Resonant Dispersive Wave Emission

• RDW emission is a nonlinear phenomenon where energy transfers from solitons or dispersive shock waves to linear dispersive waves at specific phase-matched frequencies.
• The process is modeled using the defocusing nonlinear Schrödinger equation with higher-order dispersion, allowing quantitative predictions through precise phase-matching conditions.
• This mechanism underpins applications in ultrafast optics, supercontinuum generation, fluid dynamics, and plasma physics by enabling engineered spectral control.

Resonant Dispersive Wave (RDW) emission is a fundamental nonlinear wave phenomenon occurring in dispersive media when coherent, nonlinear excitations—such as solitons or dispersive shock waves—transfer energy to linear dispersive waves at distinct, phase-matched frequencies. Initiated and sustained by higher-order (or non-convex) dispersion, RDW emission underpins a wide variety of applications in ultrafast optics, nonlinear fiber physics, plasma physics, and fluid dynamics. The phenomenon is governed by precise phase-matching conditions that link the nonlinear wave’s dynamical properties (including velocity and nonlinear phase shift) to the propagation constants set by the full, frequency-dependent dispersion relation.

1. Physical Mechanism, Governing Equations, and Phase-Matching Conditions

The essential physics of RDW emission involves resonant energy transfer from a strongly nonlinear structure—typically a soliton or a dispersive shock wave (DSW)—to linear dispersive waves at frequencies where a phase-matching (or resonance) condition is satisfied. The most general treatment appears in the context of the defocusing nonlinear Schrödinger equation (dNLSE) with a generic dispersion operator: $$i\varepsilon \partial_z \psi + d(\partial_t)\psi + |\psi|^2\psi = 0,$$ where $\varepsilon$ controls the relative strength of nonlinearity and leading-order dispersion, and $d(\partial_t)$ is typically a finite expansion in even and odd-order derivatives: $$d(\partial_t) \equiv \sum_{n} \frac{\beta_n}{n!} \varepsilon^n (i\partial_t)^n = -\frac{\varepsilon^2}{2}\partial_t^2 - i\frac{\beta_3\varepsilon^3}{6}\partial_t^3 + \frac{\beta_4\varepsilon^4}{24}\partial_t^4 + \cdots.$$ RDW emission is triggered when HOD becomes non-negligible in the nonlinear regime. The emission frequency $\omega_{RR}$ is determined by a phase-matching condition: $$\sum_{n}\frac{\beta_n}{n!}(\varepsilon\omega)^{n} - V_s (\varepsilon\omega) = \varepsilon k_{nl},\quad (1)$$ where $V_s$ is the relevant shock or soliton velocity and $k_{nl}$ encodes additional nonlinear phase shifts. For the case with dominant third-order dispersion, the key cubic equation is: $$\frac{\beta_3}{6}(\varepsilon\omega)^3 + \frac{1}{2}(\varepsilon\omega)^2 - V_s(\varepsilon\omega) + \rho_l \approx 0.$$ Here, $\rho_l$ is a density parameter and, depending on the specific regime, $V_s$ is the velocity of the dispersive shock’s leading edge or the value derived from shock jump conditions (Rankine-Hugoniot).

The physical resonance is a Doppler effect in the frame moving with the nonlinear front, and the outcome is a well-defined transfer of energy into radiation at specific frequencies—observable, for instance, as spectral peaks far from the pump.

2. Distinct Regimes and Diversity of RDW Phenomenology

The structure and spectral properties of the emitted RDW depend critically on the order and magnitude of higher-order dispersion:

• Perturbative regime (|β₃| ≲ 0.5): The DSW forms a broad fan of oscillations. The RDW frequency is dictated by the velocity of the leading edge, $V_l = 2\sqrt{\rho_r} - \sqrt{\rho_l}$. In this case, setting $V_s = V_l$ in (1) yields an accurate ω_RR.
• Non-perturbative regime (|β₃| ≈ 1): The fan structure collapses into a sharp front that can be approximated as a classical shock. The relevant velocity $V_c$ is given by the Rankine–Hugoniot jump condition: $$V_c = \frac{\beta_2(\rho_l u_l - \rho_r u_r) + \frac{\beta_3}{2} (\rho_l u_l^2 - \rho_r u_r^2)}{\rho_l - \rho_r},$$ where $u_l, u_r$ are velocity/chirp terms on each side of the discontinuity.
• Fourth-order dispersion scenario (β₄ < 0): Negative fourth-order dispersion allows the coexistence of modulational instability (MI) with DSWs. The phase-matching equation becomes quartic, leading to up to four distinct RDW frequencies. For defocusing cases (β₂ = 1), MI is “seeded” by the DSW edges and the spectral RR peaks appear symmetrically.

This variety of behaviors—ranging from resonant fans to localized, shock-like features or MI-aided emission—is governed by the explicit form of the dispersion and the shock/soliton velocities, providing a rigorous, quantitative framework for predicting RDW frequencies and spectral locations.

3. Quantitative Estimation of RDW Frequencies and Shock Velocities

For practical application, the computation sequence for predicting RDW frequencies involves:

1. Identify the dominant dispersion orders: Extract $\beta_2, \beta_3, \beta_4, ...$ from the medium/fiber or model.
2. Determine the regime:
   • Weak HOD/perturbative: Use Whitham modulation theory to compute the DSW leading edge velocity $V_l$.
   • Strong HOD/non-perturbative: Use conservation law techniques (Rankine-Hugoniot) for jump velocities $V_c$.
3. Insert velocity into the phase-matching equation: For third-order dispersion, solve:

$$\frac{\beta_3}{6} (\varepsilon \omega)^3 + \frac{1}{2} (\varepsilon \omega)^2 - V_s (\varepsilon \omega) + \rho_l \approx 0,$$

where typically $\varepsilon \omega_{RR} \approx -3\beta_2/\beta_3$ in the limit $\beta_3 V_s \rightarrow 0$.

1. Handle additional nonlinear terms (if significant): The nonlinear phase contributions (XPM, background density) can shift the RR condition, especially at higher pulse intensities or for more complex input structures.
2. Solve algebraically or numerically for $\omega_{RR}$: The roots correspond to the phase-matched emission frequencies.

This algorithmic approach makes the full RDW emission dynamics in DSWs and other nonlinear systems quantitatively predictable.

4. Relation to Broader Nonlinear Wave Physics

The RDW emission framework described for the dNLSE generalizes to a wide array of physical systems:

• Optical fibers/waveguides: HOD-induced RDW is essential in supercontinuum generation, frequency combs, and Kerr microresonators.
• Fluid dynamics: Water waves governed by KdV-type equations with higher-order corrections (extended KdV, Kawahara equation) exhibit radiating shock waves with similar resonance conditions—often captured or modeled using modulation (Whitham) theory.
• Bose–Einstein condensates and plasma physics: Nonlinear structures in dispersive media, including DSWs and “rogue” soliton-like excitations, exhibit RDW emission when group/phase velocity matching is enabled by higher-order dispersion.
• Quantum nonlinear systems: The rigorous connection between nonlinear coherent structures, resonant coupling, and emitted linear radiation is a template across nonlinear dynamics.

5. Analytical and Numerical Verification

The theoretical predictions of RDW emission have been systematically validated via:

• Direct numerical simulations: Integrations of dNLSE and its generalizations with higher-order dispersion confirm both the predicted dynamics and the spectral locations of RDW peaks under various regimes.
• Analytical asymptotics and modulation theory: Whitham theory and perturbative expansions reproduce both the DSW structures and the emitted RR, including nonlinear interactions between the DSW fans/fronts and the linear radiation. The interplay between leading-edge velocities, background density jumps, and HOD coefficients is corroborated by both full and reduced models.
• Diverse scenarios and experimental relevance: Recognition of “fan”-type RR, localized front emission, and MI-seeded spectrum underpins experiments in nonlinear fiber optics, light propagation in nonlinear crystals, and even shallow water wave tanks.

6. Implications, Limitations, and Future Directions

RDW emission is central to both understanding the limits of coherence and spectral behavior in nonlinear wave systems and to engineering applications:

• Predictive spectral engineering: By controlling input conditions and tailoring dispersion profiles, specific RDW peaks can be targeted for ultrafast pulse generation, supercontinuum extension, or selective frequency conversion.
• Extreme parameter regimes: In systems where HOD magnitudes are large, or where nonlinearities induce strong density or intensity jumps, hybrid behaviors emerge, requiring a detailed interplay of modulation theory, conservation law analysis, and numerical simulation.
• Multiple dispersive orders and control of instability: The occurrence and stability of RDW emission, particularly its coexistence or competition with MI, remain fertile topics for both mathematical and experimental inquiry.
• Generalization to dissipative and non-integrable systems: While the core framework is conservative and based on integrable (or weakly-perturbed) models, the roots of RDW emission—phase-matching between nonlinear dynamics and linear modes—translate to a broad class of physically significant settings, including driven-dissipative and complex media.

In conclusion, the comprehensive, phase-matching-based framework for RDW emission provides the basis for accurate prediction and understanding of radiative phenomena in nonlinear dispersive systems. The approach detailed in the dNLSE context is both general and adaptable, encompassing regimes from classical DSW fans through strongly localized shock fronts and accommodating the full hierarchy of dispersion-induced phenomena in modern wave physics (Conforti et al., 2013).