Nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; 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. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
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 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. Both the quantum and the classical 1D nonlinear Schrödinger equations are integrable. Of special interest is the limit of infinite strength repulsion, in which case the Lieb–Liniger model becomes the Tonks–Girardeau gas. In this limit, the bosons may, by a change of variables that is a continuum generalization of the Jordan–Wigner transformation, be transformed to a system one-dimensional noninteracting spinless fermions.
The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by in their study of optical beams.
Multi-dimensional version replaces the second spatial derivative by the Laplacian. In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.
Equation
The nonlinear Schrödinger equation is a nonlinear partial differential equation, applicable to classical and quantum mechanics.Classical equation
The classical field equation is:for the complex field ψ.
This equation arises from the Hamiltonian
with the Poisson brackets
Unlike its linear counterpart, it never describes the time evolution of a quantum state.
The case with negative κ is called focusing and allows for bright soliton solutions as well as breather solutions. It can be solved exactly by use of the inverse scattering transform, as shown by . The other case, with κ positive, is the defocusing NLS which has dark soliton solutions.
Quantum mechanics
To get the quantized version, simply replace the Poisson brackets by commutatorsand normal order the Hamiltonian
The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by Chen-Ning Yang. Quantum correlation functions also were evaluated by Korepin in 1993. The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.
Solving the equation
The nonlinear Schrödinger equation is integrable in 1d: solved it with the inverse scattering transform. The corresponding linear system of equations is known as the Zakharov–Shabat system:where
The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system:
By setting q = r* or q = − r* the nonlinear Schrödinger equation with attractive or repulsive interaction is obtained.
An alternative approach uses the Zakharov–Shabat system directly and employs the following Darboux transformation:
which leaves the system invariant.
Here, φ is another invertible matrix solution of the Zakharov–Shabat system with spectral parameter Ω:
Starting from the trivial solution U = 0 and iterating, one obtains the solutions with n solitons.
The NLS equation is a partial differential equation like the Gross–Pitaevskii equation. Usually it does not have analytic solution and the same numerical methods used to solve the Gross–Pitaevskii equation, such as the split-step Crank–Nicolson and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution.
Galilean invariance
The nonlinear Schrödinger equation is Galilean invariant in the following sense:Given a solution ψ a new solution can be obtained by replacing x with x + vt everywhere in ψ and by appending a phase factor of :
The nonlinear Schrödinger equation in fiber optics
In optics, the nonlinear Schrödinger equation occurs in the Manakov system, a model of wave propagation in fiber optics. The function ψ represents a wave and the nonlinear Schrödinger equation describes the propagation of the wave through a nonlinear medium. The second-order derivative represents the dispersion, while the κ term represents the nonlinearity. 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.
The nonlinear Schrödinger equation in water waves
For water waves, the nonlinear Schrödinger equation describes the evolution of the envelope of modulated wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian structure of water waves. In the same paper Zakharov shows, that for slowly modulated wave groups, the wave amplitude satisfies the nonlinear Schrödinger equation, approximately. The value of the nonlinearity parameter к depends on the relative water depth. For deep water, with the water depth large compared to the wave length of the water waves, к is negative and envelope solitons may occur.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.
The complex field ψ, as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves. Consider a slowly modulated carrier wave with water surface elevation η of the form:
where a and θ are the slowly modulated amplitude and phase. Further ω0 and k0 are the angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ω0 = Ω. Then
So its modulus |ψ| is the wave amplitude a, and its argument arg is the phase θ.
The relation between the physical coordinates and the coordinates, as used in the [|nonlinear Schrödinger equation given above], is given by:
Thus is a transformed coordinate system moving with the group velocity Ω' of the carrier waves,
The dispersion-relation curvature Ω" – 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 coordinates the nonlinear Schrödinger equation for water waves reads:
with and So for deep water waves.
Gauge equivalent counterpart
NLSE is gauge equivalent to the following isotropic Landau-Lifshitz equation or Heisenberg ferromagnet equationNote that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation and so on.