= Solution
The <optical Kerr effect> gives an intensity-dependent refractive index, $n=n_0+n_2I$, 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 $\phi_{\rm NL}(\mathbf r_\perp)=k_0n_2I(\mathbf r_\perp)L$, 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,
$$
i\frac{\partial A}{\partial z}+\frac1{2k}\nabla_\perp^2A+k_0n_2I A=0,\qquad k=n_0k_0.
$$
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 $\lambda^2/(n_0n_2)$, 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.
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