the nonlinear Schrodinger equation
/ SHROH-ding-er /
A pulse of light fired down an optical fibre faces the same dilemma as a water wave: dispersion wants to smear it out (different colours travel at slightly different speeds), while a nonlinear effect of the medium wants to focus it. When these balance, the pulse holds its shape over thousands of kilometres — an optical soliton, the backbone of long-distance fibre communication. The nonlinear Schrodinger equation is the universal equation governing this balance for slowly-modulated wave packets.
The equation is i u_t + u_xx + |u|^2 u = 0 (the 'focusing' cubic NLS; with a minus sign it is 'defocusing'). Here u(x, t) is complex — it is the slowly-varying envelope of an underlying oscillation, so |u|^2 is the intensity. The i u_t marks it as Schrodinger-like (the same i d/dt as quantum mechanics); u_xx is dispersion; and the cubic term |u|^2 u is the nonlinearity, which depends on the local intensity. In the focusing case the nonlinearity pulls energy toward high-intensity regions, opposing dispersion's spreading; the balance gives a bright soliton u(x,t) = A sech(A(x)) e^(i...), a self-trapped pulse. Like KdV, the one-dimensional cubic NLS is INTEGRABLE and solvable by inverse scattering, so its solitons collide cleanly. But its richer structure shows up in higher dimensions: the focusing NLS can BLOW UP in finite time (an intensity spike collapses to a point) in two or more dimensions — a model for self-focusing collapse of a laser beam.
Why so central? NLS is the canonical model for the slow modulation of ANY dispersive wave with weak nonlinearity, so it turns up in optics, water waves (rogue waves), plasma physics, and — as the Gross-Pitaevskii equation — Bose-Einstein condensates. It sits beside KdV as one of the two great integrable solitons equations, and it teaches a key contrast: defocusing NLS is globally well-behaved, but focusing NLS in high dimension can blow up. The same equation, one sign apart, lives on opposite sides of the existence-vs-blow-up divide — a vivid lesson in how delicate nonlinear PDEs are.
In one dimension the focusing NLS has the bright soliton u(x, t) = A sech(A x) e^(i A^2 t / 2): a self-trapped pulse whose width shrinks as its amplitude A grows. In two dimensions the same focusing nonlinearity instead drives self-focusing collapse — an initially smooth, intense beam can concentrate into a point and blow up in finite time, exactly the catastrophic self-focusing that can damage optical materials.
Dispersion vs. cubic nonlinearity: a soliton in 1D, possible collapse in 2D.
Sign and dimension both matter. Defocusing NLS (minus sign) is globally well-posed and has no blow-up; focusing NLS (plus sign) has bright solitons in 1D but can blow up in 2D and higher. Don't quote 'NLS' without saying which.