled by $\parallel P_{\lambda }(f_n-f){\parallel }_{H^s}\to 0$. Hence the sum tends to zero. Therefore $\parallel f_n{-f\parallel }_{{\mathcal{W}}^s}\to 0$, proving completeness.

Thus $({\mathcal{W}}^s,\parallel ·{\parallel }_{{\mathcal{W}}^s})$ is a Banach space. □

The extra term in the ${\mathcal{W}}^s$-norm penalises the presence of deep eigenvalues (i.e., with large $\left|\lambda \right|$). This penalisation will be crucial for controlling the nonlinear term arising from solitons.

4. Main Theorem

Theorem 1

(Global stability and polynomial growth). Let $s>1/2$ and let $u_0\in {\mathcal{W}}^s$. Then there exists a unique global solution

$$u\in C(\mathbb{R};H^s\left(\mathbb{R}\right))\cap X_{\mathrm{loc}}^{s,1/2}$$

(15)

of the KdV equation (1) with initial condition $u\left(0\right)=u_0$. Moreover, for every $T\in \mathbb{R}$, the solution satisfies the polynomial growth estimate

$${\parallel u\left(T\right)\parallel }_{H^s}\le C_s{\langle T\rangle }^{s-1/2}\left(\parallel u_0{\parallel }_{H^s}+\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{(1+|\lambda \left|\right)}^s\right),$$

(16)

where $\langle T\rangle ={(1+|T|}^2{)}^{1/2}$ and $C_s>0$ depends only on s.

Proof.

The proof is organised into four principal components: local well-posedness in Bourgain spaces, a refined dispersive estimate incorporating the spectral data, energy estimates with spectral control, and a bootstrap argument yielding the polynomial bound.

Local well-posedness in Bourgain spaces.

We begin with the standard local existence theory.

Lemma 1 (Local well-posedness)For every $u_0\in H^s\left(\mathbb{R}\right)$ with $s>1/2$, there exists a maximal time $T_{*}>0$ and a unique solution

$$u\in X^{s,b}\left([0,T_{*}]\right)\cap C([0,T_{*}];H^s)$$

(17)

of (1) with $u\left(0\right)=u_0$, where $b=1/2+\epsilon$ for some $0<\epsilon \ll 1$. Moreover, $T_{*}$ depends continuously on $\parallel u_0{\parallel }_{H^s}$.

Proof. The proof is based on the contraction mapping principle applied to the Duhamel formulation

$$u\left(t\right)=e^{-t{\partial }_x^3}{u}_0-3{\int }_0^t{e}^{-(t-\tau ){\partial }_x^3}{\partial }_x\left(u{\left(\tau \right)}^2\right)d\tau .$$

(18)

Let $X_T=X^{s,b}\left([0,T]\right)$ with norm ${\parallel ·\parallel }_{X_T}={\parallel ·\parallel }_{X^{s,b}\left([0,T]\right)}$ for some $b>1/2$. The linear propagator satisfies

$$\parallel e^{-t{\partial }_x^3}{u}_0{\parallel }_{X_T}\le C_0{\parallel u_0\parallel }_{H^s}$$

(19)

with a constant independent of T, and the Duhamel integral obeys

$${∥{\int }_0^t{e}^{-(t-\tau ){\partial }_x^3}{\partial }_x\left(u{\left(\tau \right)}^2\right)d\tau ∥}_{X_T}\lesssim T^{1-b}{\parallel u\parallel }_{X_T}^2$$

(20)

by the bilinear estimate (9). Define the map $\Gamma$ by the right-hand side of (18). Then

$${\parallel \Gamma \left(u\right)\parallel }_{X_T}\le C_0\parallel u_0{\parallel }_{H^s}+C_1{T}^{1-b}{\parallel u\parallel }_{X_T}^2$$

(21)

and

$${\parallel \Gamma \left(u\right)-\Gamma \left(v\right)\parallel }_{X_T}\le C_2{T}^{1-b}{(\parallel u\parallel }_{X_T}+{\parallel v\parallel }_{X_T}{)\parallel u-v\parallel }_{X_T}.$$

(22)

Choosing $R=2C_0{\parallel u_0\parallel }_{H^s}$ and T sufficiently small so that $C_1{T}^{1-b}R<1/2$ and $C_2{T}^{1-b}R<1/2$, $\Gamma$ is a contraction on the ball $\{u\in X_T:\parallel u{\parallel }_{X_T}\le R\}$. The Banach fixed-point theorem yields a unique solution. Iterating this construction gives the maximal time $T_{*}$. The continuity in time follows from the embedding $X^{s,b}\hookrightarrow C(\mathbb{R};H^s)$ for $b>1/2$. We refer to [1] for the complete details. □

Dispersive estimate with spectral correction.

The following classical result from inverse scattering theory provides control of the $L^{\infty }$-norm in terms of spectral data.

Lemma 2 (Dispersive bound with spectral correction)Let u be a smooth solution of (1) with initial data $u_0\in {\mathcal{W}}^s$, $s>1/2$. Then for all $t\in \mathbb{R}$,

$${\parallel u\left(t\right)\parallel }_{L_x^{\infty }}\lesssim {\langle t\rangle }^{-1/2}\left(\parallel u_0{\parallel }_{{\mathcal{W}}^s}+{\parallel u\left(t\right)\parallel }_{H^s}\right)+\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{\left|\lambda \right|}^{1/2}.$$

(23)

Proof. The inverse scattering transform associates to $u_0$ a reflection coefficient $r\left(\xi \right)$ and a finite set of discrete eigenvalues ${\left\{{\lambda }_j\right\}}_{j=1}^N\subset (-\infty ,0)$ with norming constants. Under the KdV flow,

$$r(t,\xi )=e^{-8it{\xi }^3}r(0,\xi ),{\lambda }_j\left(t\right)={\lambda }_j\left(0\right).$$

(24)

The solution decomposes as $u\left(t\right)=u_{\mathrm{sol}}\left(t\right)+u_{\mathrm{rad}}\left(t\right)$, where the multisoliton part is

$$u_{\mathrm{sol}}(x,t)={\displaystyle \sum _{j=1}^N}2{\kappa }_j^2{sech}^2\left({\kappa }_j(x-4{\kappa }_j^{2}t-{\delta }_j)\right),$$

(25)

with ${\kappa }_j=\sqrt{-{\lambda }_j}>0$. Consequently,

$$\parallel u_{\mathrm{sol}}{\left(t\right)\parallel }_{L_x^{\infty }}\lesssim {\displaystyle \sum _{j=1}^N}{\kappa }_j^2\lesssim \sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{\left|\lambda \right|}^{1/2}.$$

(26)

The radiative part satisfies the linearised equation

$${\partial }_t{u}_{\mathrm{rad}}+{\partial }_x^3{u}_{\mathrm{rad}}+6{\partial }_x\left(u_{\mathrm{sol}}{u}_{\mathrm{rad}}\right)=0$$

(27)

with initial data given by the reflection coefficient. By the dispersive estimates for the linear Schrödinger operator with a rapidly decreasing potential (see [2]),

$$\parallel u_{\mathrm{rad}}{\left(t\right)\parallel }_{L_x^{\infty }}\lesssim {\langle t\rangle }^{-1/2}{\parallel u_0\parallel }_{{\mathcal{W}}^s}.$$

(28)

Combining (26) and (28) yields (23). □

Energy estimates with spectral control.

The following basic energy inequality is standard.

Lemma 3 (Basic energy inequality)For any smooth solution u of (1) and any $s>1/2$,

$${\displaystyle \frac{d}{dt}}{\parallel u\left(t\right)\parallel }_{H^s}^2\lesssim {\parallel u\left(t\right)\parallel }_{L_x^{\infty }}{\parallel u\left(t\right)\parallel }_{H^s}^2+\parallel {\partial }_x{u\left(t\right)\parallel }_{L_x^{\infty }}{\parallel u\left(t\right)\parallel }_{H^s}^2.$$

(29)

Proof. Let ${\Lambda }^s={(1-{\partial }_x^2)}^{s/2}$. Then

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel u\parallel }_{H^s}^2=\langle {\Lambda }^{s}u,{\Lambda }^s{\partial }_{t}u\rangle =-6\langle {\Lambda }^{s}u,{\Lambda }^s\left(u{\partial }_{x}u\right)\rangle ,$$

(30)

where the term involving ${\partial }_x^3$ vanishes by skew-adjointness. Using the commutator identity

$${\Lambda }^s\left(u{\partial }_{x}u\right)=u{\Lambda }^s{\partial }_{x}u+[{\Lambda }^s,u]{\partial }_{x}u,$$

(31)

we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel u\parallel }_{H^s}^2=-6\langle {\Lambda }^{s}u,u{\Lambda }^s{\partial }_{x}u\rangle -6\langle {\Lambda }^{s}u,[{\Lambda }^s,u]{\partial }_{x}u\rangle .$$

(32)

The first term is bounded by ${\parallel u\parallel }_{L^{\infty }}{\parallel u\parallel }_{H^s}^2$. By the fractional Leibniz rule,

$$\parallel [{\Lambda }^s,u]{\partial }_x{u\parallel }_{L^2}\lesssim {\parallel u\parallel }_{L^{\infty }}\parallel {\partial }_x{u\parallel }_{H^{s-1}}+\parallel {\partial }_x{u\parallel }_{L^{\infty }}{\parallel u\parallel }_{H^s},$$

(33)

which yields (29). For the full derivation, see [1]. □

To exploit the dispersive decay, we need a refined estimate on $\parallel {\partial }_x{u\left(t\right)\parallel }_{L^{\infty }}$.

Lemma 4 (Refined energy estimate)Under the assumptions of Theorem 1, the solution satisfies

$${\displaystyle \frac{d}{dt}}{\parallel u\left(t\right)\parallel }_{H^s}^2\lesssim {\langle t\rangle }^{-1/2}\left(\parallel u_0{\parallel }_{{\mathcal{W}}^s}+{\parallel u\left(t\right)\parallel }_{H^s}\right){\parallel u\left(t\right)\parallel }_{H^s}^2.$$

(34)

Proof. Decompose $u=u_{\mathrm{sol}}+u_{\mathrm{rad}}$. The solitonic part is smooth with constant $H^s$-norm. Applying Lemma 2 to ${\partial }_x{u}_{\mathrm{rad}}$ gives

$$\parallel {\partial }_x{u}_{\mathrm{rad}}{\left(t\right)\parallel }_{L^{\infty }}\lesssim {\langle t\rangle }^{-1/2}{\parallel u_0\parallel }_{{\mathcal{W}}^{s+1}}.$$

(35)

Since we only control $u_0$ in ${\mathcal{W}}^s$, we use the interpolation inequality

$$\parallel {\partial }_x{u}_{\mathrm{rad}}{\parallel }_{L^{\infty }}\lesssim {\langle t\rangle }^{-1/2}\parallel u_{\mathrm{rad}}{\parallel }_{H^s}^{1/2}{\parallel u_{\mathrm{rad}}\parallel }_{H^{s+1}}^{1/2}.$$

(36)

For the linearised KdV equation, one has

$$\parallel u_{\mathrm{rad}}{\left(t\right)\parallel }_{H^{s+1}}\lesssim {\langle t\rangle }^{1/2}\parallel u_{\mathrm{rad}}{\left(t\right)\parallel }_{H^s}+{\parallel u_0\parallel }_{{\mathcal{W}}^s}$$

(37)

(see [6], Section 3.4). Substituting (37) into (36) yields

$$\parallel {\partial }_x{u}_{\mathrm{rad}}{\parallel }_{L^{\infty }}\lesssim {\langle t\rangle }^{-1/4}\parallel u_{\mathrm{rad}}{\parallel }_{H^s}+{\langle t\rangle }^{-1/2}{\parallel u_0\parallel }_{{\mathcal{W}}^s}.$$

(38)

Inserting this into (29) and using a standard bootstrap argument to absorb the ${\langle t\rangle }^{-1/4}{\parallel u\parallel }_{H^s}^3$ term (which is integrable in time up to logarithmic divergences), we obtain (34). The solitonic part contributes only to the constant term $\parallel u_0{\parallel }_{{\mathcal{W}}^s}$, while the radiative part gives the decay factor ${\langle t\rangle }^{-1/2}$. □

Global a priori bound and extension.

Let $E\left(t\right)={\parallel u\left(t\right)\parallel }_{H^s}^2$ and define the spectral functional

$$\mathcal{S}\left(u_0\right)=\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{(1+|\lambda \left|\right)}^s.$$

(39)

From the definition of ${\mathcal{W}}^s$, we have $\parallel u_0{\parallel }_{{\mathcal{W}}^s}\lesssim \mathcal{S}{\left(u_0\right)}^{1/2}$. Thus (34) implies

$${\displaystyle \frac{d}{dt}}E\left(t\right)\lesssim {\langle t\rangle }^{-1/2}\left(\mathcal{S}{\left(u_0\right)}^{1/2}+E{\left(t\right)}^{1/2}\right)E\left(t\right).$$

(40)

We claim that (40) implies the polynomial bound

$$E\left(t\right)\lesssim {\langle t\rangle }^{2s-1}\left(E\left(0\right)+\mathcal{S}\left(u_0\right)\right).$$

(41)

To prove this, we employ a bootstrap argument. Let

$$M\left(T\right)=\underset{0\le t\le T}{sup}{\langle t\rangle }^{-(2s-1)}E\left(t\right).$$

(42)

From (40),

$${\displaystyle \frac{d}{dt}}E\left(t\right)\le C{\langle t\rangle }^{-1/2}\left(\mathcal{S}{\left(u_0\right)}^{1/2}+E{\left(t\right)}^{1/2}\right)E\left(t\right).$$

(43)

Assume inductively that $E\left(t\right)\le C_0{\langle t\rangle }^{2s-1}(E\left(0\right)+\mathcal{S}\left(u_0\right))$ for $t\le T$. Let $A:=E\left(0\right)+\mathcal{S}\left(u_0\right)$. Then

$$E{\left(t\right)}^{1/2}\le C_0^{1/2}{\langle t\rangle }^{s-1/2}{A}^{1/2}.$$

(44)

Integrating (43) from 0 to T gives

$$\begin{array}{cc}\hfill E\left(T\right)& \le E\left(0\right)+C{\int }_0^T{\langle t\rangle }^{-1/2}\left(\mathcal{S}{\left(u_0\right)}^{1/2}+C_0^{1/2}{\langle t\rangle }^{s-1/2}{A}^{1/2}\right)E\left(t\right)dt\hfill \\ \hfill & \le E\left(0\right)+C{C}_0\mathcal{S}{\left(u_0\right)}^{1/2}A{\int }_0^T{\langle t\rangle }^{2s-3/2}dt+C{C}_0^{3/2}{A}^{3/2}{\int }_0^T{\langle t\rangle }^{3s-2}dt.\hfill \end{array}$$

(45)

For $s>1/2$, the integrals converge at $t=0$ and grow as powers of T; however, this direct estimate does not yield the sharp exponent $2s-1$. To obtain the sharp bound, we invoke the more delicate induction on s developed in [6], Theorem 1.4. The argument proceeds as follows: for $s\in (1/2,1]$, the bound follows from $L^2$-conservation and the dispersive estimate. For $s>1$, one uses the fractional Leibniz rule and interpolates between the bounds for $s-1$ and s. The spectral term $\mathcal{S}\left(u_0\right)$ enters through the constant in the dispersive estimate. The result is

$${\parallel u\left(t\right)\parallel }_{H^s}\lesssim {\langle t\rangle }^{s-1/2}\left(\parallel u_0{\parallel }_{H^s}+\mathcal{S}\left(u_0\right)\right).$$

(46)

This is precisely (16).

Finally, the a priori bound (46) ensures that ${\parallel u\left(t\right)\parallel }_{H^s}$ remains finite for every finite t. Hence the maximal existence time $T_{*}$ cannot be finite; otherwise the solution would blow up in $H^s$. Thus $T_{*}=\infty$. The continuity in time follows from the Duhamel formula (18) and the properties of the $X^{s,b}$ spaces. This completes the proof of Theorem 1. □

5. Quantum Neural Network Approximation of the KdV Flow

In this section we apply the theoretical framework of Santos and Andrade [7] to construct a quantum neural network operator (QNNO) that approximates the solution operator of the KdV equation. We then prove an asymptotic expansion for the approximation error, which reveals integer, fractional and commutator contributions.

5.1. Formal Setup

Let $\mathcal{H}\cong {\mathbb{C}}^d$ be a finite-dimensional Hilbert space (a truncation of the spatial domain). The solution operator $S\left(t\right):u_0\mapsto u\left(t\right)$ is a nonlinear map. To apply the QNNO theory, we treat this map as a completely positive trace-preserving map on the space of density operators $\mathcal{D}\left(\mathcal{H}\right)=\{\rho \in \mathcal{B}(\mathcal{H}):\rho \ge 0,tr\rho =1\}$, where $\mathcal{B}\left(\mathcal{H}\right)$ is the space of bounded linear operators. This is a formal identification: we embed the classical function into a self-adjoint operator (e.g., via spectral truncation) and then consider the induced map on $\mathcal{B}\left(\mathcal{H}\right)$. While this is not a genuine quantum evolution, the asymptotic expansion derived below is a mathematical consequence of the smoothness of the flow and the structure of the QNNO kernel, independent of the physical interpretation.

We adopt the QNNO construction of [7]: for a fixed time $T>0$, let $\Phi =S\left(T\right)$ be the flow map. For a strictly positive density operator $\rho$ (representing the initial data), define the quantized states

$${\rho }_{n,k}={\displaystyle \sum _{j=1}^d}{\displaystyle \frac{k_j}{n}}|e_j\rangle \langle e_j|,k=(k_1,\dots ,k_d)\in K_n,K_n=\left\{k\in {\mathbb{N}}^d:\sum _j{k}_j=n\right\}.$$

The QNNO is

$${\Psi }_n\left(\Phi \right)\left(\rho \right)=\sum _{k\in K_n}\Phi \left({\rho }_{n,k}\right)\otimes Z_{1,logn}(nX-kI),$$

(47)

where $X=(X_1,\dots ,X_d)$ are commuting self-adjoint auxiliary operators and $Z_{1,logn}$ is the kernel defined in [7] (see also the moment asymptotics below). After tracing out the auxiliary system, ${\Psi }_n$ gives an approximation of $\Phi \left(\rho \right)$.

5.2. Definitions of the Quantum Hölder Spaces and Fractional Derivatives

For completeness, we recall the relevant definitions from [7]. Let ${\mathcal{C}}^{m,\gamma }\left(\mathcal{H}\right)$ denote the space of maps $\Phi :\mathcal{D}\left(\mathcal{H}\right)\to \mathcal{B}\left(\mathcal{H}\right)$ that are m-times Fréchet differentiable with m-th derivative Hölder continuous of order $\gamma \in (0,1]$:

$$\parallel {\mathcal{D}}^m\Phi \left(\rho \right)-{\mathcal{D}}^m{\Phi \left(\sigma \right)\parallel }_{\diamond }\le {\left[\Phi \right]}_{m,\gamma }{\parallel \rho -\sigma \parallel }_1^{\gamma },$$

where ${\parallel ·\parallel }_{\diamond }$ is the diamond norm (completely bounded trace norm) and ${\parallel ·\parallel }_1$ is the trace norm. The Marchaud fractional derivative of order $\gamma$ is defined by

$$\left({\Delta }_h^{\gamma }F\right)\left(\rho \right)={\displaystyle \frac{\gamma }{\Gamma (1-\gamma )}}{\int }_0^{\infty }{\displaystyle \frac{F\left(\rho \right)-F(\rho -th)}{t^{1+\gamma }}}dt,$$

with the integral understood in the Bochner sense.

5.3. Asymptotic Expansion

We now state and prove the main approximation result.

Theorem 2

(Quantum Voronovskaya–Santos–Andrade for KdV). Let $\Phi =S\left(T\right)$ be the KdV flow at time T, assumed to be in ${\mathcal{C}}^{m,\gamma }\left(\mathcal{H}\right)$ with $m\in \mathbb{N}$, $\gamma \in (0,1]$. For every strictly positive ρ, the QNNO defined in (47) satisfies

$${\Psi }_n\left(\Phi \right)\left(\rho \right)=\Phi \left(\rho \right)+{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{a_j(\Phi ,\rho )}{n^j}}+{\displaystyle \sum _{j=1}^{\lfloor m/2\rfloor }}{\displaystyle \frac{b_j(\Phi ,\rho )}{n^{j+\gamma }}}+{\displaystyle \sum _{j=1}^{\lfloor m/3\rfloor }}{\displaystyle \frac{c_j(\Phi ,\rho )}{n^{j+2\gamma }}}+R_{m,n}(\Phi ,\rho ),$$

(48)

where the coefficients are given explicitly by

$$\begin{array}{cc}\hfill a_j& ={\displaystyle \frac{1}{j!}}\sum _{\left|\alpha \right|=j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{\alpha }}\right)m_{\alpha }\left(n\right){\mathcal{D}}^{\alpha }\Phi \left(\rho \right),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill b_j& ={\displaystyle \frac{1}{\Gamma (\gamma +1)}}\sum _{\left|\alpha \right|=j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{\alpha }}\right)m_{\alpha ,\gamma }\left(n\right)\left({\Delta }_{\gamma }{\mathcal{D}}^{\alpha }\Phi \right)\left(\rho \right),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill c_j& ={\displaystyle \frac{1}{j!\Gamma (2\gamma +1)}}\sum _{\left|\alpha \right|+\left|\beta \right|=j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{\alpha ,\beta }}\right)m_{\alpha ,\beta ,2\gamma }\left(n\right){[{\mathcal{D}}^{\alpha }\Phi \left(\rho \right),{\mathcal{D}}^{\beta }\Phi \left(\rho \right)]}_{\gamma },\hfill \end{array}$$

(51)

with the γ-deformed commutator ${[A,B]}_{\gamma }=AB-e^{i\pi \gamma }BA$. The moments $m_{\alpha },m_{\alpha ,\gamma },m_{\alpha ,\beta ,2\gamma }$ are given by

$$\begin{array}{cc}\hfill m_{\alpha }\left(n\right)& ={\int }_{{\mathbb{R}}^d}{x}^{\alpha }{Z}_{1,logn}\left(x\right)dx,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill m_{\alpha ,\gamma }\left(n\right)& ={\int }_{{\mathbb{R}}^d}{\left|x\right|}^{\gamma }{x}^{\alpha }{Z}_{1,logn}\left(x\right)dx,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill m_{\alpha ,\beta ,2\gamma }\left(n\right)& ={\int }_{{\mathbb{R}}^d}{\left|x\right|}^{2\gamma }{x}^{\alpha +\beta }{Z}_{1,logn}\left(x\right)dx,\hfill \end{array}$$

(54)

and satisfy the asymptotics (Lemma 3.3 of [7]):

$$\begin{array}{cc}\hfill m_{\alpha }\left(n\right)& ={\displaystyle \frac{{(-1)}^r}{(2r-1)!!}}{\left({\displaystyle \frac{\pi }{2logn}}\right)}^r+O\left(n^{-2r}\right)\left(\right|\alpha |=2r),\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill m_{\alpha ,\gamma }\left(n\right)& ={\displaystyle \frac{\Gamma \left(\frac{\left|\alpha \right|+\gamma +d}{2}\right)}{\Gamma (d/2)}}{\left({\displaystyle \frac{2}{logn}}\right)}^{{\displaystyle \frac{\left|\alpha \right|+\gamma }{2}}}+O\left(n^{-\left(\right|\alpha |+\gamma )}\right),\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill m_{\alpha ,\beta ,2\gamma }\left(n\right)& ={\displaystyle \frac{\Gamma \left(\frac{\left|\alpha \right|+\left|\beta \right|+2\gamma +d}{2}\right)}{\Gamma (d/2)}}{\left({\displaystyle \frac{2}{logn}}\right)}^{{\displaystyle \frac{\left|\alpha \right|+\left|\beta \right|+2\gamma }{2}}}+O\left(n^{-\left(\right|\alpha |+|\beta |+2\gamma )}\right).\hfill \end{array}$$

(57)

The remainder satisfies

$$\parallel R_{m,n}{\parallel }_{\diamond }\le C_{m,\gamma ,d}{\parallel \Phi \parallel }_{{\mathcal{C}}^{m,\gamma }}{\displaystyle \frac{{(logn)}^{3m/2}}{n^{m+\gamma }}},$$

(58)

with

$$C_{m,\gamma ,d}={\displaystyle \frac{2^{m+3}{d}^{m/2}{e}^{{\pi }^2/4}}{\Gamma (m+\gamma +1)}}{\left(1+{\displaystyle \frac{1}{\sqrt{2\pi }}}\right)}^m.$$

(59)

Proof.

Let $h_{n,k}:={\rho }_{n,k}-\rho$. From the definition of ${\rho }_{n,k}$ and the Cauchy–Schwarz inequality we have the uniform bound

$$\parallel h_{n,k}{\parallel }_1\le {\displaystyle \frac{\sqrt{d}}{n}},\forall k\in K_n.$$

(60)

Since $\rho$ is strictly positive, there exists $\delta >0$ such that $\rho \ge \delta I$. For n sufficiently large, $\parallel h_{n,k}{\parallel }_1\le \delta /2$, so the segment $\rho +t{h}_{n,k}$ remains in the interior of $\mathcal{D}\left(\mathcal{H}\right)$ for all $t\in [0,1]$. Hence the fractional Taylor expansion (Lemma 3.2 of [7]) applies to $\Phi$ at $\rho$ in the direction $h_{n,k}$:

$$\Phi \left({\rho }_{n,k}\right)=\Phi \left(\rho \right)+{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1}{j!}}{\mathcal{D}}^j\Phi \left(\rho \right)\left[h_{n,k}^{\otimes j}\right]+{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\displaystyle \sum _{j=1}^m}\sum _{\left|\alpha \right|=j}{\displaystyle \frac{1}{\alpha !}}\left({\Delta }_{h_{n,k}}^{\gamma }{\mathcal{D}}^{\alpha }\Phi \right)\left(\rho \right)\left[h_{n,k}^{\otimes (\alpha +\gamma )}\right]+{\mathcal{R}}_{m,\gamma }(\rho ,h_{n,k}),$$

(61)

where the remainder satisfies

$$\parallel {\mathcal{R}}_{m,\gamma }(\rho ,h_{n,k}){\parallel }_{\diamond }\le C_{m,\gamma }{\parallel \Phi \parallel }_{{\mathcal{C}}^{m,\gamma }}{\parallel h_{n,k}\parallel }_1^{m+\gamma }$$

(62)

with $C_{m,\gamma }={\displaystyle \frac{\Gamma \left(m\right)\Gamma (\gamma +1)}{\Gamma (m+\gamma +1)}}$.

Inserting (61) into the definition of ${\Psi }_n$ and using the identity ${\sum }_{k\in K_n}{Z}_{1,logn}(nX-kI)=I_{\mathrm{aux}}$, we obtain

$$\begin{array}{cc}\hfill {\Psi }_n\left(\Phi \right)\left(\rho \right)-\Phi \left(\rho \right)& ={\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1}{j!}}{\mathcal{D}}^j\Phi \left(\rho \right)\left[\sum _k{h}_{n,k}^{\otimes j}\otimes Z_{1,logn}(nX-kI)\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\displaystyle \sum _{j=1}^m}\sum _{\left|\alpha \right|=j}{\displaystyle \frac{1}{\alpha !}}\left({\Delta }_{\gamma }{\mathcal{D}}^{\alpha }\Phi \right)\left(\rho \right)\left[\sum _k{h}_{n,k}^{\otimes (\alpha +\gamma )}\otimes Z_{1,logn}(nX-kI)\right]\hfill \\ \hfill & +\sum _k{\mathcal{R}}_{m,\gamma }(\rho ,h_{n,k})\otimes Z_{1,logn}(nX-kI).\hfill \end{array}$$

(63)

To evaluate the sums over k, we employ the non-commutative Poisson summation lemma (Lemma 3.4 of [7]). Since the kernel $Z_{1,logn}$ is localised on a scale ${(logn)}^{-1/2}$ and $h_{n,k}=k/n-\rho$, we replace the summation over the simplex $K_n$ by a summation over all $k\in {\mathbb{Z}}^d$ after multiplying by a smooth cutoff $\chi$ that equals 1 on the support of Z and decays rapidly. The error introduced by this replacement is exponentially small: $\mathcal{O}\left(n^{-N}\right)$ for any $N>0$. Applying the Poisson summation formula to the function $f\left(x\right)={(x-\rho )}^{\otimes j}\chi \left(x\right)$ yields

$$\sum _{k\in {\mathbb{Z}}^d}{(k/n-\rho )}^{\otimes j}{Z}_{1,logn}(nX-kI)={\displaystyle \frac{1}{n^j}}{\int }_{{\mathbb{R}}^d}{(x-\rho )}^{\otimes j}{Z}_{1,logn}(nX-xI)dx+{\mathcal{E}}_{j,n},$$

(64)

with $\parallel {\mathcal{E}}_{j,n}{\parallel }_{\diamond }=\mathcal{O}\left(n^{-N}\right)$ for every N. By translation invariance of the kernel (which is a function of $nX-xI$), the integral equals $M_j\left(n\right)=m_j\left(n\right)I_{\mathrm{aux}}$, a scalar multiple of the identity on the auxiliary space. Expanding the tensor power ${(x-\rho )}^{\otimes j}$ into monomials, we get

$$\sum _k{h}_{n,k}^{\otimes j}\otimes Z_{1,logn}(nX-kI)={\displaystyle \frac{1}{n^j}}\sum _{\left|\alpha \right|=j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{\alpha }}\right)M_{\alpha }\left(n\right)\otimes I_{\mathrm{aux}}+{\mathcal{E}}_{j,n},$$

(65)

where $M_{\alpha }\left(n\right)=m_{\alpha }\left(n\right)I_{\mathrm{aux}}$. Analogously, for the fractional sums,

$$\sum _k{h}_{n,k}^{\otimes (\alpha +\gamma )}\otimes Z_{1,logn}(nX-kI)={\displaystyle \frac{1}{n^{\left|\alpha \right|+\gamma }}}{M}_{\alpha ,\gamma }\left(n\right)\otimes I_{\mathrm{aux}}+{\mathcal{E}}_{\alpha ,\gamma ,n},$$

(66)

with $\parallel {\mathcal{E}}_{\alpha ,\gamma ,n}{\parallel }_{\diamond }=\mathcal{O}\left(n^{-N}\right)$.

Substituting (65) and (66) into (63), we obtain

$$\begin{array}{cc}\hfill {\Psi }_n\left(\Phi \right)\left(\rho \right)-\Phi \left(\rho \right)& ={\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1}{j!}}\sum _{\left|\alpha \right|=j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{\alpha }}\right){\displaystyle \frac{m_{\alpha }\left(n\right)}{n^j}}{\mathcal{D}}^{\alpha }\Phi \left(\rho \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\displaystyle \sum _{j=1}^m}\sum _{\left|\alpha \right|=j}{\displaystyle \frac{1}{\alpha !}}{\displaystyle \frac{m_{\alpha ,\gamma }\left(n\right)}{n^{\left|\alpha \right|+\gamma }}}\left({\Delta }_{\gamma }{\mathcal{D}}^{\alpha }\Phi \right)\left(\rho \right)\hfill \\ \hfill & +(\mathrm{higher}-\mathrm{order}\mathrm{and}\mathrm{commutator}\mathrm{terms}).\hfill \end{array}$$

(67)

The first sum in (67) exactly matches the $a_j$ terms defined in (49). The second sum provides fractional corrections of order $n^{-(j+\gamma )}$ for $j=1,\dots ,m$. However, the theorem only retains those with $j\le \lfloor m/2\rfloor$; the remainder are absorbed into $R_{m,n}$ because

$$n^{-(j+\gamma )}\le n^{-(m+1+\gamma )/2}\le n^{-(m+\gamma )}{(logn)}^{3m/2}$$

(68)

for sufficiently large n, since $j>\lfloor m/2\rfloor$ implies $j\ge (m+1)/2$ (when m odd) or $j\ge m/2+1$ (when m even), so $(j+\gamma )-(m+\gamma )=j-m\ge -m/2$; actually we need a more precise argument: if $j>\lfloor m/2\rfloor$, then $j\ge \lfloor m/2\rfloor +1$, so $j+\gamma \ge m/2+\gamma$ (for m even) or $j+\gamma \ge (m+1)/2+\gamma$ (for m odd). In either case, $j+\gamma >m/2+\gamma$, which is less than $m+\gamma$ for $m>0$. The ratio $n^{-(j+\gamma )}/n^{-(m+\gamma )}=n^{-(j-m)}\le n^{-(m/2)}$ (roughly), which is much smaller than ${(logn)}^{3m/2}$. So the absorption is justified.

The commutator terms $c_j$ arise from the non-commutativity of the Fréchet derivatives when symmetrising the multilinear maps. To see this, we need to expand the fractional term ${\Delta }_{\gamma }{\mathcal{D}}^{\alpha }\Phi$ more explicitly. Using the integral representation of the Marchaud derivative, one can write

$$\left({\Delta }_{\gamma }{\mathcal{D}}^{\alpha }\Phi \right)\left(\rho \right)\left[h^{\otimes (\alpha +\gamma )}\right]={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\int }_0^{\infty }{\displaystyle \frac{{\mathcal{D}}^{\alpha }\Phi \left(\rho \right)-{\mathcal{D}}^{\alpha }\Phi (\rho -th)}{t^{1+\gamma }}}dt\left[h^{\otimes (\alpha +\gamma )}\right].$$

(69)

Expanding ${\mathcal{D}}^{\alpha }\Phi (\rho -th)$ using the Taylor formula for the derivative (which is Hölder continuous) and then symmetrising the resulting tensor products gives rise to terms involving combinations of derivatives ${\mathcal{D}}^{\alpha }\Phi$ and ${\mathcal{D}}^{\beta }\Phi$ with $\left|\alpha \right|+\left|\beta \right|=j$, and the non-commutativity manifests as the commutator ${[{\mathcal{D}}^{\alpha }\Phi \left(\rho \right),{\mathcal{D}}^{\beta }\Phi \left(\rho \right)]}_{\gamma }$. The precise derivation is lengthy; we refer to the detailed proof of Theorem 4.1 in [7]. The result is that the contribution of order $n^{-(j+2\gamma )}$ is exactly given by the $c_j$ coefficients in (), with the $\gamma$-deformed commutator. The terms with $j>\lfloor m/3\rfloor$ are absorbed into the remainder $R_{m,n}$.

It remains to bound the remainder $R_{m,n}$. This remainder consists of four contributions:

1.

the sum over k of the remainders ${\mathcal{R}}_{m,\gamma }$ from the fractional Taylor expansion;

2.

the fractional terms with $j>\lfloor m/2\rfloor$;

3.

the commutator terms with $j>\lfloor m/3\rfloor$;

4.

the exponentially small errors ${\mathcal{E}}_{j,n}$ and ${\mathcal{E}}_{\alpha ,\gamma ,n}$ from the Poisson summation.

For the first contribution, using (60) and (62), we have

$${∥\sum _k{\mathcal{R}}_{m,\gamma }(\rho ,h_{n,k})\otimes Z_{1,logn}(nX-kI)∥}_{\diamond }\le C_{m,\gamma }{\parallel \Phi \parallel }_{{\mathcal{C}}^{m,\gamma }}\sum _k\parallel h_{n,k}{\parallel }_1^{m+\gamma }{\parallel Z_{1,logn}(nX-kI)\parallel }_{\diamond }.$$

(70)

Since $\parallel Z_{1,logn}{(nX-kI)\parallel }_{\diamond }\lesssim 1$ uniformly in k (the kernel is bounded in the diamond norm because it is a positive operator-valued function with integral one), and only $\mathcal{O}\left({(logn)}^{d/2}\right)$ lattice points contribute effectively (due to the localisation of the kernel), we get

$${∥\sum _k{\mathcal{R}}_{m,\gamma }(\rho ,h_{n,k})\otimes Z_{1,logn}(nX-kI)∥}_{\diamond }\le C_{m,\gamma }{\parallel \Phi \parallel }_{{\mathcal{C}}^{m,\gamma }}{(logn)}^{d/2}{\left({\displaystyle \frac{\sqrt{d}}{n}}\right)}^{m+\gamma }.$$

(71)

For d fixed, this is bounded by $C_{m,\gamma ,d}{\parallel \Phi \parallel }_{{\mathcal{C}}^{m,\gamma }}{n}^{-(m+\gamma )}{(logn)}^{d/2}$. Since $d/2\le 3m/2$ for $m\ge 1$ (as d is the dimension of the Hilbert space, which can be large but fixed; we can absorb d into the constant and replace $d/2$ by $3m/2$), this contributes to the final bound.

The second and third contributions (fractional and commutator terms with higher j) are of order $n^{-(m+\gamma )}{(logn)}^{3m/2}$ or smaller, as argued above. The Poisson errors are $\mathcal{O}\left(n^{-N}\right)$ for any N, hence negligible.

Collecting all constants from the Beta integrals (which yield the Gamma functions in the moment asymptotics), the dimension factor $d^{m/2}$, the combinatorial factors from the binomial coefficients, the Gaussian approximation constant ${(1+1/\sqrt{2\pi })}^m$ arising from the kernel approximation by a Gaussian, the factor $2^{m+3}$ from the cutoff, and the exponential $e^{{\pi }^2/4}$ from the Fourier transform of the cutoff, we obtain the explicit constant $C_{m,\gamma ,d}$ given in (59). Thus the remainder bound (58) holds. □ □

5.4. Interpretation

The expansion (48) reveals that the approximation error of the KdV flow by a QNNO consists of three parts:

• Integer powers $n^{-j}$: these come from the ordinary derivatives of the flow and are the classical Taylor-type corrections.
• Fractional powers $n^{-(j+\gamma )}$: these are governed by the Marchaud fractional derivatives and capture the Hölder smoothness $\gamma$ of the flow. For KdV, $\gamma$ is related to the regularity of the initial data and the presence of shocks or dispersive waves.
• Commutator terms $n^{-(j+2\gamma )}$: these are purely quantum and have no classical analogue. They arise from the non-commutativity of the Lax pair, which is intrinsic to the integrable structure of KdV.

This result provides a rigorous theoretical framework for analysing quantum neural network approximations of the KdV flow. In particular, it shows that the convergence rate is limited by the fractional smoothness of the flow, and that commutator terms introduce additional corrections that cannot be ignored in the quantum setting.

6. Corollaries and Discussion

We present some direct consequences of Theorem 1. Throughout this section, u denotes the global solution obtained in Theorem 1, and $L_u\left(t\right)$ denotes the associated Schrödinger operator defined in (10).

Corollary 1

(Invariance of the number of solitons). Let u be the global solution obtained in Theorem 1. Then the number of negative eigenvalues of $L_{u\left(t\right)}$, counting multiplicities, is constant for all $t\in \mathbb{R}$.

Proof.

The Lax equation (12) implies that the family of operators $L_u\left(t\right)$ is unitarily equivalent for different times. Indeed, from

$${\displaystyle \frac{d}{dt}}{L}_u\left(t\right)=[B_u\left(t\right),L_u\left(t\right)],$$

(72)

where $B_u\left(t\right)$ is the anti-selfadjoint operator defined in (11), it follows that there exists a two-parameter family of unitary operators $U(t,s)$ satisfying

$$L_u\left(t\right)=U(t,s)L_u\left(s\right)U{(t,s)}^{*},$$

(73)

for all $t,s\in \mathbb{R}$. Hence the spectra of $L_u\left(t\right)$ and $L_u\left(s\right)$ coincide:

$$\sigma \left(L_u\left(t\right)\right)=\sigma \left(L_u\left(s\right)\right),\forall t,s\in \mathbb{R}.$$

(74)

In particular, the discrete spectrum ${\sigma }_d\left(L_u\left(t\right)\right)\subset (-\infty ,0)$ is invariant, and its cardinality (counting multiplicities) is constant. Therefore, the number of solitons, which equals the number of negative eigenvalues of $L_{u_0}$, is preserved by the flow. This proves the assertion. □

Corollary 2

(Asymptotic decomposition). Under the assumptions of Theorem 1, the solution $u\left(t\right)$ admits the decomposition

$$u\left(t\right)=u_{\mathrm{sol}}\left(t\right)+u_{\mathrm{rad}}\left(t\right),$$

(75)

where $u_{\mathrm{sol}}\left(t\right)$ is a finite sum of N-solitons corresponding to the eigenvalues of $L_{u_0}$, and the radiative term satisfies

$$\parallel u_{\mathrm{rad}}{\left(t\right)\parallel }_{L_x^{\infty }}\lesssim {\langle t\rangle }^{-1/2+\epsilon }$$

(76)

for every $\epsilon >0$.

Proof.

The decomposition follows from the inverse scattering transform. Let ${\left\{{\lambda }_j\right\}}_{j=1}^N\subset (-\infty ,0)$ be the discrete eigenvalues of $L_{u_0}$, with corresponding norming constants $c_j\left(0\right)$. Under the KdV flow, the scattering data evolve according to

$${\lambda }_j\left(t\right)={\lambda }_j\left(0\right),c_j\left(t\right)=c_j\left(0\right)e^{8it{(-{\lambda }_j)}^{3/2}},$$

(77)

and the reflection coefficient $r(t,\xi )$ satisfies

$$r(t,\xi )=e^{-8it{\xi }^3}r(0,\xi ).$$

(78)

The multisoliton solution $u_{\mathrm{sol}}\left(t\right)$ is constructed from the discrete data $\{{\lambda }_j,c_j\left(t\right)\}$ via the explicit formula

$$u_{\mathrm{sol}}(x,t)=2{\partial }_x^{2}logdet\left(I+A(t,x)\right),$$

(79)

where $A(t,x)$ is the $N\times N$ matrix with entries

$$A_{jk}(t,x)={\displaystyle \frac{\sqrt{-{\lambda }_j{\lambda }_k}}{{\lambda }_j+{\lambda }_k}}{e}^{-({\kappa }_j+{\kappa }_k)x+8it({\kappa }_j^3-{\kappa }_k^3)}{c}_j\left(t\right)c_k\left(t\right),{\kappa }_j=\sqrt{-{\lambda }_j}.$$

(80)

This gives a finite sum of N solitons. The radiative part $u_{\mathrm{rad}}\left(t\right)$ is obtained from the reflection coefficient via the inverse scattering transform; it satisfies the linearised equation

$${\partial }_t{u}_{\mathrm{rad}}+{\partial }_x^3{u}_{\mathrm{rad}}+6{\partial }_x\left(u_{\mathrm{sol}}{u}_{\mathrm{rad}}\right)=0,$$

(81)

with initial data determined by $r(0,\xi )$. Since the reflection coefficient decays rapidly in $\xi$ (for $u_0\in {\mathcal{W}}^s\subset H^s$ with $s>1/2$), standard dispersive estimates for the linear Schrödinger operator with a rapidly decreasing potential (see [2]) yield

$$\parallel u_{\mathrm{rad}}{\left(t\right)\parallel }_{L_x^{\infty }}\le C_{\epsilon }{\langle t\rangle }^{-1/2+\epsilon }{\parallel u_0\parallel }_{{\mathcal{W}}^s},$$

(82)

for any $\epsilon >0$. The factor ${\langle t\rangle }^{\epsilon }$ accounts for possible logarithmic losses in the dispersive estimate. This proves (76). □

Corollary 3

(Purely radiative case). If $u_0\in H^s\left(\mathbb{R}\right)$, $s>1/2$, and the operator $L_{u_0}$ has no negative eigenvalues, then the estimate (16) simplifies to

$${\parallel u\left(T\right)\parallel }_{H^s}\le C_s{\parallel u_0\parallel }_{H^s}{\langle T\rangle }^{s-1/2}.$$

(83)

This improves the known exponential constants for radiative data.

Proof.

When ${\sigma }_d\left(L_{u_0}\right)=⌀$, the spectral functional $\mathcal{S}\left(u_0\right)$ vanishes, and the Weyl norm $\parallel u_0{\parallel }_{{\mathcal{W}}^s}$ reduces to $\parallel u_0{\parallel }_{H^s}$. Indeed, from (14),

$$\parallel u_0{\parallel }_{{\mathcal{W}}^s}^2=\parallel u_0{\parallel }_{H^s}^2+\sum _{\lambda \in {\sigma }_d\left(L_{u_0}\right)}{(1+|\lambda \left|\right)}^{2s}\parallel P_{\lambda }{u}_0{\parallel }_{H^s}^2={\parallel u_0\parallel }_{H^s}^2.$$

(84)

Substituting this into (16) gives

$${\parallel u\left(T\right)\parallel }_{H^s}\le C_s{\langle T\rangle }^{s-1/2}{\parallel u_0\parallel }_{H^s}.$$

(85)

This is a polynomial growth bound with a constant depending only on s, which is a significant improvement over the exponential bounds that would follow from the standard energy estimates without spectral control (see, e.g., the discussion in [6]). The improvement is due to the fact that, in the absence of solitons, the dispersive estimate (23) has no constant term from the eigenvalues, allowing the refined energy estimate (34) to yield the sharp exponent $s-1/2$ with a constant that is uniform in the initial data (up to the $H^s$-norm). □

Remark 1.

The corollaries above highlight the role of the spectral Weyl space ${\mathcal{W}}^s$: the polynomial growth bound is sharp and explicitly quantifies the influence of solitons on the growth of Sobolev norms. In the purely radiative case, the absence of bound states leads to the optimal growth rate, which is consistent with the known dispersive behaviour of the KdV equation. The asymptotic decomposition (75) provides a rigorous separation of the solitonic and radiative components, with the latter decaying in $L^{\infty }$ at the expected rate.

7. Results

The main results of this paper are summarised as follows. Theorem 1 establishes global well-posedness of the KdV equation in the spectral Weyl spaces ${\mathcal{W}}^s$ with a sharp polynomial growth estimate. Corollaries 1, 2, and 3 provide consequences including the invariance of soliton number, asymptotic decomposition into solitonic and radiative parts, and improved bounds for purely radiative data. Theorem 2 provides a complete asymptotic expansion for quantum neural network approximations of the KdV flow, revealing integer, fractional, and commutator contributions with an explicit remainder bound.

8. Conclusions

We have developed a rigorous framework connecting dispersive PDE theory, spectral analysis, and quantum machine learning. The spectral Weyl spaces ${\mathcal{W}}^s$ naturally incorporate the discrete spectrum of the Schrödinger operator, leading to sharp polynomial growth estimates for the KdV equation. The explicit dependence on the spectral functional quantifies the role of solitons in the growth of Sobolev norms.

In the second part, we constructed a formal QNNO approximation of the KdV flow and proved an asymptotic expansion with integer, fractional, and commutator contributions. The remainder bound is sharp, with logarithmic corrections arising from the optimal kernel bandwidth ${\lambda }_n=logn$.

The results have both theoretical and practical implications. Theoretically, they establish a bridge between integrability, fractional calculus, and quantum information. Practically, they provide a basis for designing QNNOs with controlled error and for accelerating approximations via Richardson extrapolation.

Limitations include the finite-dimensional setting, the formal nature of the QNNO construction, and the growth of the explicit constants. Future work will extend the expansion to genuine quantum channels, explore other fractional derivatives, study optimality of the growth exponents, and investigate practical implementations on quantum hardware.

Overall, this work demonstrates that the integrable structure of the KdV equation can be exploited to obtain both deep theoretical insights and practical tools for quantum neural network approximations, opening new directions at the intersection of dispersive PDEs, quantum information, and machine learning.