Continuous binary Darboux transformation as an abstract framework for KdV soliton gases (2512.12495v1)

Abstract: We present a unified operator-theoretic framework for constructing deterministic KdV soliton gases and step-type KdV solutions. Starting from Dyson's determinantal formula, we obtain a broad class of reflectionless solutions and describe their basic spectral and analytic properties, including their interpretation as deterministic soliton gases. We then introduce a continuous binary Darboux transformation that acts directly on the scattering data and generates general step-type solutions, with particular emphasis on reflectionless hydraulic-jump-type profiles modelling a soliton condensate on the left and vacuum on the right. The paper is methodological in nature: our goal is not to develop a full kinetic or probabilistic theory, but to show how classical tools from spectral and scattering theory can be combined into a conceptually simple framework that accommodates both reflectionless and non-reflectionless soliton gas configurations, including step-like backgrounds.

• The paper presents a continuous binary Darboux transformation as an innovative operator-theoretic method to design deterministic KdV soliton gases.
• The framework generalizes Dyson’s determinantal formula to yield both reflectionless multi-soliton and step-type soliton condensate–vacuum configurations.
• The method ensures unique spectral mapping and analytical control over soliton injection and evaporation in non-decaying, asymmetric backgrounds.

Operator-Theoretic Framework for KdV Soliton Gases via Continuous Binary Darboux Transformation

Introduction and Context

The paper "Continuous binary Darboux transformation as an abstract framework for KdV soliton gases" (2512.12495) presents a comprehensive operator-theoretic framework for constructing deterministic soliton gas solutions and step-type configurations of the Korteweg–de Vries (KdV) equation. The methodology consolidates classical spectral and scattering theory techniques by leveraging Dyson's determinantal formula and, crucially, develops a continuous version of the binary Darboux transformation acting on scattering data. This approach offers a unified treatment of both reflectionless and more general non-reflectionless KdV backgrounds, extending well beyond the symmetric, spatially decaying scenario traditionally considered in soliton gas theory.

The KdV soliton gas paradigm treats large ensembles of KdV solitons as a thermodynamic limit, giving rise to macroscopic spectral descriptions and kinetic equations. However, physical water wave systems, motivating a significant body of integrable hydrodynamics research, frequently possess step-like (asymmetric) backgrounds, mandating a robust extension of soliton gas frameworks to nontrivial boundary conditions. This work addresses the methodological gap by exhibiting how classical operator methods—previously applied to reflectionless and symmetric settings—can accommodate general step-type backgrounds and, in particular, deterministic soliton condensate–vacuum configurations.

Dyson's Formula and Reflectionless Potentials

Central to the framework is Dyson's determinantal formula, which constructs solutions to KdV of the form:

$$q_{\sigma}(x, t) = -2 \partial_x^2 \log \det (I + \mathbb{K}_{x,t})$$

where $\mathbb{K}_{x,t}$ is a Hankel operator built from a compactly supported nonnegative measure %%%%2%%%% on $[0, \infty)$. For atomic (discrete) measures, the formula specializes to Kay–Moses multi-soliton solutions; for absolutely continuous or singular continuous $\sigma$, it yields generalized, possibly singular, reflectionless potentials. The correspondence with tau-function approaches and integrable operator theory is explicit: when $\mathrm{d}\sigma$ is positive and the integrability condition $\int \frac{\mathrm{d}\sigma(k)}{k} < \infty$ holds, the resulting $q_{\sigma}$ is a global, regular, reflectionless KdV solution.

Reflectionless potentials constructed via this method possess a purely singular negative spectrum (supported on $-\operatorname{Supp}(\sigma)^2$), and standard properties such as analyticity in strips, normal families (in the sense of Montel), and exponential decay as $x \to +\infty$ when $0 \not\in \operatorname{Supp}(\sigma)$. The solution class comprehensively encodes deterministic soliton gases in the sense of Zakharov's original construction—a deterministic, non-stochastic, prescription of spectral density for soliton ensembles.

Uniqueness of these reflectionless solutions is significant: the mapping between spectral data and solution profile is injective, ensuring that deterministic soliton gases generated in this manner retain a rigid correspondence between spectral content and physical realization. This property distinguishes them from generic step-like or almost periodic solutions, where uniqueness can fail without additional regularity.

Deterministic Soliton Gases and Step-Type Configurations

This framework interprets any bounded reflectionless KdV solution produced via Dyson’s formula as a deterministic soliton gas, generalizing the traditional requirement of continuous spectral intervals with smooth densities. Especially notable are reflectionless step-like profiles that interpolate between a condensate (maximally dense gas) and vacuum. For instance, the spectral measure

$$\mathrm{d}\sigma(k) = 2 \frac{k}{h} \sqrt{h^2 - k^2} \, \mathrm{d}k, \quad 0 \leq k \leq h$$

leads to a solution $q_{\sigma}(x, t)$ converging to $-h^2$ as $x \to -\infty$ and to zero as $x \to +\infty$. This explicit, analytic soliton condensate–vacuum interface matches the band density of states for one-gap finite-gap KdV potentials and demonstrates that the continuous operator construction can realize physical boundary-matching profiles known to arise in dispersive shock wave dynamics.

Structurally, these deterministic gases have several direct implications:

• The profile is completely determined by any nontrivial segment (due to analyticity and uniqueness).
• Solutions never bifurcate, and soliton crowding (pileup) is uniformly excluded.
• Reflectionless step-like potentials serve as analytical models for undular bores and soliton condensate–vacuum interfaces, bridging finite-gap theory and kinetic soliton gas approaches.

Continuous Binary Darboux Transformation

The main methodological advance is the introduction of a continuous analog of the binary Darboux transformation operating at the level of scattering data. For a step-type KdV solution $q(x, t)$ with scattering data $(R, \mathrm{d}\rho)$ and a signed measure $\sigma$ satisfying certain integrability conditions, the transformation modifies the negative spectrum:

$$q_{\sigma}(x, t) = q(x, t) - 2 \partial_x^2 \log \det (I + \mathbb{K}_{x, t})$$

where $\mathbb{K}_{x, t}$ integrates Jost solutions against $\mathrm{d}\sigma$. The result is a new step-type KdV solution whose reflection coefficient remains unchanged but whose spectral measure is shifted from $\mathrm{d}\rho$ to $\mathrm{d}\rho + \mathrm{d}\sigma$.

Several technical properties follow:

• The negative spectrum can be arbitrarily designed, allowing soliton injection and removal, with invertibility of the transformation.
• The change in potential decays rapidly at $+\infty$ if $0 \not\in \operatorname{Supp}(\sigma)$.
• Embedded eigenvalues arise naturally if $\mathrm{d}\sigma$ overlaps the existing negative spectrum.
• The transformation's applicability to non-reflectionless, asymmetric (step-type) backgrounds provides a rigorous and flexible operator-theoretic tool for constructing deterministic soliton gases on realistic physical backgrounds.

Implications and Open Problems

This operator-theoretic consolidation has several theoretical and practical consequences. It offers a transparent, mathematically rigorous platform to construct KdV deterministic soliton gases and interface solutions with prescribed spectral content, bridging finite-gap spectral theory, deterministic superpositions, and scattering-based approaches.

Notably, the framework opens avenues for:

• Modeling local and nonlocal interactions in soliton condensates, including dynamical "evaporation" of local soliton density under injection of new spectral components.
• Extending integrable turbulence theory to asymmetric settings and possibly building deterministic analogs of kinetic and statistical soliton gas equations on step-like backgrounds.
• Systematic construction of explicit, analytically tractable solutions for physical problems involving steps, hydraulic jumps, and dispersive shocks in hydrodynamic regimes.

Outstanding problems include rigorous characterization of soliton injection/evaporation phenomena, generalizations to left-scattering or non-scalar settings, and extension to higher-genus/finite-gap backgrounds or measures with singular continuous components.

Conclusion

The paper presents a unified and abstract framework for constructing KdV soliton gases using a continuous binary Darboux transformation, subsuming classical approaches and substantially expanding the range of analytically tractable KdV solutions. Deterministic soliton gases, step-type profiles, and soliton condensates are systematically realized using operator and spectral-theoretic tools, with robust extensibility to physically and mathematically significant configurations. This methodology provides a solid base for future exploration of both deterministic and statistical integrable hydrodynamics in asymmetric and non-decaying settings.