We construct a Lagrangian density:
L=12(tI2c2I2)4I4+J(x,t)I+J(x,t)I\mathcal{L} = \frac{1}{2} \left( \left| \partial_t I \right|^2 - c^2 \left| \nabla I \right|^2 \right) - \frac{\lambda}{4} |I|^4 + J(x,t) I^* + J^*(x,t) IL=21(tI2c2I2)4I4+J(x,t)I+J(x,t)I
where:
I(x,t)CI(x,t) \in \mathbb{C}I(x,t)C: complex-valued field (information excitation),
ccc: effective propagation speed in the medium,
>0\lambda > 0>0: nonlinear self-interaction constant,
J(x,t)J(x,t)J(x,t): external source or blink excitation field.
2. Euler--Lagrange Equation
The Euler--Lagrange equation for a complex scalar field is:
LIt(L(tI))+(L(I))=0\frac{\partial \mathcal{L}}{\partial I^*} - \partial_t \left( \frac{\partial \mathcal{L}}{\partial (\partial_t I^*)} \right) + \nabla \cdot \left( \frac{\partial \mathcal{L}}{\partial (\nabla I^*)} \right) = 0ILt((tI)L)+((I)L)=0
Evaluating the functional derivatives yields:
Ic22I+I2I=J(x,t)\ddot{I} - c^2 \nabla^2 I + \lambda |I|^2 I = J(x,t)Ic22I+I2I=J(x,t)
This is the core governing equation for a self-interacting information field under external blink excitation.
3. Incorporating Dissipation
To introduce dissipative effects (e.g., due to medium friction, decoherence, or information leakage), we phenomenologically add a damping term I\gamma \dot{I}I, leading to:
