a. Assume a quasi-stationary solution:
I(x,t)A(x)eitI(x,t) \sim A(x)e^{-i\omega t}I(x,t)A(x)eit, where A(x)A(x)A(x) is a complex amplitude envelope.
b. Transform into Fourier space:
A^(k)=B^(k)2+i+c2k2(A2A)^(k)2+i+c2k2\hat{A}(k) = \frac{\hat{B}(k)}{-\omega^2 + i\gamma \omega + c^2 k^2} - \frac{\lambda \widehat{(|A|^2 A)}(k)}{-\omega^2 + i\gamma \omega + c^2 k^2}A^(k)=2+i+c2k2B^(k)2+i+c2k2(A2A)(k)
The denominator represents the dispersion relation, modified by damping and mass terms.
c. Iterative update scheme:
Iteratively renormalize the spectral amplitude A^(k)\hat{A}(k)A^(k) to enforce a target power or norm constraint.
d. Back-transform to real space:
Recover spatially localized solutions with self-consistent spectral profiles.
This process allows us to extract stable eigenmodes, filter out unstable growth channels, and track energy localization across scales.
2. Mapping the Stability Landscape
By varying key parameters---particularly the nonlinear coefficient \lambda, damping rate \gamma, and pulse energy input---we build a phase diagram of solution regimes:
Stable Soliton Regime: Localized pulses retain form over time, exhibiting spectral plateaus and robust phase-locking.
Breathing / Quasi-periodic Regime: Solutions oscillate in amplitude, but maintain spatial coherence.
Turbulent Regime: Energy rapidly disperses; the spectrum becomes broad and incoherent.
Collapse / Blow-up Regime: Excessive nonlinear focusing leads to numerical divergence or singular energy spikes.
Each regime can be characterized by spectral entropy, Lyapunov exponents, and renormalized spectral width, yielding a multi-dimensional stability landscape.
3. Spectral Indicators of Emergence and Order
To monitor the transition from blink-induced chaos to ordered structures, we use several spectral observables:
By tracking these metrics, we determine how emergent geometries are encoded in spectral structure, particularly as certain Fourier modes dominate, representing the core frequencies of localized spatial zones.
4. Spectral Signatures of Topological States
Topological excitations such as vortices and domain walls exhibit distinct spectral fingerprints:
Vortices: Ring-like spectral features, angular phase winding.
Domain walls: Step-function-like spatial profiles yield sinc-shaped spectral peaks.
Multi-soliton states: Multi-lobed spectral envelopes, phase-coherent peaks.
This allows us to classify emergent structures spectrally, supporting topological interpretations without direct geometric assumptions.
5. From Spectral to Physical Geometry
