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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 (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 (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).
For water waves, the nonlinear Schrödinger equation describes the evolution of the
In theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fiber
An optical fiber, or optical fibre, is a flexible glass or plastic fiber that can transmit light from one end to the other. Such fibers find wide usage in fiber-optic communications, where they permit transmission over longer distances and at ...
s, planar waveguides and hot rubidium vapors
and to Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einste ...
s confined to highly anisotropic
Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves
In fluid dynamics, gravity waves are waves in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the oc ...
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 Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
In quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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 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.
Definition
The nonlinear Schrödinger equation is anonlinear partial differential equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear system, nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have b ...
, applicable to classical and quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
Classical equation
The classical field equation (indimensionless
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
form) is:. Originally in: ''Teoreticheskaya i Matematicheskaya Fizika'' 19(3): 332–343. June 1974.
for the complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
field ''ψ''(''x'',''t'').
This equation arises from the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
:Quantum mechanics
To get the quantized version, simply replace the Poisson brackets by commutators : and normal order the Hamiltonian : The quantum version was solved byBethe ansatz
In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenv ...
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.
Solution
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 (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 nsoliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s. This can be achieved via direct numerical simulation using, for example, the split-step method In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivari ...
.
This method has been implemented on both CPU and GPU.
Applications
Fiber optics
Inoptics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, 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
Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems, including optical, radio, atmospheric, and magnetohydrodynamic systems. As a prototype behav ...
, stimulated Raman scattering, optical solitons,
ultrashort pulses, etc.
Water waves
For water waves, the nonlinear Schrödinger equation describes the evolution of the envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card.
Traditional envelopes are made from sheets of paper cut to one o ...
of modulated wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
structure of water waves. In the same paper Zakharov shows that, for slowly modulated wave groups, the wave amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
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
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card.
Traditional envelopes are made from sheets of paper cut to one o ...
soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s may occur. Additionally, the group velocity of these envelope solitons could be increased by an acceleration induced by an external time-dependent water flow.
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 wave A rogue wave is an abnormally large ocean wave.
Rogue wave may also refer to:
* Optical rogue waves, are rare pulses of light analogous to rogue or freak ocean waves.
* Rogue Wave Software, a software company
* Rogue Wave (band), an American in ...
s.
The complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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
In telecommunications, a carrier wave, carrier signal, or just carrier, is a periodic waveform (usually sinusoidal) that conveys information through a process called ''modulation''. One or more of the wave's properties, such as amplitude or freq ...
with water surface elevation
The elevation of a geographic location (geography), ''location'' is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational equipotenti ...
''η'' of the form:
:
where ''a''(''x''0, ''t''0) and ''θ''(''x''0, ''t''0) are the slowly modulated amplitude and phase. Further ''ω''0 and ''k''0 are the (constant) angular frequency
In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
and wavenumber
In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...
of the carrier waves, which have to satisfy the dispersion relation ''ω''0 = Ω(''k''0). Then
:
So its modulus , ''ψ'', is the wave amplitude ''a'', and its argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
arg(''ψ'') is the phase ''θ''.
The relation between the physical coordinates (''x''0, ''t''0) 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
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space.
For example, if a stone is thro ...
Ω'(''k''0) of the carrier waves,
The dispersion-relation curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
Ω"(''k''0) – 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:
: so
where ''g'' is the acceleration due to gravity at the Earth's surface.
In the original (''x''0, ''t''0) coordinates the nonlinear Schrödinger equation for water waves reads:
:
with (i.e. the complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
of ) and So for deep water waves.
Vortices
showed that the work of on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, used this correspondence to show that breather solutions can also arise for a vortex filament.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 : :Gauge equivalent counterpart
NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation : 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 thecurvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of a particular - connection on being equal to zero.
Explicitly, with coordinates on , the connection components are given by
where the are the Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
.
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