Modulational instability: Difference between revisions

Revision as of 14:00, 3 January 2017

Revision as of 16:57, 3 January 2017

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where the perturbation has been assumed to be small, such that <math>\varepsilon^2\ll P.</math> The [[complex conjugate]] of <math>\varepsilon</math> is denoted as <math>\varepsilon^*.</math> Instability can now be discovered by searching for solutions of the perturbation equation which grow exponentially. This can be done using a trial function of the general form

:<math>\varepsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m^* z + i \omega_m t},</math>

where <math>\omega_mk_m</math> and <math>k_m\omega_m</math> are the [[frequencywavenumber]] and (real-valued) [[wavenumberangular frequency]] of a perturbation, and <math>c_1</math> and <math>c_2</math> are constants. The nonlinear Schrödinger equation is constructed by removing the [[carrier wave]] of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore, <math>\omega_m</math> and <math>k_m</math> don't represent absolute frequencies and wavenumbers, but the ''difference'' between these and those of the initial beam of light. It can be shown that the trial function is valid, subject to the condition

:<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}.</math>