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For the complex Ginzburg–Landau equation $A_t=A+(1+i{c}_1)A_{xx}-(1+i{c}_3)|A{|}^{2}A$, the uniform oscillation has a neutral phase mode whose growth rate is $\lambda =-(1+c_1{c}_3)k^2+O(k^4)$. Negative phase diffusion therefore causes a long-wavelength modulational instability. The Benjamin-Feir stability condition is the complementary positive-diffusion condition. On a finite domain an unstable wavenumber must actually be allowed by the boundary conditions.

For $A_t=A+(1+ib)A_{xx}-(1+ic)|A{|}^{2}A$, the uniform oscillation $A=e^{-ict}$ has phase diffusion coefficient $D=1+bc$. Its amplitude-phase dispersion relation is ${\lambda }^2+(2+2k^2)\lambda +2D{k}^2+(1+b^2)k^4=0$. Thus $D<0$ gives a long-wavelength modulational instability, whereas $D>0$ gives linear stability of all nonzero Fourier modes. On a fixed periodic domain, instability requires an allowed wavenumber with $k^2<-2D/(1+b^2)$.

For $-i{A}_t+\chi A_{xx}-\mu A+\delta A|A{|}^2=0$ with positive coefficients, linearize around $A=\sqrt{\mu /\delta }$ using real amplitude and phase perturbations. Their two-by-two dispersion relation gives the displayed equation. The modulational instability band is $0<K^2<2\mu /\chi$, with maximum growth rate $\mu$ at $K^2=\mu /\chi$.

Write $r=\sqrt{\mu /\delta }$. At first order the amplitude-phase perturbation is $A=r+\epsilon (A_1+ir\theta )+O({\epsilon }^2)$. Expanding the cubic term in the focusing nonlinear Schrodinger equation gives $A|A{|}^2=r^3+\epsilon (3r^2{A}_1+i{r}^3\theta )+O({\epsilon }^2)$. The constant terms cancel because $\delta r^2=\mu$. Separating real and imaginary parts yieldsFor the Fourier normal modes, these equations give $-i\Omega {\hat{A}}_1=-\chi r{K}^2\hat{\theta }$ and $-ir\Omega \hat{\theta }+(2\mu -\chi K^2){\hat{A}}_1=0$. Their determinant vanishes exactly whenThe focusing nonlinear Schrodinger modulation dispersion is negative for $0<K^2<2\mu /\chi$. One of the two frequencies then has positive imaginary part and grows exponentially: the constant state has modulational instability. The growth rate is $\sqrt{\chi K^2(2\mu -\chi K^2)}$, maximized at $K^2=\mu /\chi$ with value $\mu$. Modes beyond the unstable band have real frequencies. The zero-wavenumber phase mode is neutral, with possible linear phase drift from a uniform amplitude perturbation.

The optical Kerr effect gives an intensity-dependent refractive index, $n=n_0+n_{2}I$, in the simplest instantaneous isotropic model. A beam whose intensity varies transversely therefore writes a transverse index profile into the medium. Over length $L$ its nonlinear phase is approximately ${\varphi }_{\mathrm{NL}}({\mathbf{r}}_{\perp })=k_0{n}_{2}I({\mathbf{r}}_{\perp })L$, with $k_0=\omega /c$.

For a bright-centered beam, positive $n_2$ raises the index at the center and produces self-focusing; negative $n_2$ produces self-defocusing. The medium acts as an intensity-dependent lens. This is a transverse effect, distinct from temporal self-phase modulation broadening a pulse's spectrum, though both originate in the same nonlinear index. Kerr lenses can provide intensity-dependent mode selection in a laser, and a beam's index profile can guide or deflect a second beam through cross-phase modulation.

The competition with diffraction appears in a paraxial model,In one transverse dimension, focusing can balance diffraction to support a bright spatial optical soliton under suitable boundary conditions. Transverse perturbations can also grow through modulational instability, breaking a broad beam into localized structures. In an ideal two-transverse-dimensional cubic model, sufficiently strong focusing permits collapse; the critical-power scale is proportional to ${\lambda }^2/(n_0{n}_2)$, with its numerical coefficient depending on the convention and profile used.

A real medium does not develop an actual infinite intensity. Saturation, multiphoton absorption, ionization, nonparaxial propagation and damage can arrest or invalidate the simple model. Filament formation therefore requires more physics than inserting a real constant $n_2$ into a lens formula. The useful applications exploit a controllable index profile; the same positive feedback can be a serious limitation on transporting high-power beams.