Reflectionless Hydraulic-Jump Profiles

• Reflectionless hydraulic‐jump‐type profiles are special fluid transitions that ensure non‐radiative state changes by exactly matching conservation laws and characteristic speeds.
• Key methodologies involve two-layer Boussinesq systems, variable-coefficient shallow-water models, and integrable KdV frameworks employing inverse scattering and Darboux transformations.
• These profiles prevent reflected waves and radiative losses, offering practical solutions for engineering designs that require precise energy conservation in fluid systems.

A reflectionless hydraulic-jump-type profile describes a transition in a fluid system (or dispersive wave system) between two distinct states, which does not generate reflected waves or radiative losses under the relevant evolution equations. These profiles manifest as special solutions in various models, including two-layer Boussinesq shallow-water equations, linearized variable-coefficient shallow-water flows, and integrable dispersive equations such as KdV, where the "reflectionless" property signifies exact transmission of disturbances through the profile, often corresponding to conservative hydraulic jumps or soliton condensate steps. The underlying mathematical structure involves exact conservation laws, characteristic velocity continuity, and explicit spectral constructions.

1. Governing Models and Conservation Laws

Reflectionless hydraulic-jump-type profiles arise in several mathematical frameworks:

• Two-layer Boussinesq shallow-water systems feature two immiscible fluids with slightly different densities and enforce the rigid-lid, Boussinesq, and hydrostatic approximations. The key variables are the interface displacement $\eta(x,t)$ and the layer-wise velocity difference (shear) $\theta(x,t)$. The governing evolution equations admit four locally conservative forms: circulation, kinematic (mass-difference), impulse, and energy conservation, with explicit expressions for each. Importantly, the shallow-water equations exhibit invariance under interchange of interface height and shear, leading to symmetries fundamental for reflectionless profiles (Priede, 2022).
• Linear shallow-water theory with variable geometry and current considers flows in ducts or channels with spatially varying depth $H(x)$, width $W(x)$, and mean flow velocity $U(x)$. The reflectionless property for linear perturbations depends on fine-tuned relationships among these variables, such as particular ordinary differential equations (ODEs) connecting $c(x) = \sqrt{g H(x)}$ and $U(x)$ (Churilov et al., 2021).
• Integrable dispersive systems (KdV type) allow construction of reflectionless step-like solutions through operator-theoretic and inverse scattering methods. The KdV equation admits a hierarchy of reflectionless profiles generated via Dyson’s determinantal formula and continuous binary Darboux transformations (Rybkin, 13 Dec 2025).

2. Reflectionless Condition and Jump Structure

The reflectionless character of a hydraulic-jump-type profile is defined by the continuity of characteristics (or eigenspeeds) and the absence of wave reflection or radiation:

• In two-layer systems, the "cross-symmetry" condition ($\eta_+ = \theta_-, \theta_+ = \eta_-$ for upstream/downstream values) ensures the automatic conservation of impulse and energy, provided that mass and circulation conservation laws are satisfied. This symmetry implies that the characteristic velocities

$$\lambda_\pm = -\eta\,\theta \pm \frac{1}{2}\sqrt{(1-\eta^2)(1-\theta^2)}$$

are identical on both sides of the jump, prohibiting reflection of linear disturbances (Priede, 2022).

• For linear shallow-water flow with variable coefficients, reflectionlessness requires that the pair $(c(x), U(x))$ satisfy

$$\frac{d}{dx}[c(x)\,U(x)] = \frac{2D\,c^{5/2}(x)\,U^{3/2}(x)}{c^2(x) - U^2(x)},$$

with $D=0$ yielding the simpler condition $c(x)U(x) = \text{const}$. When this holds, excitations propagate freely through hydraulic-jump-like transitions, such as Froude-number changes, without partial reflection (Churilov et al., 2021).

• For KdV and soliton gases, the operator-theoretic construction ensures that the reflection coefficient $R(k)$ vanishes on the continuous spectrum, with the spectral measure filling a specified band. Step-like profiles display a "hydraulic-jump" from a soliton condensate on one side to vacuum on the other, with no radiative tail (Rybkin, 13 Dec 2025).

3. Explicit Construction of Reflectionless Hydraulic-Jump-Type Solutions

Two-layer Boussinesq Systems

• The Rankine–Hugoniot conditions for the four conservation laws (mass-difference, shear, impulse, energy) pin down possible jump configurations.
• Imposing the cross-symmetry and combining with mass and circulation conservation gives the jump propagation speed:

$$s = \frac{d\xi}{dt} = \pm \frac{1}{2}(1 + \eta_+\,\eta_-).$$

• The dispersive (non-hydrostatic) solibore equivalent, retaining the leading Green–Naghdi correction, yields the ODE

$$\left(\frac{d\eta}{d\xi}\right)^2 = \frac{3}{4c^2}(\eta_+-\eta)^2(\eta-\eta_-)^2,$$

with a tanh-profile solution

$$\eta(\xi) = H_0 + \Delta H \tanh (k\xi),\quad k = \frac{\sqrt{3}\Delta H}{2c}.$$

Here, $H_0 = (\eta_++\eta_-)/2$, $\Delta H = (\eta_+-\eta_-)/2$ (Priede, 2022).

Linear Shallow-Water Theory

• Hydraulic-jump-type, reflectionless profiles are built by enforcing the reflectionless ODE for $(c(x), U(x))$. Several classes of profiles are constructed:
  • Arbitrary $c(x)$ with $U(x) = \kappa / c(x)$ and corresponding adjustments to $W(x)$.
  • Constant wave speed or power-law couplings lead to quartic equations for auxiliary variables and explicit matching of subcritical and supercritical branches.
• The general reflectionless wave solution is

$$\varphi(x,t) = a(x) \left[ \Psi^+\left( t - \int^x \frac{dz}{U(z)+c(z)} \right) + \Psi^-\left( t - \int^x \frac{dz}{U(z)-c(z)} \right) \right],$$

with $a(x) = \sqrt{c(x) U(x)}$ and arbitrary traveling waveforms $\Psi^\pm$ (Churilov et al., 2021).

KdV and Soliton Condensates

• Dyson’s determinantal construction starts from a nonnegative, compactly supported measure $\sigma(k)$ and produces a reflectionless KdV solution via

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

where $\mathbb{K}_{x, t}$ is a Hankel operator with kernel parameterized by $\sigma(k)$.

• The continuous binary Darboux transformation allows further, step-type hydraulic-jump solutions with explicit Fredholm integral formulas for the potential profile. For the specific case

$$d\sigma(k) = 2 \frac{k}{h} \sqrt{h^2 - k^2}\,\mathbf{1}_{[0, h]}(k)\,dk,$$

the resulting profile satisfies

$$\lim_{x \to -\infty} q_\sigma(x, t) = -h^2, \quad \lim_{x \to +\infty} q_\sigma(x, t) = 0,$$

and is strictly reflectionless (Rybkin, 13 Dec 2025).

4. Spectral, Asymptotic, and Analytic Properties

• In two-layer and linear models, the reflectionless condition is associated with continuity of all hyperbolic characteristics or wave modes, precluding both partial reflection and dispersive radiation.
• In the operator-theoretic KdV construction, potentials are analytic in a strip $|\Im z| < 1/h$, with universal bounds $|q_\sigma(x+iy, t)| \le 2h^2(1-h|y|)^{-2}$ and, for the step-type profile, exponential decay at $x \to +\infty$.
• The spectral measure for KdV fills the band $[-h^2, 0]$, where the density

$$\rho(k) = \frac{2k}{h} \sqrt{h^2 - k^2},\quad k \in [0, h]$$

matches the density of states for the one-gap elliptic solution; thus, the jump profile may be interpreted as a soliton condensate band abutting vacuum (Rybkin, 13 Dec 2025).

5. Physical Interpretation and Applicability

• Reflectionless hydraulic-jump-type profiles correspond to non-radiating, lossless transitions such as internal solibores or smooth transitions between different background densities or currents.
• They are of primary importance for identifying transitions that do not generate undular tails or turbulent dissipation, in stark contrast to generic hydraulic jumps or internal bores where partial reflection, short-wave radiation, or mixing are unavoidable (Priede, 2022).
• In engineering, such profiles are relevant for channel design where mitigation of wave impact or energy loss is crucial, provided the linearized regime is adequate (Churilov et al., 2021).

6. Connections to Soliton Gases and Special Cases

• The construction of reflectionless KdV jump profiles generalizes to the soliton gas framework: replacing the continuous measure $\rho(k)$ by discrete approximations yields finite $N$-soliton Kay–Moses determinants whose large-$N$ limit recovers the continuous step profile.
• In the pure one-gap elliptic (cnoidal wave) case, these reflectionless profiles correspond to exact finite-gap solutions, and the soliton condensate density agrees precisely with the density of states of the underlying periodic solution.
• The continuous binary Darboux transformation framework accommodates both step-type and more general soliton gas backgrounds, providing an abstract operator-theoretic foundation for these phenomena (Rybkin, 13 Dec 2025).

7. Limitations and Practical Considerations

• Two-layer and linear shallow-water models: Reflectionless hydraulic-jump-type profiles require precise matching of upstream/downstream states or exact balancing of variable coefficients, limiting their occurrence in practical nonlinear, turbulent, or multidimensional regimes. In the nonlinear regime, the RL-ODE may become singular at critical points, with possible weak defects unless the double root condition is enforced (Churilov et al., 2021).
• Dispersive (KdV-type) models: Reflectionless jumps are robust in the sense of scattering data, but real physical realizations are subject to finite-gap effects, internal mixing, and complicated boundary conditions.

The reflectionless hydraulic-jump-type profile unifies key concepts in the study of conservative, non-radiative transitions across a spectrum of hydrodynamic and dispersive wave systems, providing foundational understanding and explicit solution techniques for non-dissipative jumps and soliton condensates.