Nonlinear Schrödinger Equation
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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 natural phenomena. This is in contrast to experimental physics, which uses experi ...
, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
variation of the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. 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 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 densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
s confined to highly anisotropic 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 generated 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 a ...
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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, 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 a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, 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 The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensiona ...
, 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
.


Classical equation

The classical field equation (in
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
form) is:. Originally in: ''Teoreticheskaya i Matematicheskaya Fizika'' 19(3): 332–343. June 1974. for the complex field ''ψ''(''x'',''t''). This equation arises from the Hamiltonian :H=\int \mathrmx \left Poisson_bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
s :<math>\=\=0_\,.html" ;"title="\partial_x\psi, ^2+, \psi, ^4\right] with the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
s :\=\=0 \, :\=i\delta(x-y). \, 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 Bright may refer to: Common meanings *Bright, an adjective meaning giving off or reflecting illumination; see Brightness *Bright, an adjective meaning someone with intelligence People *Bright (surname) *Bright (given name) *Bright, the stage name ...
solutions (localized in space, and having spatial attenuation towards infinity) as well as
breather In physics, a breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards ...
solutions. It can be solved exactly by use of the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
, as shown by (see
below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...
). The other case, with κ positive, is the defocusing NLS which has
dark soliton Modified Mexican hat, Modified Morlet and Dark soliton or Darklet wavelets are derived from hyperbolic (sech) (bright soliton) and hyperbolic tangent (tanh) (dark soliton) pulses. These functions are derived intuitively from the solutions of the no ...
solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).


Quantum mechanics

To get the quantized version, simply replace the Poisson brackets by commutators :\begin psi(x),\psi(y) &= psi^*(x),\psi^*(y)= 0\\ psi^*(x),\psi(y)&= -\delta(x-y) \end and normal order the Hamiltonian :H=\int dx \left partial_x\psi^\dagger\partial_x\psi+\psi^\dagger\psi^\dagger\psi\psi\right The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by
Chen-Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge the ...
. 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 In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
. The corresponding linear system of equations is known as the Zakharov–Shabat system: : \begin \phi_x &= J\phi\Lambda + U\phi \\ \phi_t &= 2J\phi\Lambda^2 + 2U\phi\Lambda + \left(JU^2 - JU_x\right)\phi, \end where : \Lambda = \begin \lambda_1&0\\ 0&\lambda_2 \end, \quad J = i\sigma_z = \begin i & 0 \\ 0 & -i \end, \quad U = i \begin 0 & q \\ r & 0 \end. The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system: : \phi_ = \phi_ \quad \Rightarrow \quad U_t = -JU_ + 2JU^2 U \quad \Leftrightarrow \quad \begin iq_t = q_ + 2qrq \\ ir_t = -r_ - 2qrr. \end 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: : \begin \phi \to \phi &= \phi\Lambda - \sigma\phi \\ U \to U &= U + , \sigma\\ \sigma &= \varphi\Omega\varphi^ \end which leaves the system invariant. Here, ''φ'' is another invertible matrix solution (different from ''ϕ'') of the Zakharov–Shabat system with spectral parameter Ω: : \begin \varphi_x &= J\varphi\Omega + U\varphi \\ \varphi_t &= 2J\varphi\Omega^2 + 2U\varphi\Omega + \left(JU^2 - JU_x\right)\varphi. \end Starting from the trivial solution ''U'' = 0 and iterating, one obtains the solutions with ''n''
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
s. 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 Fourier may refer to: People named Fourier *Joseph Fourier (1768–1830), French mathematician and physicist *Charles Fourier (1772–1837), French utopian socialist thinker * Peter Fourier (1565–1640), French saint in the Roman Catholic Church ...
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''(''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 e^\,: :\psi(x,t) \mapsto \psi_(x,t)=\psi(x+vt,t)\; e^.


The nonlinear Schrödinger equation in fiber optics

In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
, the nonlinear Schrödinger equation occurs in the
Manakov system Maxwell's Equations, when converted to cylindrical coordinates, and with the boundary conditions for an optical fiber while including birefringence as an effect taken into account, will yield the coupled nonlinear Schrödinger equations. After emp ...
, 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 Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation ...
, four-wave mixing,
second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
, stimulated Raman scattering,
optical solitons In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons: * spatial solitons: ...
, ultrashort pulses, etc.


The nonlinear Schrödinger equation in water waves

For
water waves In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
, 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 or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
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 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 a ...
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 or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
s may occur. Furthermore, these envelope solitons may be accelerated under 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 Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s. 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 In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
with water surface
elevation The elevation of a geographic 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 surface (see Geodetic datum § ...
''η'' of the form: : \eta = a(x_0,t_0)\; \cos \left k_0\, x_0 - \omega_0\, t_0 - \theta(x_0,t_0) \right 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 "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
and
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the carrier waves, which have to satisfy the dispersion relation ''ω''0 = Ω(''k''0). Then : \psi = a\; \exp \left( i \theta \right). So its modulus , ''ψ'', is the wave amplitude ''a'', and its
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
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: : x = k_0 \left x_0 - \Omega'(k_0)\; t_0 \right \quad t = k_0^2 \left -\Omega''(k_0) \right; t_0 Thus (''x, t'') is a transformed coordinate system moving with the group velocity Ω'(''k''0) of the carrier waves, The dispersion-relation
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
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: :\kappa = - 2 k_0^2, \quad \Omega(k_0) = \sqrt = \omega_0 \,\! so \Omega'(k_0) = \frac \frac, \quad \Omega''(k_0) = -\frac \frac, \,\! 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: :i\, \partial_ A + i\, \Omega'(k_0)\, \partial_ A + \tfrac12 \Omega''(k_0)\, \partial_ A - \nu\, , A, ^2\, A = 0, with A=\psi^* (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, then) the complex conjugate of a + bi is equal to a - ...
of \psi) and \nu=\kappa\, k_0^2\, \Omega''(k_0). So \nu = \tfrac12 \omega_0 k_0^2 for deep water waves.


Gauge equivalent counterpart

NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation :\vec_t=\vec\wedge \vec_. \qquad 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

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.


See also

*
AKNS system In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: . ...
*
Eckhaus equation In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various ...
* Quartic interaction for a related model in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
* Soliton (optics) *
Logarithmic Schrödinger equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum m ...


Notes


References


Notes


Other

* * * * *


External links

*
Tutorial lecture on Nonlinear Schrodinger Equation (video)

Nonlinear Schrodinger Equation with a Cubic Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation with a Power-Law Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation of General Form
at EqWorld: The World of Mathematical Equations.
Mathematical aspects of the nonlinear Schrödinger equation
at Dispersive Wiki {{DEFAULTSORT:Nonlinear Schrodinger equation Partial differential equations Exactly solvable models Schrödinger equation Integrable systems>\partial_x\psi, ^2+, \psi, ^4\right/math> with the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
s :\=\=0 \, :\=i\delta(x-y). \, 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 Bright may refer to: Common meanings *Bright, an adjective meaning giving off or reflecting illumination; see Brightness *Bright, an adjective meaning someone with intelligence People *Bright (surname) *Bright (given name) *Bright, the stage name ...
solutions (localized in space, and having spatial attenuation towards infinity) as well as
breather In physics, a breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards ...
solutions. It can be solved exactly by use of the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
, as shown by (see
below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...
). The other case, with κ positive, is the defocusing NLS which has
dark soliton Modified Mexican hat, Modified Morlet and Dark soliton or Darklet wavelets are derived from hyperbolic (sech) (bright soliton) and hyperbolic tangent (tanh) (dark soliton) pulses. These functions are derived intuitively from the solutions of the no ...
solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).


Quantum mechanics

To get the quantized version, simply replace the Poisson brackets by commutators :\begin psi(x),\psi(y) &= psi^*(x),\psi^*(y)= 0\\ psi^*(x),\psi(y)&= -\delta(x-y) \end and normal order the Hamiltonian :H=\int dx \left partial_x\psi^\dagger\partial_x\psi+\psi^\dagger\psi^\dagger\psi\psi\right The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by
Chen-Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge the ...
. 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 In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
. The corresponding linear system of equations is known as the Zakharov–Shabat system: : \begin \phi_x &= J\phi\Lambda + U\phi \\ \phi_t &= 2J\phi\Lambda^2 + 2U\phi\Lambda + \left(JU^2 - JU_x\right)\phi, \end where : \Lambda = \begin \lambda_1&0\\ 0&\lambda_2 \end, \quad J = i\sigma_z = \begin i & 0 \\ 0 & -i \end, \quad U = i \begin 0 & q \\ r & 0 \end. The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system: : \phi_ = \phi_ \quad \Rightarrow \quad U_t = -JU_ + 2JU^2 U \quad \Leftrightarrow \quad \begin iq_t = q_ + 2qrq \\ ir_t = -r_ - 2qrr. \end 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: : \begin \phi \to \phi &= \phi\Lambda - \sigma\phi \\ U \to U &= U + , \sigma\\ \sigma &= \varphi\Omega\varphi^ \end which leaves the system invariant. Here, ''φ'' is another invertible matrix solution (different from ''ϕ'') of the Zakharov–Shabat system with spectral parameter Ω: : \begin \varphi_x &= J\varphi\Omega + U\varphi \\ \varphi_t &= 2J\varphi\Omega^2 + 2U\varphi\Omega + \left(JU^2 - JU_x\right)\varphi. \end Starting from the trivial solution ''U'' = 0 and iterating, one obtains the solutions with ''n''
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
s. 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 Fourier may refer to: People named Fourier *Joseph Fourier (1768–1830), French mathematician and physicist *Charles Fourier (1772–1837), French utopian socialist thinker * Peter Fourier (1565–1640), French saint in the Roman Catholic Church ...
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''(''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 e^\,: :\psi(x,t) \mapsto \psi_(x,t)=\psi(x+vt,t)\; e^.


The nonlinear Schrödinger equation in fiber optics

In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
, the nonlinear Schrödinger equation occurs in the
Manakov system Maxwell's Equations, when converted to cylindrical coordinates, and with the boundary conditions for an optical fiber while including birefringence as an effect taken into account, will yield the coupled nonlinear Schrödinger equations. After emp ...
, 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 Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation ...
, four-wave mixing,
second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
, stimulated Raman scattering,
optical solitons In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons: * spatial solitons: ...
, ultrashort pulses, etc.


The nonlinear Schrödinger equation in water waves

For
water waves In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
, 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 or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
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 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 a ...
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 or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
s may occur. Furthermore, these envelope solitons may be accelerated under 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 Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s. 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 In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
with water surface
elevation The elevation of a geographic 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 surface (see Geodetic datum § ...
''η'' of the form: : \eta = a(x_0,t_0)\; \cos \left k_0\, x_0 - \omega_0\, t_0 - \theta(x_0,t_0) \right 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 "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
and
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the carrier waves, which have to satisfy the dispersion relation ''ω''0 = Ω(''k''0). Then : \psi = a\; \exp \left( i \theta \right). So its modulus , ''ψ'', is the wave amplitude ''a'', and its
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
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: : x = k_0 \left x_0 - \Omega'(k_0)\; t_0 \right \quad t = k_0^2 \left -\Omega''(k_0) \right; t_0 Thus (''x, t'') is a transformed coordinate system moving with the group velocity Ω'(''k''0) of the carrier waves, The dispersion-relation
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
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: :\kappa = - 2 k_0^2, \quad \Omega(k_0) = \sqrt = \omega_0 \,\! so \Omega'(k_0) = \frac \frac, \quad \Omega''(k_0) = -\frac \frac, \,\! 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: :i\, \partial_ A + i\, \Omega'(k_0)\, \partial_ A + \tfrac12 \Omega''(k_0)\, \partial_ A - \nu\, , A, ^2\, A = 0, with A=\psi^* (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, then) the complex conjugate of a + bi is equal to a - ...
of \psi) and \nu=\kappa\, k_0^2\, \Omega''(k_0). So \nu = \tfrac12 \omega_0 k_0^2 for deep water waves.


Gauge equivalent counterpart

NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation :\vec_t=\vec\wedge \vec_. \qquad 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

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.


See also

*
AKNS system In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: . ...
*
Eckhaus equation In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various ...
* Quartic interaction for a related model in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
* Soliton (optics) *
Logarithmic Schrödinger equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum m ...


Notes


References


Notes


Other

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External links

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Tutorial lecture on Nonlinear Schrodinger Equation (video)

Nonlinear Schrodinger Equation with a Cubic Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation with a Power-Law Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation of General Form
at EqWorld: The World of Mathematical Equations.
Mathematical aspects of the nonlinear Schrödinger equation
at Dispersive Wiki {{DEFAULTSORT:Nonlinear Schrodinger equation Partial differential equations Exactly solvable models Schrödinger equation Integrable systems