In-Band Supratransmission in Nonlinear Lattices

• In-band supratransmission is a nonlinear effect where energy is transmitted inside a forbidden band gap once a critical driving amplitude destabilizes localized, evanescent modes.
• The phenomenon relies on nonlinear instabilities and bifurcation dynamics, analyzed via Schrödinger-type equations, Jacobian spectra, and saddle-node/Hopf bifurcations.
• Applications span photonics, phononics, and metamaterials, leveraging controlled energy transmission in systems like flat band lattices, Josephson junctions, and saturable nonlinear media.

In-band supratransmission is a nonlinear wave phenomenon in discrete lattices and multicomponent systems where energy is transmitted at a driving frequency inside a band that, in the linear regime, is non-propagating due to forbidden band gaps or flat band localization. Contrary to linear predictions, above a critical driving amplitude, nonlinear instabilities destabilize evanescent modes and trigger sudden transitions to energy transmission, often accompanied by the generation of localized nonlinear excitations such as gap solitons, breathers, kinks, or modulated wavepackets. Recent advances clarify in-band supratransmission mechanisms in flat band networks, saturable nonlinear media, oscillator arrays, Josephson junctions, and multicomponent systems, with implications for photonics, phononics, and engineered metamaterials.

1. Fundamental Mechanism and Mathematical Formulation

In-band supratransmission occurs when a discrete or multicomponent nonlinear system is driven harmonically at a frequency coinciding with a forbidden band gap or a flat band—regions where linear transport is suppressed (zero group velocity, exponential evanescence, or destructive interference). The dynamics are described by nonlinear generalizations of Schrödinger-type equations, sine-Gordon chains, or multi-field wave equations. For example, in a nonlinear flat band lattice with Kerr nonlinearity, edge driving at flat band frequency leads to time-independent nonlinear equations such as:

$i\dot{A}_n = \dots + \gamma |A_n|^2 A_n,$

where $\gamma$ quantifies cubic nonlinearity and similar terms appear for other sublattice components.

A characteristic feature is the existence of a localized evanescent edge mode under weak driving. As driving amplitude $f_0$ increases, bifurcation diagrams constructed by plotting the norm of the steady-state solution versus $f_0$ reveal a turning point or onset of instability. At the critical threshold $f_0^{\text{th}}$, the evanescent solution either ceases to exist (saddle-node bifurcation) or loses stability (real or complex eigenvalues cross into the unstable regime), prompting nonlinear plane waves or soliton formation and the onset of supratransmission (Kusdiantara et al., 17 Sep 2025, Susanto et al., 2023).

2. Role of Nonlinearity

Nonlinearity is both necessary and determinative for supratransmission:

• Cubic (Kerr-type) nonlinearities ($\gamma|A_n|^2A_n$) modify localized edge and bulk modes, enabling the destabilization of flat band-confined states. The nonlinearity counteracts destructive interference, shifts energy thresholds, and alters dispersion relations.
• For defocusing nonlinearity ($\gamma<0$), evanescent edge states are more likely to destabilize via quartet eigenvalue instabilities, resulting in an abrupt surge in transmission (as observed in sawtooth and diamond lattices) (Susanto et al., 2023, Kusdiantara et al., 17 Sep 2025).
• Higher-order nonlinearities, such as cubic–quartic terms, further affect modulation instability (MI) gain profiles and the threshold for supratransmission, supporting transitions from bright solitons to chaotic wavepackets (Houwe et al., 2023).
• Saturable nonlinearities constrain nonlinearity at high intensity (e.g., $\psi_n/(1+|\psi_n|^2)$), producing distinct bifurcation structures and threshold formulas for waveguide arrays (Susanto et al., 2019).

The critical amplitude thresholds are derived via static analysis, bifurcation theory, and expansion methods, for example:

$$A_{\text{thr}} = \gamma/2 + (2-\Delta) + \frac{(\Delta-1)(\Delta-3)}{\gamma} + O(1/\gamma^2)$$

for discrete NLS equations with saturable nonlinearity (Susanto et al., 2019).

3. Flat Band and Lattice Topological Effects

Flat bands, defined by constant dispersion ($\omega(k)$ independent of $k$), present zero group velocity and prevent linear wave transmission. In-band supratransmission leverages these localized states:

• In diamond lattices, high connectivity produces a broad dispersive spectrum, allowing thresholds for supratransmission to be tuned (Kusdiantara et al., 17 Sep 2025).
• Stub lattices, with reduced connectivity ($c_1 = 0$), maintain a flat band but also support topologically protected edge states. The existence and stability of these edge states can lead to additional dynamic features, such as spectral beating below threshold or multi-modal transitions at threshold (Kusdiantara et al., 17 Sep 2025).
• Nearly flat bands retain similar phenomenology; supratransmission remains robust even when the flat band is slightly dispersive (Susanto et al., 2023).
• Evansescent edge modes, present at flat band frequencies, serve as "barriers" to energy propagation until destabilized by nonlinearity, typically at a critical amplitude (Susanto et al., 2023).

Topology (lattice connectivity, unit cell structure) systematically shapes both the forbidden regions and the energy thresholds for supratransmission, and it determines whether the onset is governed by mode disappearance (turning point) or instability.

4. Bifurcation, Instability, and Numerical Characterization

Analysis proceeds by constructing steady-state solutions under time-harmonic driving (often at the boundary):

• Bifurcation diagrams capture solution norm versus driving amplitude, revealing transitions via saddle-node bifurcations (loss of steady solution) or Hopf/oscillatory instabilities (complex eigenvalue quartets) (Kusdiantara et al., 17 Sep 2025).
• Linearization of the governing equations about steady states yields Jacobian matrices whose spectra are used to determine stability regions. Stability is lost when eigenvalues cross the imaginary axis, marking the threshold for supratransmission.
• Time-dependent simulations validate the bifurcation analysis: below threshold, energy remains at the boundary; above threshold, power transmission surges and localized wavepackets emerge and propagate.
• Fourier and spectral analysis of site dynamics demonstrates a transition from discrete resonance (sub-threshold) to broadband noise or excitation (above threshold), especially apparent in flat/stub band models (Susanto et al., 2023, Kusdiantara et al., 17 Sep 2025).

5. Extensions to Multicomponent, Disordered, and Damped Systems

• In multicomponent systems (e.g., coupled three-wave resonant interactions), an asymptotic expansion rooted in linear evanescence is constructed. Thresholds are determined via vanishing Jacobian conditions on the driving amplitude–parameter map, generalizable to arbitrary component number via determinant singularity (Anghel-Vasilescu et al., 2010).
• In disordered periodic lattices, supratransmission threshold (ensemble average) is robust against disorder, but transmitted energy is reduced due to Anderson localization; damping further influences bifurcation types and transmission regime (Yousefzadeh et al., 2015).
• Hysteretic damping alters wavepacket spreading properties — local models yield uniform spreading, while nonlocal/difference-dependent damping models produce fast, localized packets and breather arrest phenomena, relevant for energy attenuation and rapid localization (Bountis et al., 2021).

6. Physical Realizations and Applications

Physical systems in which in-band supratransmission has been observed or predicted include:

• Photonic and phononic lattices with engineered flat bands (diamond, stub, sawtooth, FPU ladder systems) (Kusdiantara et al., 17 Sep 2025, Susanto et al., 2023, Malishava et al., 2015).
• Arrays of coupled oscillators and Josephson junctions, supporting inbound digital signal transfer via nonlinear supratransmission (Macías-Díaz et al., 2011, Santis et al., 2022, Guarcello et al., 2015).
• Long Josephson junctions: in-band supratransmission generates travelling breathers whose formation, decay, and probability are set by damping, bias, and noise (Santis et al., 2022).
• Saturable nonlinear waveguide arrays with both site- and intersite-centered localized states and distinct critical thresholds provide photonic platforms for supratransmission with controlled nonlinear dissipation (Susanto et al., 2019).
• Chains with cubic–quartic nonlinearities, supporting the creation and transformation of long-lived modulated waves—including transitions between bright and chaotic solitons through the supratransmission regime (Houwe et al., 2023).

Applications can be realized in optical switches, amplifiers, power limiters, phonon transistors, digital signal transmission devices, and mechanical metamaterials for energy management.

7. Outlook and Research Directions

Recent results urge a broader re-examination of energy transport in nonlinear lattices, focusing on:

• The interplay of topology (flatness, connectivity, disorder), nonlinearity, and boundary conditions in setting transmission thresholds and mode selection.
• The detailed bifurcation structure and stability analysis of edge and bulk modes, particularly in high-component systems.
• The effect of dissipation, hysteretic damping, and noise on threshold dynamics and mode lifetimes in supratransmission.
• Extension to systems with embedded flat bands, higher-order nonlinearities, and topologically protected edge states for robust, tunable wave control.
• The exploration of supratransmission in multi-dimensional and higher-rank lattices for advanced photonic and phononic device engineering.

These developments position in-band supratransmission as a foundational mechanism for next-generation devices in wave-based computation, energy harvesting, and dynamic signal processing.