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.[6]
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.[7]
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). The other case, with κ positive, is the defocusing NLS which has
dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).[9]
Quantum mechanics
To get the
quantized version, simply replace the Poisson brackets by commutators
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.[6] The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.[10]
Solving the equation
The nonlinear Schrödinger equation is integrable in 1d: Zakharov and Shabat (
1972) 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 (different from ϕ) of the Zakharov–Shabat system with spectral parameter Ω:
Starting from the trivial solution U = 0 and iterating, one obtains the solutions with nsolitons.
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[11] and
Fourier spectral[12] methods, are used for its solution. There are different Fortran and C programs for its solution.[13][14]
Galilean invariance
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 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.[15] 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
envelopesolitons may occur. Additionally, the group velocity of these envelope solitons could be increased by an acceleration induced by an external time-dependent water flow.[16]
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.[17]
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
elevationη of the form:
where a(x0, t0) and θ(x0, t0) are the slowly modulated amplitude and
phase. 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.
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:
Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the
Ishimori equation and so on.
Zero-curvature formulation
The NLSE is equivalent to the
curvature of a particular -
connection on being equal to zero.[19]
Explicitly, with coordinates on , the connection components are given by
where the are the
Pauli matrices.
Then the zero-curvature equation
is equivalent to the NLSE . The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined .
The pair of matrices and are also known as a
Lax pair for the NLSE, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.
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.
^
abcdMalomed, Boris (2005), "Nonlinear Schrödinger Equations", in Scott, Alwyn (ed.), Encyclopedia of Nonlinear Science, New York: Routledge, pp. 639–643
^Pitaevskii, L.; Stringari, S. (2003), Bose-Einstein Condensation, Oxford, U.K.: Clarendon
^Gurevich, A. V. (1978), Nonlinear Phenomena in the Ionosphere, Berlin: Springer
^
abKorepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1993). Quantum Inverse Scattering Method and Correlation Functions. Cambridge, U.K.: Cambridge University Press.
ISBN978-0-521-58646-7.
^
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.
S2CID121253212. 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.
S2CID55755251. Originally in: Zhurnal Prikdadnoi Mekhaniki i Tekhnicheskoi Fiziki 9 (2): 86–94, 1968.]