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remove incorrect statement: recurrence often occurs |
→Mathematical derivation of gain spectrum: c_2=c_1^* |
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:<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>k_m</math> and <math>\omega_m</math> are the [[wavenumber]] and (real-valued) [[angular 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, provided <math>c_2=c_1^*</math> and subject to the condition
:<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}.</math>
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