Coupling of Coaxial Roton Spiral Waves

In this section we derive a compact coupling model for self-stable coaxial cylindrical (spiral) LEDO-waves, as they occur in the Roton-Model. We consider two Roton-spiral waves located on different ring radii, and compute how strongly the outer spiral wave can couple inward to the inner spiral wave. We also identify the geometric and frequency conditions for forward coupling, resonance, and the possibility of a backward-running component.

Coaxial Roton Spirals

Radial Gaussian Overlap

Conclusion:

Strong coupling of two Roton-Resonance waves requires nearly matching radii and small-integer resonance conditions. A backward-running LEDO component (repulsion) meight potentially become possible when the inner spirals frequency forms a specific periodic modulation.

1. Geometry of a Single Roton Spiral Wave

A self-stable Roton spiral wave positioned on a ring of radius $R$ around the $z$-axis can be written in cylindrical coordinates $(\rho,\theta,z)$ as

$$ \Psi_R(\rho,\theta,z,t) = A_R, f_R(\rho), e^{i(m\theta + kz - \omega t)}, $$

where:

• $R$ — ring radius of the spiral wave
• $m$ — azimuthal mode number (number of windings per full rotation)
• $k$ — longitudinal wavenumber (spiral rise along $z$)
• $\omega$ — temporal frequency
• $A_R$ — amplitude
• $f_R(\rho)$ — radial ring profile

A Gaussian profile is convenient:

$$ f_R(\rho) = \exp!\left( -,\frac{(\rho - R)^2}{2\sigma_R^2} \right), $$

with $\sigma_R$ the radial “thickness” of the cylindrical wavefront.

2. Inner Spiral Wave on a Smaller Radius

A second Roton spiral wave on a smaller radius $r < R$ is modeled analogously:

$$ \Psi_r(\rho,\theta,z,t) = A_r, f_r(\rho), e^{i(m_r\theta + k_r z - \omega_r t)}, $$

with

$$ f_r(\rho) = \exp!\left( -,\frac{(\rho - r)^2}{2\sigma_r^2} \right). $$

The goal is to determine how strongly $\Psi_R$ couples into $\Psi_r$.

3. Coupling Strength via Mode Overlap

The natural definition of the coupling coefficient is the mode-overlap integral:

$$ K_{R\to r} \propto \int \Psi_R(\rho,\theta,z,t), \Psi_r^{!*}(\rho,\theta,z,t), dV , $$

with the cylindrical volume element

$$ dV = \rho, d\rho, d\theta, dz $$

Inserting the two spiral waves gives

$$ \Psi_R \Psi_r^{!} = A_R A_r^{!}, f_R(\rho) f_r(\rho), e^{ i[(m-m_r)\theta + (k-k_r)z - (\omega-\omega_r)t]} $$

Integration over $\theta$ and $z$ yields:

• Azimuthal matching: $m = m_r$
• Longitudinal forward coupling: $k \approx k_r$
• Longitudinal backward component: $k \approx -k_r$
• Temporal resonance: $\omega \approx \omega_r$

Coaxial Roton Spirals

Radial Gaussian Overlap

The remaining radial overlap integral is

$$ I_\rho = \int_0^\infty \rho, f_R(\rho) f_r(\rho), d\rho . $$

For Gaussian ring-profiles:

$$ I_\rho \propto \exp!\left[ -,\frac{(R - r)^2}{2(\sigma_R^2 + \sigma_r^2)} \right]. $$

Thus:

$$ K_{R\to r} = K_0 , \exp!\left[ -,\frac{(R-r)^2}{2\sigma_{\text{eff}}^2} \right], F_{\text{res}}(m,m_r,k,k_r,\omega,\omega_r), $$

with $\sigma_{\text{eff}}^2 = \sigma_R^2 + \sigma_r^2$.

4. Geometric Requirement for Significant Coupling

Significant coupling requires

$$ |R - r| ,\lesssim, \sigma_{\text{eff}}. $$

Typical assumptions ($\sigma_R \sim \sigma_r \sim 0.1R$) give

• Strong coupling for $|R-r|/R \lesssim 0.1$.
• Rapid suppression for $|R-r| \gtrsim 2\sigma_{\text{eff}}$.

5. Resonance Conditions

1. Temporal Resonance $$ \omega_R \approx \omega_r. $$

2. Azimuthal Resonance $$ mR \approx m_r r. $$

3. Longitudinal Resonance Forward: $$ k \approx k_r. $$

Backward: $$ k \approx -k_r. $$

6. Condition for a Backward-Running Component

If the inner spiral forms a periodic modulation of period $\Lambda$, define

$$ G = \frac{2\pi}{\Lambda}. $$

Backward scattering occurs if

$$ 2k_R \approx G, \qquad k_r \approx -k_R. $$

7. Summary

The coupling of coaxial Roton spiral waves is determined by radial overlap and resonance conditions:

$$ K_{R\to r} \sim \exp!\left[ -,\frac{(R-r)^2}{2\sigma_{\text{eff}}^2} \right] \cdot F_{\text{res}}(m,m_r,k,k_r,\omega,\omega_r). $$

Conclusion: Strong coupling requires nearly matching radii and small-integer resonance conditions. A backward-running LEDO component becomes possible when the inner spiral forms a periodic modulation satisfying $k_r \approx -k_R$.