Nonlinear Schrödinger equation

The nonlinear Schrödinger equation

i\epsilon u_t+\epsilon^2u_{xx}+\rho |u|^2 u=0, \quad\epsilon\ll1

appears in many applications in the water waves or nonlinear optics when the modulational dependence is studied. This means that one is interested in the modulation of a plane wave with complex amplitude u in a given setting.

The equation is focusing for 𝜌 = 1 and defocusing for 𝜌 = -1. The limit 𝜀 → 0 is known as the semiclassical limit. A dispersive shock wave can appear in this case.

The first video (above) shows the solution to the defocusing NLS equation for the initial data

u(x,0)=exp(-x^2)

for 𝜀 = 0.5.

The video shows the solution to the focusing NLS equation for the initial data

u(x,0)=exp(-x^2)

for 𝜀 = 0.1.

For details and references, see 0704.0501.