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In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) 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 (zero-viscosity) 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 (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. 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 (also called the hard-core Bose gas, or impenetrable Bose 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[nb 1] 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 R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) 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.
Unlike its linear counterpart, it never describes the time evolution of a quantum state.
The case with negative ? 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). The other case, with ? positive, is the defocusing NLS which has dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).
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.
A hyperbolic secant (sech) envelope soliton for surface waves on deep water. Blue line: water waves. Red line: envelope soliton.
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 ? depends on the relative water depth. For deep water, with the water depth large compared to the wave length of the water waves, ? is negative and envelopesolitons may occur.
For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameter ? 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 ?, 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? of the form:
where a(x0, t0) and ?(x0, t0) are the slowly modulated amplitude and phase. Further ?0 and k0 are the (constant) angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ?0 = ?(k0). Then
So its modulus |?| is the wave amplitude a, and its argument arg(?) is the phase ?.
Thus (x, t) is a transformed coordinate system moving with the group velocity ?'(k0) of the carrier waves,
The dispersion-relation curvature ?"(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:
Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation and so on.
Relation to vortices
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.
^A possible source of confusion here is the spin-statistics theorem, which demands that fermions have half-integer spin; however, it is a theorem of relativistic 3+1-dimensional quantum field theories, and thus is not applicable in this 1D, nonrelativistic case.
^ abV.E. Zakharov; S.V. Manakov (1974). "On the complete integrability of a nonlinear Schrödinger equation". Journal of Theoretical and Mathematical Physics. 19 (3): 551-559. Bibcode:1974TMP....19..551Z. doi:10.1007/BF01035568. Originally in: Teoreticheskaya i Matematicheskaya Fizika19(3): 332-343. June 1974.
^Ablowitz, M.J. (2011), Nonlinear dispersive waves. Asymptotic analysis and solitons, Cambridge University Press, pp. 152-156, ISBN978-1-107-01254-7
^V. E. Zakharov (1968). "Stability of periodic waves of finite amplitude on the surface of a deep fluid". Journal of Applied Mechanics and Technical Physics. 9 (2): 190-194. Bibcode:1968JAMTP...9..190Z. doi:10.1007/BF00913182. Originally in: Zhurnal Prikdadnoi Mekhaniki i Tekhnicheskoi Fiziki 9 (2): 86-94, 1968.]