To ensure physical realizability and guide experimental implementation across different platforms (spin lattices, magnonic crystals, optical cavities), we perform a dimensional analysis of the governing equation:
I+Ic22I+I2I=B(x,t)\ddot{I} + \gamma \dot{I} - c^2 \nabla^2 I + \lambda |I|^2 I = B(\vec{x}, t)I+Ic22I+I2I=B(x,t)
We define the fundamental dimensions of each parameter:
Non-Dimensionalization
We define dimensionless variables:
x=x/L0\vec{x}' = \vec{x}/L_0x=x/L0
t=t/T0t' = t/T_0t=t/T0
I=I/I0I' = I/I_0I=I/I0
Substituting into the equation and dividing through by the coefficient of I\ddot{I}I, we obtain:
d2Idt2+dIdt2I+I2I=B(x,t)\frac{d^2 I'}{dt'^2} + \Gamma \frac{d I'}{dt'} - \nabla'^2 I' + \Lambda |I'|^2 I' = \mathcal{B}(\vec{x}', t')dt2d2I+dtdI2I+I2I=B(x,t)
Where the dimensionless parameters become:
=T0\Gamma = \gamma T_0=T0
=I02T02\Lambda = \lambda I_0^2 T_0^2=I02T02
B=BL02T02/I0\mathcal{B} = B L_0^2 T_0^2 / I_0B=BL02T02/I0
This enables tuning of the dominant dynamical regime:
Linear regime: 1\Lambda \ll 11
Nonlinear, weakly dissipative: 1\Lambda \sim 11, <1\Gamma < 1<1
Highly dissipative / resonant: 1\Gamma \gg 11
Characteristic Scales
