d2I~dt~2+~dI~dt~2I~+~I~2I~=B~(x~,t~)\frac{d^2 \tilde{I}}{d\tilde{t}^2} + \tilde{\gamma} \frac{d \tilde{I}}{d\tilde{t}} - \nabla^2 \tilde{I} + \tilde{\lambda} |\tilde{I}|^2 \tilde{I} = \tilde{B}(\tilde{x}, \tilde{t})dt~2d2I~+~dt~dI~2I~+~I~2I~=B~(x~,t~)
This form is convenient for simulation and stability analysis.
5. Stationary Solutions and Linear Stability
Set B(x,t)=0B(x,t) = 0B(x,t)=0 and consider stationary solutions I(x,t)=(x)eitI(x,t) = \phi(x) e^{-i \omega t}I(x,t)=(x)eit, leading to:
2ic22+2=0-\omega^2 \phi - i \gamma \omega \phi - c^2 \nabla^2 \phi + \lambda |\phi|^2 \phi = 02ic22+2=0
This is a nonlinear eigenvalue problem. The solutions include:
Localized solitonic modes,
Energy bubbles or curvature spikes,
Topological excitations under certain boundary/topology constraints.
6. Connection to Curvature and Geometry
Under energy-density localization, we define an effective metric perturbation:
R(x,t)I(x,t)2R(x,t) \sim \kappa |I(x,t)|^2R(x,t)I(x,t)2
Where:
R(x,t)R(x,t)R(x,t): emergent scalar curvature,
\kappa: proportionality constant linking energy density to curvature,
Suggesting geometry emerges from localized nonlinear excitation.
