Marchaud-Type Singular Integral Representation

• Marchaud-type singular integrals are nonlocal operators defined via weighted difference quotients, essential for extending classical derivatives to fractional orders.
• The approach underpins the spectral time-fractional KdV equation by employing a bilinear Hirota method to preserve soliton structures while altering temporal dispersion.
• Explicit one- and two-soliton solutions are derived with a precise fractional dispersion law, demonstrating the method’s practical impact on integrable systems.

The spectral time-fractional Korteweg–de Vries (KdV) equation generalizes the integrable KdV model by replacing the classical time derivative with a spectral (Fourier-multiplier) fractional derivative of order $0<\alpha\leq 1$. This construction, built via a fractional extension of the Hirota bilinear calculus, directly alters the temporal dispersive character of soliton solutions while preserving the spatial algebraic structure and soliton interaction properties. The approach exploits well-defined nonlocal operators—specifically, the spectral fractional derivative and its bilinear extension—to derive explicit one- and two-soliton $\tau$-functions and the corresponding dispersion relation $\omega^\alpha = -k^3$ (Ray, 24 Jan 2026).

1. Spectral Fractional Derivative: Definition and Properties

Let $f\in S(\mathbb{R})$ and $0<\alpha\leq 1$. The spectral fractional derivative $D_\xi^\alpha$ is defined through its Fourier transform by

$$\mathcal{F}[D_\xi^\alpha f](k) = (i k)^\alpha \hat{f}(k),$$

where the principal branch of $(i k)^\alpha$ is used. This operator extends by density to $D_\xi^\alpha: H^s\to H^{s-\alpha}$ for any $s\in\mathbb{R}$.

For $0<\alpha<1$, the Marchaud-type singular integral representation holds: $$D_\xi^\alpha f(\xi) = C_\alpha \int_0^\infty [f(\xi) - f(\xi-y)] y^{-1-\alpha}\, dy, \qquad C_\alpha = \frac{\alpha}{\Gamma(1-\alpha)} = -\frac{1}{\Gamma(-\alpha)}.$$

Bilinear Operator

Defined for $f,g\in S(\mathbb{R})$ as

$$D_\xi^\alpha f\cdot g := (D_\xi^\alpha f)g - f(D_\xi^\alpha g),$$

this operator is equivalently realized through a two-variable extension $D_{\xi_1}^\alpha - D_{\xi_2}^\alpha$ on $f(\xi_1)g(\xi_2)$ with evaluation at $\xi_1 = \xi_2 = \xi$. Its Fourier symbol is $(i k_1)^\alpha - (i k_2)^\alpha$, leading to the key form: $$D_\xi^\alpha f\cdot g(\xi) = (2\pi)^{-2} \int_{\mathbb{R}^2} [(i k_1)^\alpha - (i k_2)^\alpha] \hat{f}(k_1)\hat{g}(k_2) e^{i(k_1+k_2)\xi}\, dk_1\, dk_2.$$ For $0<\alpha<1$, the Marchaud-kernel representation generalizes to

$$D_\xi^\alpha f\cdot g(\xi) = C_\alpha \int_0^\infty [f(\xi) g(\xi-y) - f(\xi-y) g(\xi)] y^{-1-\alpha} dy.$$

Algebraic and Sobolev Properties

• Bilinear in $(f,g)$.
• Skew-symmetric: $D_\xi^\alpha f\cdot g = -D_\xi^\alpha g\cdot f$.
• Diagonal vanishing: $D_\xi^\alpha f\cdot f = 0$.
• Bounded: $D_\xi^\alpha : H^s\times H^s\to H^{s-\alpha}$, for $s > 1/2$.
• Classical limit: $D_\xi^\alpha f\cdot g \to D_\xi f\cdot g$ as $\alpha \to 1^-$.

2. Bilinear Formulation of the Spectral Time-Fractional KdV Equation

The spectral time-fractional KdV equation on $\mathbb{R}_x\times\mathbb{R}_t$ is

$$D_t^\alpha u + u_{xxx} + 6uu_x = 0, \quad 0<\alpha\leq 1,$$

where $D_t^\alpha$ is the spectral fractional derivative in $t$. The Hirota $\tau$-function ansatz is applied: $$u(x,t) = 2 \partial_x^2 (\log \tau(x,t)), \qquad \tau > 0.$$ The equivalent bilinear equation is

$[D_x D_t^\alpha + D_x^4]\ \tau\cdot\tau = 0,$

where $D_x$ and $D_x^4$ are Hirota operators in $x$, and $D_t^\alpha$ is the fractional bilinear operator in $t$. This reduction is justified by explicit calculation: the local terms are captured by

$$D_x^2 \tau\cdot\tau = 2\tau^2(\log\tau)_{xx}, \qquad D_x^4 \tau\cdot\tau = 2\tau^2 \left((\log\tau)_{xxxx} + 6((\log\tau)_{xx})^2\right),$$

while the mixed term satisfies

$$D_x D_t^\alpha \tau\cdot\tau = 2[\tau D_t^\alpha \tau_x - \tau_x D_t^\alpha \tau],$$

which, under standard manipulations, reduces to $2D_t^\alpha (\log\tau)_x$. Normalizing by $\tau^2$ yields the equation for $u$.

3. Fractional Dispersion Relation

To determine the dispersion law, consider the single exponential ansatz: $$\tau(x,t) = 1 + e^\theta, \quad \theta = k x + \omega t + \delta.$$ Off-diagonal contributions give

$$D_x^n e^{0\cdot x}\cdot e^{k x} = (-k)^n e^{k x}, \qquad D_t^\alpha e^{0\cdot t}\cdot e^{\omega t} = -\omega^\alpha e^{\omega t}.$$

Thus,

$$(D_x D_t^\alpha + D_x^4)(1\cdot e^\theta) = k\omega^\alpha e^\theta + k^4 e^\theta.$$

Symmetry yields the factor $2[k\omega^\alpha + k^4] e^\theta$, leading to the fractional dispersion relation: $\omega^\alpha = -k^3.$ When $\alpha=1$, the classical cubic $\omega=-k^3$ is recovered. For $0<\alpha<1$, the phase speed $\omega/k$ is a nonlinear function of $\alpha$, and wave packet dispersion corresponds to a time-rescaled fractional power: $\omega\sim k^{3/\alpha}$. This suggests that the fractional order parameter $\alpha$ directly modifies the temporal scaling of dispersive propagation while preserving the cubic dependence on $k$ structure.

4. Explicit Soliton Solutions

One-Soliton $\tau$-Function

With

$\tau(x,t) = 1 + e^{k x + \omega t + \delta},$

and dispersion $\omega^\alpha=-k^3$, the associated solution is

$$u(x,t) = 2 \partial_x^2 \log (1 + e^{k x + \omega t + \delta}) = 2k^2 \,\operatorname{sech}^2 \left(\frac{k x + \omega t + \delta}{2}\right),$$

a localized pulse with speed determined by the fractional dispersion relation.

Two-Soliton $\tau$-Function

The Hirota ansatz leads to

$$\tau(x,t) = 1 + e^{\theta_1} + e^{\theta_2} + A_{12} e^{\theta_1 + \theta_2},$$

with $\theta_j = k_j x + \omega_j t + \delta_j$ and $\omega_j^\alpha = -k_j^3$ for $j=1,2$. The interaction coefficient is

$$A_{12} = \left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2,$$

exactly as in the classical KdV case. The formal two-soliton solution for $u$ is then $u=2\partial_x^2\log\tau(x,t)$. In the classical limit $\alpha\to 1^-$, this construction and coefficient reduce directly to the standard KdV soliton result.

5. Summary of Analytical Features

$\begin{array}{lcl} \text{Spectral Fractional Operator} & D_\xi^\alpha f & (i\,k)^\alpha \hat{f}(k) \ \text{Marchaud Integral (0<\alpha<1)} & D_\xi^\alpha f(\xi) & C_\alpha \int_0^\infty [f(\xi) - f(\xi-y)] y^{-1-\alpha} dy \ \text{Bilinear Hirota Operator} & D_\xi^\alpha f\cdot g & (D_\xi^\alpha f)g - f(D_\xi^\alpha g) \ \text{Bilinear KdV Form} & D_x D_t^\alpha \tau\cdot\tau + D_x^4 \tau\cdot\tau & = 0 \ \text{Dispersion Law} & \omega^\alpha & = -k^3 \ \end{array}$

These results demonstrate that the spectral time-fractional KdV equation retains the structure of soliton solutions, with explicit $\tau$-functions and interaction coefficients matching the classical case. A plausible implication is that soliton interaction properties are dictated by the spatial algebraic structure, while the temporal evolution is sensitive to the fractional order. The construction unifies nonlocal calculus and integrable systems while admitting limit transitions to standard models (Ray, 24 Jan 2026).