I+Ic22I+I2I=B(x,t)\boxed{ \ddot{I} + \gamma \dot{I} - c^2 \nabla^2 I + \lambda |I|^2 I = B(x,t) }I+Ic22I+I2I=B(x,t)
Where:
\gamma: damping coefficient,
B(x,t)B(x,t)B(x,t): source term, analogous to the Blink impulse or sustained excitation.
4. Normalization and Scaling Analysis
Let:
x=x0x~x = x_0 \tilde{x}x=x0x~,t=t0t~t = t_0 \tilde{t}t=t0t~,I=I0I~I = I_0 \tilde{I}I=I0I~,
Then the governing equation becomes (after substitution and simplification):
I0t02d2I~dt~2+I0t0dI~dt~c2I0x022I~+I03I~2I~=B(x,t)\frac{I_0}{t_0^2} \frac{d^2 \tilde{I}}{d\tilde{t}^2} + \frac{\gamma I_0}{t_0} \frac{d \tilde{I}}{d\tilde{t}} - \frac{c^2 I_0}{x_0^2} \nabla^2 \tilde{I} + \lambda I_0^3 |\tilde{I}|^2 \tilde{I} = B(x,t)t02I0dt~2d2I~+t0I0dt~dI~x02c2I02I~+I03I~2I~=B(x,t)
Choosing characteristic scales such that:
I0t02=c2I0x02x0t0=c\frac{I_0}{t_0^2} = \frac{c^2 I_0}{x_0^2} \Rightarrow \frac{x_0}{t_0} = ct02I0=x02c2I0t0x0=c
And defining normalized dimensionless parameters:
~=t0,~=I02t02\tilde{\gamma} = \gamma t_0,\quad \tilde{\lambda} = \lambda I_0^2 t_0^2~=t0,~=I02t02
We obtain the dimensionless equation:
