Nonlinear Schrödinger equation

It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides[2] and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime.

[3] Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water;[2] the Langmuir waves in hot plasmas;[2] the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere;[4] the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains;[5] and many others.

More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion.

[2] Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state.

In quantum mechanics, the 1D NLSE is a special case of the classical nonlinear Schrödinger field, which in turn is a classical limit of a quantum Schrödinger field.

Conversely, when the classical Schrödinger field is canonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantum nonlinear Schrödinger equation″) that describes bosonic point particles with delta-function interactions — the particles either repel or attract when they are at the same point.

In fact, when the number of particles is finite, this quantum field theory is equivalent to the Lieb–Liniger model.

This equation arises from the Hamiltonian[9] with the Poisson brackets Unlike its linear counterpart, it never describes the time evolution of a quantum state.

[citation needed] The case with negative κ is called focusing and allows for bright soliton solutions (localized in space, and having spatial attenuation towards infinity) as well as breather solutions.

It can be solved exactly by use of the inverse scattering transform, as shown by Zakharov & Shabat (1972) (see below).

[10] To get the quantized version, simply replace the Poisson brackets by commutators and normal order the Hamiltonian The quantum version was solved by Bethe ansatz by Lieb and Liniger.

[7] The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.

[11] The nonlinear Schrödinger equation is integrable in 1d: Zakharov and Shabat (1972) solved it with the inverse scattering transform.

The equation models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second-harmonic generation, stimulated Raman scattering, optical solitons, ultrashort pulses, etc.

In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian structure of water waves.

[13] The value of the nonlinearity parameter к depends on the relative water depth.

Additionally, the group velocity of these envelope solitons could be increased by an acceleration induced by an external time-dependent water flow.

[14] For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameter к is positive and wave groups with envelope solitons do not exist.

In shallow water surface-elevation solitons or waves of translation do exist, but they are not governed by the nonlinear Schrödinger equation.

The nonlinear Schrödinger equation is thought to be important for explaining the formation of rogue waves.

[15] The complex field ψ, as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves.

Further ω0 and k0 are the (constant) angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ω0 = Ω(k0).

Then So its modulus |ψ| is the wave amplitude a, and its argument arg(ψ) is the phase θ.

The relation between the physical coordinates (x0, t0) and the (x, t) coordinates, as used in the nonlinear Schrödinger equation given above, is given by: Thus (x, t) is a transformed coordinate system moving with the group velocity Ω'(k0) of the carrier waves, The dispersion-relation curvature Ω"(k0) – representing group velocity dispersion – is always negative for water waves under the action of gravity, for any water depth.

For waves on the water surface of deep water, the coefficients of importance for the nonlinear Schrödinger equation are: where g is the acceleration due to gravity at the Earth's surface.

In the original (x0, t0) coordinates the nonlinear Schrödinger equation for water waves reads:[16] with

Hasimoto (1972) showed that the work of da Rios (1906) on vortex filaments is closely related to the nonlinear Schrödinger equation.

Subsequently, Salman (2013) used this correspondence to show that breather solutions can also arise for a vortex filament.

The nonlinear Schrödinger equation is Galilean invariant in the following sense: Given a solution ψ(x, t) a new solution can be obtained by replacing x with x + vt everywhere in ψ(x, t) and by appending a phase factor of

The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined