Linearizing the Lugiato-Lefever Partial Differential Equation

[BearShark]

Homework Statement

Given the Lugiato-Lefever equation, linearize the equation and determine the dynamics near a stationary solution by looking for a stationary solution with a small perturbation.

Relevant Equations

The Lugiato-Lefever Equation is given by:
$$ \frac {d\psi} {dt} = -(1 + i \alpha)\psi + \frac {i} {2} \frac {d^2\psi} {dx^2} + i * \psi^2 \bar \psi + F $$

It can be shown that when looking for stationary solutions in both time and space, the following is a solution:
$$ \psi_e = \frac F {1 + i * (\alpha - \rho)} $$

When $\rho$ is the solution of the polynomial:
$$ F ^ 2 = (1 + (\rho - \alpha) ^ 2)\rho $$

I started by substituting the following anzatz:
$$ \psi = \psi_e + \psi_1 $$

When $|\psi_1| \ll 1$. Substituting the above into the equation yields:
$$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i (\bar \psi_1 \psi_1 ^2 + \bar \psi_e \psi _1 ^2 + 2\bar\psi_e \psi_e \psi_1 + \bar \psi_1 \psi_e ^2 + 2\bar\psi_1 \psi_e \psi_1) $$

I am interested in a first order treatment, so i will take out the terms quadratic in $\psi_1$:
$$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i ( 2\bar\psi_e \psi_e \psi_1 + \bar \psi_1 \psi_e ^2) $$

I know $\psi_e$ as it is the stationary solution from above, and for $\psi_1$ I can look for solutions of the form:
$$ \psi_1 = e^{i\omega t}*(a * e^{ikx} + \bar b * e^{-ikx})$$

After doing that and equating the coefficients of the backwards and forwards propagating plane waves, I arrived at the following equations:
$$ i\omega \bar b = \bar b (-(1+i\alpha) - \frac i 2 k^2 + i2\bar\psi_e\psi_e) + \bar a (i\bar\psi_1\psi_e^2) $$

$$ i\omega a = a(-(1+i\alpha) - \frac i 2 k^2 + i2\bar\psi_e\psi_e) + b(i\bar\psi_`1\psi_e^2) $$

This is almost what I wanted, but the first equation is on the conjugates of a and b rather than on a and b so I am not sure how to proceed.

 

[mark726]

Hello! I would like to provide some feedback on your work so far. First of all, it is great that you have started by substituting an anzatz to simplify the equation. This is a common approach in physics and can often lead to more manageable equations.

I also appreciate that you are interested in a first order treatment, as this is a good starting point for analyzing a problem. However, it is important to note that taking out terms quadratic in $\psi_1$ may lead to a loss of accuracy in your calculations. It may be worth considering a higher order treatment if possible.

Moving on to your solutions for $\psi_1$, it is important to note that you have assumed a specific form for this function. While this may work for some problems, it may not be applicable to all situations. It is always a good idea to check the validity of your assumptions and make sure they are appropriate for the problem at hand.

Finally, in regards to your equations, it is important to double check your calculations and make sure that the equations are consistent. It seems like there may be a mistake in the first equation, which is why you are getting conjugates of a and b instead of just a and b. I would recommend going back and double checking your work to ensure accuracy.

Overall, your approach is a good starting point, but it is important to be mindful of the assumptions and calculations involved. Keep up the good work!