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Nonlinear Schrodinger Equation Dispersion Relation

  1. May 17, 2017 #1
    The Nonlinear Schrodinger 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!
     
  2. jcsd
  3. May 19, 2017 #2
    Your expression of the NSD is incorrect. The Schrodinger equation relates the first derivative in time of the wave function to its second derivative in space.
     
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