View source for Nonlinear Schrödinger equation

{{dablink|1=This page is about the equation i&psi;<sub>t</sub> = −½&psi;<sub>xx</sub> + &kappa;{{!}}&psi;{{!}}<sup>2</sup>&psi;. For the equation i&psi;<sub>t</sub> = −½&psi;<sub>xx</sub> + V(x)&psi; + &kappa;{{!}}&psi;{{!}}<sup>2</sup>&psi;, used in Bose–Einstein condensate theory, see [[Gross–Pitaevskii equation]].}}

[[File:NLS breathers - Akhmediev Peregrine and Kuznetsov-Ma breather.jpg|thumb|right|[[Absolute value]] of the [[complex number|complex]] [[envelope (waves)|envelope]] of exact analytical [[breather]] solutions of the nonlinear Schrödinger (NLS) equation in [[nondimensionalization|nondimensional]] form. (A) The Akhmediev breather; (B) the [[Peregrine soliton|Peregrine breather]]; (C) the Kuznetsov–Ma breather.<ref>Figure 1 from: {{citation |first1=M. |last1=Onorato |first2=D. |last2=Proment |first3=G. |last3=Clauss |author3-link=Günther Clauss |first4=M. |last4=Klein |year=2013 |title=Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test |journal=PLOS ONE |volume=8 |issue=2 |pages=e54629 |nopp=yes |doi=10.1371/journal.pone.0054629 |pmc=3566097|bibcode = 2013PLoSO...854629O |pmid=23405086}}</ref>]]

In [[theoretical physics]], the (one-dimensional) '''nonlinear Schrödinger equation''' ('''NLSE''') is a [[nonlinear partial differential equation|nonlinear]] variation of the [[Schrödinger equation]].  It is a [[Classical field theory|classical field equation]] whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides<ref name=Enc_Nonlin_Sci>{{Citation
| last  = Malomed
| first = Boris
| contribution = Nonlinear Schrödinger Equations
| editor-last  = Scott
| editor-first = Alwyn
| title = Encyclopedia of Nonlinear Science
| pages  = 639–643
| publisher = Routledge
| place = New York
| date = 2005
}}</ref> and to [[Bose–Einstein condensate|Bose-Einstein condensates]] confined to highly anisotropic cigar-shaped traps, in the mean-field regime.<ref name=Pitaevskii_and_Stringari>{{Citation
| last1  = Pitaevskii
| first1 = L.
| last2  = Stringari
| first2 = S.
| title     = Bose-Einstein Condensation 
| publisher = Clarendon
| place     = Oxford, U.K.
| year    = 2003
}}</ref> Additionally, the equation appears in the studies of small-amplitude [[gravity waves]] on the surface of deep inviscid (zero-viscosity) water;<ref name="Enc_Nonlin_Sci" /> the [[Langmuir waves]] in hot plasmas;<ref name="Enc_Nonlin_Sci" /> the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere;<ref name=Gurevich>{{Citation
| last1  = Gurevich
| first1 = A. V.
| title     = Nonlinear Phenomena in the Ionosphere
| publisher = Springer
| place     = Berlin
| year    = 1978
}}</ref> the propagation of [[Davydov soliton|Davydov's alpha-helix solitons]], which are responsible for energy transport along molecular chains;<ref name="Balakrishnan_1985">{{Cite journal | last1 = Balakrishnan | first1 = R. | title = Soliton propagation in nonuniform media | doi = 10.1103/PhysRevA.32.1144 | journal = Physical Review A | volume = 32 | issue = 2 | pages = 1144–1149 | year = 1985 | pmid =  9896172| pmc = |bibcode = 1985PhRvA..32.1144B }}</ref> 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 relation|dispersion]].<ref name="Enc_Nonlin_Sci" /> 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 [[second quantization|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{{refn|group=nb|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.}} fermions.<ref name="Korepin_1993">{{cite book |last1= Korepin |first1= V. E. | last2= Bogoliubov |first2= N. M. | last3= Izergin |first3= A. G. | title= Quantum Inverse Scattering Method and Correlation Functions |publisher= Cambridge University Press |year= 1993 | place = Cambridge, U.K. |isbn=978-0-521-58646-7 |doi=10.2277/0521586461 }}</ref>

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 {{harvs|txt|year=1964|first1=R. Y. |last1=Chiao|first2= E.|last2= Garmire|first3= C. H. |last3=Townes|loc=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.<ref name=Falkovich>{{cite book | author=G. Falkovich| year=2011 | title=Fluid Mechanics (A short course for physicists)|url=http://www.cambridge.org/gb/knowledge/isbn/item6173728/?site_locale=en_GB | publisher=Cambridge University Press | isbn=978-1-107-00575-4 }}</ref>

== Equation ==
The nonlinear Schrödinger equation is a [[nonlinear partial differential equation]], applicable to [[classical mechanics|classical]] and [[quantum mechanics]].

=== Classical equation ===
The classical field equation (in [[dimensionless]] form) is:<ref name=Zakharov_Manakov>{{Cite journal |  author1 = V.E. Zakharov | authorlink1 = Vladimir E. Zakharov | author2 = S.V. Manakov | title = On the complete integrability of a nonlinear Schrödinger equation | journal = Journal of Theoretical and Mathematical Physics | volume = 19 | issue = 3 | year = 1974 | doi = 10.1007/BF01035568 | pages = 551–559 |  postscript = none |bibcode = 1974TMP....19..551Z }}. Originally in: ''Teoreticheskaya i Matematicheskaya Fizika'' '''19'''(3): 332–343. June 1974.</ref>

{{Equation box 1
|indent=:
|title='''Nonlinear Schrödinger equation''' ''(Classical field theory)''
|equation=<math>i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

for the [[complex number|complex]] field ''ψ''(''x'',''t'').

This equation arises from the [[Hamiltonian (quantum mechanics)|Hamiltonian]]<ref name=Zakharov_Manakov/>

:<math>H=\int \mathrm{d}x \left[{1\over 2}|\partial_x\psi|^2+{\kappa \over 2}|\psi|^4\right]</math>

with the [[Poisson bracket]]s

:<math>\{\psi(x),\psi(y)\}=\{\psi^*(x),\psi^*(y)\}=0 \, </math>

:<math>\{\psi^*(x),\psi(y)\}=i\delta(x-y). \,</math>

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 {{harvtxt|Zakharov|Shabat|1972}} (see [[#Solving the equation|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).<ref>{{citation | title = Nonlinear dispersive waves. Asymptotic analysis and solitons | first = M.J. | last = Ablowitz | year = 2011 | publisher = Cambridge University Press | isbn = 978-1-107-01254-7 | pages = 152–156 }}</ref>

=== Quantum mechanics ===

To get the [[Schrödinger field|quantized version]], simply replace the Poisson brackets by commutators

:<math>\begin{align}
  {}[\psi(x),\psi(y)]   &= [\psi^*(x),\psi^*(y)] = 0\\
  {}[\psi^*(x),\psi(y)] &= -\delta(x-y)
\end{align}</math>

and [[normal order]] the Hamiltonian

:<math>H=\int dx \left[{1\over 2}\partial_x\psi^\dagger\partial_x\psi+{\kappa \over 2}\psi^\dagger\psi^\dagger\psi\psi\right].</math>

The quantum version was  solved by [[Bethe ansatz]] by [[Lieb-Liniger Model|Lieb and Liniger]]. Thermodynamics was described by [[Chen Nin Yang]]. Quantum correlation functions also were evaluated by Korepin in 1993.<ref name="Korepin_1993" /> The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.<ref>http://insti.physics.sunysb.edu/~korepin/davis.pdf</ref>

== Solving the equation ==
The nonlinear Schrödinger equation is integrable in 1d: {{harvs|txt|last1=Zakharov|last2=Shabat|year=1972}} solved it with the [[inverse scattering transform]]. The corresponding linear system of equations is known as the [[Zakharov–Shabat system]]:

:<math> \begin{align}
\phi_x &= J\phi\Lambda+U\phi \\
\phi_t &= 2J\phi\Lambda^2+2U\phi\Lambda+(JU^2-JU_x)\phi,
\end{align} </math>
where
:<math>
\Lambda =
\begin{pmatrix}
\lambda_1&0\\
0&\lambda_2
\end{pmatrix}
, \quad
J = i\sigma_z =
\begin{pmatrix}
i&0\\
0&-i
\end{pmatrix}
, \quad
U = i
\begin{pmatrix}
0&q\\
r&0
\end{pmatrix}.</math>

The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system:

:<math> \phi_{xt}=\phi_{tx} \quad \Rightarrow \quad U_t=-JU_{xx}+2JU^2U
\quad \Leftrightarrow \quad 
\begin{cases}
iq_t=q_{xx}+2qrq \\
ir_t=-r_{xx}-2qrr. 
\end{cases}
\,</math>

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]]:

:<math> \begin{align}
& \phi \to \phi[1]=\phi\Lambda-\sigma\phi \\
& U \to U[1]=U+[J,\sigma] \\
& \sigma = \varphi\Omega\varphi^{-1} 
\end{align} </math>

which leaves the system invariant.

Here, ''φ'' is another invertible matrix solution (different from ''ϕ'') of the Zakharov–Shabat system with spectral parameter Ω:
:<math> \begin{align}
\varphi_x &= J\varphi\Omega+U\varphi \\
\varphi_t &= 2J\varphi\Omega^2+2U\varphi\Omega+(JU^2-JU_x)\varphi.
\end{align} </math>

Starting from the trivial solution ''U'' = 0 and iterating, one obtains the solutions with ''n'' [[soliton]]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 method|Crank–Nicolson]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari 
 |year=2009
 |title=Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal=Comput. Phys. Commun.
 |volume=180 |issue=3 |pages=1888–1912
 |doi=10.1016/j.cpc.2009.04.015
|bibcode=2009CoPhC.180.1888M|arxiv=0904.3131}}</ref> and [[Spectral method|Fourier spectral]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari 
 |year=2003
 |title=Bose-Einstein condensation dynamics in three dimensions by the pseudo-spectral and finite-difference methods
 |journal=J. Phys. B
 |volume=36 |issue= 12|pages=2501–2514
 |doi=10.1088/0953-4075/36/12/310
 |bibcode=2003JPhB...36.2501M|arxiv=cond-mat/0210177}}</ref> methods, are used for its solution.  There are different Fortran and C programs for [[its solution]]<ref>{{cite journal
 |author=D. Vudragovic
 |display-authors=etal
 |year=2012
 |title=C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=183 |issue=9 |pages=2021–2025
 |doi=10.1016/j.cpc.2012.03.022
  |bibcode=2012CoPhC.183.2021V|arxiv=1206.1361}}</ref><ref>
{{cite journal
 |author=L. E. Young-S.
 |display-authors=etal
 |year=2016
 |title=OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=204 |issue=9 |pages=209–213
 |doi=10.1016/j.cpc.2016.03.015
 |bibcode=2016CoPhC.204..209Y|arxiv=1605.03958}}</ref>.

== Galilean invariance ==
The nonlinear Schrödinger equation is [[Galilean invariance|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 <math>e^{-iv(x+vt/2)}\,</math>:

:<math>\psi(x,t) \mapsto \psi_{[v]}(x,t)=\psi(x+vt,t)\; e^{-iv(x+vt/2)}.</math>

== 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]], [[Raman scattering|stimulated Raman scattering]], [[Soliton (optics)|optical solitons]],
[[Ultrashort pulse|ultrashort pulses]], etc.

== The nonlinear Schrödinger equation in water waves ==<!-- [[group velocity]] points here -->
[[File:Sech soliton.svg|thumb|right|A [[hyperbolic function|hyperbolic secant (sech)]] envelope soliton for surface waves on deep water.<br>Blue line: water waves.<br>Red line: envelope soliton.]] 
For [[ocean surface waves|water waves]], the nonlinear Schrödinger equation describes the evolution of the [[envelope (mathematics)|envelope]] of [[modulation|modulated]] wave groups. In a paper in 1968, [[Vladimir E. Zakharov]] describes the [[Hamiltonian mechanics|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.<ref>{{cite journal | title= Stability of periodic waves of finite amplitude on the surface of a deep fluid | author= V. E. Zakharov | authorlink = Vladimir E. Zakharov | year=1968 | journal= Journal of Applied Mechanics and Technical Physics | volume=9 | issue=2 | pages=190–194 | doi=10.1007/BF00913182 |bibcode = 1968JAMTP...9..190Z }} Originally in: ''Zhurnal Prikdadnoi Mekhaniki i Tekhnicheskoi Fiziki'' 9 (2): 86–94, 1968.]</ref> 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 (mathematics)|envelope]] [[soliton]]s 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 [[wave of translation|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]]s.<ref>{{cite journal |   title=Oceanic rogue waves | last1=Dysthe | first1=K. | last2=Krogstad |first2=H.E. | last3=Müller | first3=P. | journal=Annual Review of Fluid Mechanics | volume=40 | issue=1 | pages=287–310 | year=2008 | doi=10.1146/annurev.fluid.40.111406.102203| bibcode=2008AnRFM..40..287D }}</ref>

The [[complex number|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:
:<math>
  \eta = a(x_0,t_0)\; \cos \left[ k_0\, x_0 - \omega_0\, t_0 - \theta(x_0,t_0) \right],
</math>
where ''a''(''x''<sub>0</sub>, ''t''<sub>0</sub>) and ''θ''(''x''<sub>0</sub>, ''t''<sub>0</sub>) are the slowly modulated amplitude and [[phase (waves)|phase]]. Further ''ω''<sub>0</sub> and ''k''<sub>0</sub> are the (constant) [[angular frequency]] and [[wavenumber]] of the carrier waves, which have to satisfy the [[dispersion (water waves)|dispersion]] relation ''ω''<sub>0</sub> = Ω(''k''<sub>0</sub>). Then

:<math> \psi = a\; \exp \left( i \theta \right). </math>

So its [[Complex number#Polar form|modulus]] |''ψ''| is the wave amplitude ''a'', and its [[Complex number#Polar form|argument]] arg(''ψ'') is the phase ''θ''.

The relation between the physical coordinates (''x''<sub>0</sub>, ''t''<sub>0</sub>) and the (''x, t'') coordinates, as used in the [[#Equation|nonlinear Schrödinger equation given above]], is given by:

:<math> x = k_0 \left[ x_0 - \Omega'(k_0)\; t_0 \right], \quad t = k_0^2 \left[ -\Omega''(k_0) \right]\; t_0 </math>

Thus (''x, t'') is a transformed coordinate system moving with the [[group velocity]] Ω'(''k''<sub>0</sub>) of the carrier waves,
The dispersion-relation [[curvature]] Ω"(''k''<sub>0</sub>) – 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:

:<math>\kappa = - 2 k_0^2, \quad \Omega(k_0) = \sqrt{g k_0} = \omega_0 \,\!</math> {{pad|1em}} so {{pad|1em}} <math>\Omega'(k_0) = \frac{1}{2} \frac{\omega_0}{k_0}, \quad \Omega''(k_0) = -\frac{1}{4} \frac{\omega_0}{k_0^2}, \,\!</math>

where ''g'' is the [[Earth's gravity|acceleration due to gravity]] at the Earth's surface.

In the original (''x''<sub>0</sub>,''t''<sub>0</sub>) coordinates the nonlinear Schrödinger equation for water waves reads:<ref>{{Cite book | first = G.B. | last = Whitham | author-link = Gerald B. Whitham | year = 1974 | title = Linear and nonlinear waves | publisher = Wiley-Interscience | isbn = 0-471-94090-9 | pages = 601–606 & 489–491 }}</ref>

:<math>i\, \partial_{t_0} A + i\, \Omega'(k_0)\, \partial_{x_0} A + \tfrac12 \Omega''(k_0)\, \partial_{x_0 x_0} A - \nu\, |A|^2\, A = 0,</math>

with <math>A=\psi^*</math> (i.e. the [[complex conjugate]] of <math>\psi</math>) and <math>\nu=\kappa\, k_0^2\, \Omega''(k_0).</math> So <math>\nu = \tfrac12 \omega_0 k_0^2</math> for deep water waves.

== Gauge equivalent counterpart ==
NLSE (1) is gauge equivalent to  the following isotropic [[Landau–Lifshitz model|Landau-Lifshitz equation]] (LLE) or [[Heisenberg ferromagnet]] equation

:<math>\vec{S}_t=\vec{S}\wedge \vec{S}_{xx}. \qquad </math>

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 ==
{{harvtxt|Hasimoto|1972}} showed that the work of {{harvs|txt|last=da Rios|authorlink=Luigi Sante Da Rios |year=1906}} on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, {{harvtxt|Salman|2013}} used this correspondence to show that breather solutions can also arise for a vortex filament.

== See also ==
* [[AKNS system]]
* [[Eckhaus equation]]
* [[Quartic interaction]] for a related model in [[quantum field theory]]
* [[Soliton (optics)]]
* [[Logarithmic Schrödinger equation]]

== Notes ==
{{reflist|group=nb}}

== References ==

=== Notes ===
{{reflist|30em}}

=== Other ===
{{refbegin}}
*{{citation|journal=Phys. Rev. Lett. |volume=13|issue=15|pages= 479–482 |year=1964
|title=Self-Trapping of Optical Beams|first1=R. Y. |last1=Chiao|first2= E.|last2= Garmire|first3= C. H. |last3=Townes|doi= 10.1103/PhysRevLett.13.479|bibcode = 1964PhRvL..13..479C }}
*{{citation|jfm=37.0764.01 
|first=Luigi Sante |last=da Rios
|title=Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque|language=Italian
|journal= Rendiconti del Circolo Matematico di Palermo|volume= 22|pages= 117–135 |year=1906|doi=10.1007/BF03018608|url=https://zenodo.org/record/1713176 }}
*{{citation| title=A soliton on a vortex filament|first=Hidenori|last= Hasimoto|journal=Journal of Fluid Mechanics|volume= 51 |issue=3|year= 1972|pages= 477–485|doi=10.1017/S0022112072002307|bibcode = 1972JFM....51..477H }}
*{{citation| title=Breathers on Quantized Superfluid Vortices|first=Hayder|last= Salman|journal=Phys. Rev. Lett.|volume= 111 |issue=16|year= 2013|pages= 165301|doi=10.1103/PhysRevLett.111.165301|bibcode=2013PhRvL.111p5301S|arxiv = 1307.7531|pmid=24182275}}
*{{citation|mr=0406174 |last1=Zakharov|first1= V. E.|last2= Shabat|first2= A. B.
|title=Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media 
|journal=Journal of Experimental and Theoretical Physics|volume= 34 |year=1972|issue= 1|pages= 62–69|url=http://www.jetp.ac.ru/cgi-bin/e/index/e/34/1/p62?a=list|bibcode = 1972JETP...34...62Z }}
{{refend}}

== External links ==
*{{scholarpedia|urlname=Nonlinear_Schrodinger_systems:_continuous_and_discrete|title=Nonlinear Schrodinger systems}}
* [http://online.kitp.ucsb.edu/online/turbulence11/falkovich/ Tutorial lecture on Nonlinear Schrodinger Equation (video)].
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1401.pdf Nonlinear Schrodinger Equation with a Cubic Nonlinearity] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1402.pdf Nonlinear Schrodinger Equation with a Power-Law Nonlinearity] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf Nonlinear Schrodinger Equation of General Form] at EqWorld: The World of Mathematical Equations.
* [http://wiki.math.toronto.edu/DispersiveWiki/index.php/NLS Mathematical aspects of the nonlinear Schrödinger equation] at Dispersive Wiki

{{DEFAULTSORT:Nonlinear Schrodinger equation}}
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[[Category:Exactly solvable models]]
[[Category:Schrödinger equation]]
[[Category:Integrable systems]]