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^{ [2] } and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime.^{ [3] } Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water;^{ [2] } the Langmuir waves in hot plasmas;^{ [2] } the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere;^{ [4] } the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains;^{ [5] } 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.^{ [2] } 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.

- Equation
- Classical equation
- Quantum mechanics
- Solving the equation
- Galilean invariance
- The nonlinear Schrödinger equation in fiber optics
- The nonlinear Schrödinger equation in water waves
- Gauge equivalent counterpart
- Relation to vortices
- See also
- Notes
- References
- Notes 2
- Other
- External links

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,andC. 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] }

The nonlinear Schrödinger equation is a nonlinear partial differential equation, applicable to classical and quantum mechanics.

The classical field equation (in dimensionless form) is:^{ [8] }

for the complex field *ψ*(*x*,*t*).

This equation arises from the Hamiltonian ^{ [8] }

with the Poisson brackets

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 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] }

To get the quantized version, simply replace the Poisson brackets by commutators

and normal order the Hamiltonian

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] }

The nonlinear Schrödinger equation is integrable in 1d: ZakharovandShabat ( 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 *n* solitons.

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] }

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 :

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.

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 envelope solitons may occur. Furthermore, these envelope solitons may be accelerated under 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 elevation *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 and wavenumber 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 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 Ω'(*k*_{0}) of the carrier waves, The dispersion-relation curvature 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:^{ [18] }

with (i.e. the complex conjugate of ) and So for deep water waves.

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.

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.

In physics, **interference** is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called **interferograms**.

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

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 the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

A **wave function** in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters *ψ* and Ψ.

The **Klein–Gordon equation** is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

The **sine-Gordon equation** is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions.

In mathematics, the **Korteweg–De Vries (KdV) equation** is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq and rediscovered by Diederik Korteweg and Gustav de Vries (1895).

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

The **Wheeler–DeWitt equation** is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group".

In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favoured than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by Kurt Symanzik (1981).

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

The **Gross–Pitaevskii equation** describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

In quantum mechanics and quantum field theory, a **Schrödinger field**, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-body Schrödinger equation for identical particles, the field theory is more suitable for situations where the particle number changes.

In quantum field theory, a **non-topological soliton** (**NTS**) is a soliton field configuration possessing, contrary to a topological one, a conserved Noether charge and stable against transformation into usual particles of this field for the following reason. For fixed charge *Q*, the mass sum of *Q* free particles exceeds the energy (mass) of the NTS so that the latter is energetically favorable to exist.

The **fractional Schrödinger equation** is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term *fractional Schrödinger equation* was coined by Nick Laskin.

The **Vakhitov–Kolokolov stability criterion** is a condition for linear stability of solitary wave solutions to a wide class of **U**(1)-invariant Hamiltonian systems, named after Soviet scientists Aleksandr Kolokolov and Nazib Vakhitov . The condition for linear stability of a solitary wave with frequency has the form

The **Gausson** is a soliton which is the solution of the logarithmic Schrödinger equation, which describes a quantum particle in a possible nonlinear quantum mechanics. The logarithmic Schrödinger equation preserves the dimensional homogeneity of the equation, i.e. the product of the independent solutions in one dimension remain the solution in multiple dimensions. While the nonlinearity alone cannot cause the quantum entanglement between dimensions, the logarithmic Schrödinger equation can be solved by the separation of variables.

In quantum probability, the **Belavkin equation**, also known as **Belavkin-Schrödinger equation**, **quantum filtering equation**, **stochastic master equation**, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988.

The **Kundu equation** is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu as

The **Schamel equation (S-equation)** is a nonlinear partial differential equation of first order in time and third order in space. Similar to a Korteweg de Vries equation (KdV), it describes the development of a localized, coherent wave structure that propagates in a nonlinear dispersive medium. It was first derived in 1973 by Hans Schamel to describe the effects of electron trapping in the trough of the potential of a solitary electrostatic wave structure travelling with ion acoustic speed in a two-component plasma. It now applies to various localized pulse dynamics such as:

- ↑ Figure 1 from: Onorato, M.; Proment, D.; Clauss, G.; Klein, M. (2013), "Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test",
*PLOS One*,**8**(2): e54629, Bibcode:2013PLoSO...854629O, doi:10.1371/journal.pone.0054629, PMC 3566097 , PMID 23405086 - 1 2 3 4 Malomed, 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 - ↑ Balakrishnan, R. (1985). "Soliton propagation in nonuniform media".
*Physical Review A*.**32**(2): 1144–1149. Bibcode:1985PhRvA..32.1144B. doi:10.1103/PhysRevA.32.1144. PMID 9896172. - 1 2 Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1993).
*Quantum Inverse Scattering Method and Correlation Functions*. Cambridge, U.K.: Cambridge University Press. doi:10.2277/0521586461. ISBN 978-0-521-58646-7. - ↑ G. Falkovich (2011).
*Fluid Mechanics (A short course for physicists)*. Cambridge University Press. ISBN 978-1-107-00575-4. - 1 2 V.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 Fizika***19**(3): 332–343. June 1974. - ↑ Ablowitz, M.J. (2011),
*Nonlinear dispersive waves. Asymptotic analysis and solitons*, Cambridge University Press, pp. 152–156, ISBN 978-1-107-01254-7 - ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2012-05-16. Retrieved 2011-09-04.CS1 maint: archived copy as title (link)
- ↑ P. Muruganandam and S. K. Adhikari (2009). "Fortran Programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap".
*Comput. Phys. Commun*.**180**(3): 1888–1912. arXiv: 0904.3131 . Bibcode:2009CoPhC.180.1888M. doi:10.1016/j.cpc.2009.04.015. - ↑ P. Muruganandam and S. K. Adhikari (2003). "Bose-Einstein condensation dynamics in three dimensions by the pseudo-spectral and finite-difference methods".
*J. Phys. B*.**36**(12): 2501–2514. arXiv: cond-mat/0210177 . Bibcode:2003JPhB...36.2501M. doi:10.1088/0953-4075/36/12/310. - ↑ D. Vudragovic; et al. (2012). "C Programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap".
*Comput. Phys. Commun*.**183**(9): 2021–2025. arXiv: 1206.1361 . Bibcode:2012CoPhC.183.2021V. doi:10.1016/j.cpc.2012.03.022. - ↑ L. E. Young-S.; et al. (2016). "OpenMP Fortran and C Programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap".
*Comput. Phys. Commun*.**204**(9): 209–213. arXiv: 1605.03958 . Bibcode:2016CoPhC.204..209Y. doi:10.1016/j.cpc.2016.03.015. - ↑ 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.] - ↑ G. G. Rozenman, A. Arie, L. Shemer (2019). "Observation of accelerating solitary wavepackets".
*Phys. Rev. E*.**101**(5): 050201. doi:10.1103/PhysRevE.101.050201. PMID 32575227. - ↑ Dysthe, K.; Krogstad, H.E.; Müller, P. (2008). "Oceanic rogue waves".
*Annual Review of Fluid Mechanics*.**40**(1): 287–310. Bibcode:2008AnRFM..40..287D. doi:10.1146/annurev.fluid.40.111406.102203. - ↑ Whitham, G.B. (1974).
*Linear and nonlinear waves*. Wiley-Interscience. pp. 601–606 & 489–491. ISBN 0-471-94090-9.

- Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964), "Self-Trapping of Optical Beams",
*Phys. Rev. Lett.*,**13**(15): 479–482, Bibcode:1964PhRvL..13..479C, doi:10.1103/PhysRevLett.13.479 - da Rios, Luigi Sante (1906), "Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque",
*Rendiconti del Circolo Matematico di Palermo*(in Italian),**22**: 117–135, doi:10.1007/BF03018608, JFM 37.0764.01 - Hasimoto, Hidenori (1972), "A soliton on a vortex filament",
*Journal of Fluid Mechanics*,**51**(3): 477–485, Bibcode:1972JFM....51..477H, doi:10.1017/S0022112072002307 - Salman, Hayder (2013), "Breathers on Quantized Superfluid Vortices",
*Phys. Rev. Lett.*,**111**(16): 165301, arXiv: 1307.7531 , Bibcode:2013PhRvL.111p5301S, doi:10.1103/PhysRevLett.111.165301, PMID 24182275 - Zakharov, V. E.; Shabat, A. B. (1972), "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media",
*Journal of Experimental and Theoretical Physics*,**34**(1): 62–69, Bibcode:1972JETP...34...62Z, MR 0406174

- "Nonlinear Schrodinger systems".
*Scholarpedia*. - 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

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