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The Nonlinear Schrödinger Equation (NSE) is presented as:
$$i\frac{∂A}{∂z} = \frac{1}{2}β_2\frac{∂^2A}{∂t^2}-\gamma|A^2|A$$
The steady state solution
$$A(z)$$
Can be derived as an Ansatz given by:
$$ A(z) = \rho(z)e^{i\phi(z)}$$
By substituting and solving the ODE, the steady state solution results in:
$$A(z)=A_0e^{i\gamma {A_0}^2 z}$$
The quest at hand seeks to determine if the solution is stable by introducing a perturbation such that time dependent solution is:
$$A(z,t) = (A_0 + \epsilon(z,t))e^{i\gamma {A_0}^2 z}$$
By pluggin it back into the NSE and retaining only the linear terms in the perturbation, the linearized equation results in:
$$i\frac{∂\epsilon}{∂z} = \frac{1}{2}β_2\frac{∂^2\epsilon}{∂t^2}-\gamma{A_0}^2(\epsilon + \epsilon^*)$$
Supposing that the solution is of the form:
$$\epsilon(z,t) = \epsilon_1cos(\kappa z - \omega t) + i\epsilon_2sin(\kappa z - \omega t) $$
A side goal of this homework task is to also determine the dispersion relation that I can get by substituting the supposed solution. But here I ran into a problem: my attempt at determining the dispersion relation gave me the subsequent expression
$$ -i\kappa \epsilon_1sin(\kappa z - \omega t) -\kappa \epsilon_2cos(\kappa z - \omega t) + \frac{1}{2} \epsilon_1 \omega^2 \beta_2cos(\kappa z - \omega t) + i\frac{1}{2} \epsilon_2 \omega^2 \beta_2sin(\kappa z - \omega t) + 2 \epsilon_1 \gamma {A_0}^2cos(\kappa z - \omega t) = 0 $$
Supposedly, the solution is right but the dispersion relation gives me distinct expressions:
$$ \kappa = \frac{\epsilon_2}{2\epsilon_1} \omega^2 \beta_2 $$
And also,
$$ \kappa = \frac{\epsilon_1}{2\epsilon_2} \omega^2 \beta_2 + 2\frac{\epsilon_1}{\epsilon_2} \gamma {A_0}^2 $$
Moreover, the wavevector is to be complex because the next question in my homework asks be to calculated the gain in power given by:
$$ g(\omega) = 2Im(\kappa) $$
I have checked the math of the linear equation I've obtained and I don't seem to I have missed anything but it is surely strange to have different real expressions for the wavevector!
Can anyone help me figure this out?
Thanks in advance!
$$i\frac{∂A}{∂z} = \frac{1}{2}β_2\frac{∂^2A}{∂t^2}-\gamma|A^2|A$$
The steady state solution
$$A(z)$$
Can be derived as an Ansatz given by:
$$ A(z) = \rho(z)e^{i\phi(z)}$$
By substituting and solving the ODE, the steady state solution results in:
$$A(z)=A_0e^{i\gamma {A_0}^2 z}$$
The quest at hand seeks to determine if the solution is stable by introducing a perturbation such that time dependent solution is:
$$A(z,t) = (A_0 + \epsilon(z,t))e^{i\gamma {A_0}^2 z}$$
By pluggin it back into the NSE and retaining only the linear terms in the perturbation, the linearized equation results in:
$$i\frac{∂\epsilon}{∂z} = \frac{1}{2}β_2\frac{∂^2\epsilon}{∂t^2}-\gamma{A_0}^2(\epsilon + \epsilon^*)$$
Supposing that the solution is of the form:
$$\epsilon(z,t) = \epsilon_1cos(\kappa z - \omega t) + i\epsilon_2sin(\kappa z - \omega t) $$
A side goal of this homework task is to also determine the dispersion relation that I can get by substituting the supposed solution. But here I ran into a problem: my attempt at determining the dispersion relation gave me the subsequent expression
$$ -i\kappa \epsilon_1sin(\kappa z - \omega t) -\kappa \epsilon_2cos(\kappa z - \omega t) + \frac{1}{2} \epsilon_1 \omega^2 \beta_2cos(\kappa z - \omega t) + i\frac{1}{2} \epsilon_2 \omega^2 \beta_2sin(\kappa z - \omega t) + 2 \epsilon_1 \gamma {A_0}^2cos(\kappa z - \omega t) = 0 $$
Supposedly, the solution is right but the dispersion relation gives me distinct expressions:
$$ \kappa = \frac{\epsilon_2}{2\epsilon_1} \omega^2 \beta_2 $$
And also,
$$ \kappa = \frac{\epsilon_1}{2\epsilon_2} \omega^2 \beta_2 + 2\frac{\epsilon_1}{\epsilon_2} \gamma {A_0}^2 $$
Moreover, the wavevector is to be complex because the next question in my homework asks be to calculated the gain in power given by:
$$ g(\omega) = 2Im(\kappa) $$
I have checked the math of the linear equation I've obtained and I don't seem to I have missed anything but it is surely strange to have different real expressions for the wavevector!
Can anyone help me figure this out?
Thanks in advance!