Intheoretical physics, the (one-dimensional)nonlinear Schrödinger equation (NLSE) is anonlinear variation of theSchrödinger equation. It is aclassical field equation whose principal applications are to the propagation of light in nonlinearoptical fibers, planarwaveguides^[2] and hot rubidium vapors^[3]and toBose–Einstein condensates confined to highlyanisotropic, cigar-shapedtraps, in themean-field regime.^[4] Additionally, the equation appears in the studies of small-amplitudegravity waves on the surface of deepinviscid (zero-viscosity) water;^[2] theLangmuir waves in hotplasmas;^[2] the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere;^[5] the propagation ofDavydov's alpha-helix solitons, which are responsible for energy transport along molecular chains;^[6] and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packetsof quasi-monochromatic waves in weakly nonlinear media that havedispersion.^[2] Unlike the linearSchrödinger equation, the NLSE never describes the time evolution of a quantum state.^{[citation needed]} The 1D NLSE is an example of anintegrable model.

Inquantum mechanics, the 1D NLSE is a special case of the classical nonlinearSchrödinger field, which in turn is a classical limit of a quantum Schrödinger field. Conversely, when the classical Schrödinger field iscanonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantumnonlinear 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 theLieb–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 theTonks–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 theJordan–Wigner transformation, be transformed to a system one-dimensional noninteracting spinless^[7] fermions.^[8]

The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of theGinzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) in their study of optical beams.

Multi-dimensional version replaces the second spatial derivative by the Laplacian. In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.^[9]

The nonlinear Schrödinger equation is anonlinear partial differential equation, applicable toclassical andquantum mechanics.

The classical field equation (indimensionless form) is:^[10]

for thecomplex fieldψ(x,t).

This equation arises from theHamiltonian^[10]

${\displaystyle H=\int \mathrm{d}x\left[\frac{1}{2}|{\mathrm{\partial }}_x\psi {|}^2+\frac{\kappa }{2}|\psi {|}^4\right]}$

with thePoisson brackets

${\displaystyle \{\psi (x),\psi (y)\}=\{{\psi }^{\ast }(x),{\psi }^{\ast }(y)\}=0}$

${\displaystyle \{{\psi }^{\ast }(x),\psi (y)\}=i\delta (x-y).}$

Unlike its linear counterpart, it never describes the time evolution of a quantum state.^{[citation needed]}

The case with negative κ is called focusing and allows forbright soliton solutions (localized in space, and having spatial attenuation towards infinity) as well asbreather solutions. It can besolved exactly by use of theinverse scattering transform, as shown byZakharov & Shabat (1972) (seebelow). The other case, with κ positive, is the defocusing NLS which hasdark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).^[11]

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

andnormal order the Hamiltonian

${\displaystyle H=\int dx\left[\frac{1}{2}{\mathrm{\partial }}_x{\psi }^{†}{\mathrm{\partial }}_x\psi +\frac{\kappa }{2}{\psi }^{†}{\psi }^{†}\psi \psi \right].}$

The quantum version was solved byBethe ansatz byLieb and Liniger. Thermodynamics was described byChen-Ning Yang. Quantum correlation functions also were evaluated by Korepin in 1993.^[8] The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.^[12]

The nonlinear Schrödinger equation is integrable in 1d: Zakharov and Shabat (1972) solved it with theinverse scattering transform. The corresponding linear system of equations is known as theZakharov–Shabat system:

where

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

${\displaystyle {\varphi }_{xt}={\varphi }_{tx}\Rightarrow U_t=-J{U}_{xx}+2J{U}^{2}U\iff \{\begin{array}{l}i{q}_t=q_{xx}+2qrq\\ i{r}_t=-r_{xx}-2qrr.\end{array}}$

By settingq =r* orq = −r* the nonlinear Schrödinger equation with attractive or repulsive interaction is obtained.

An alternative approach uses the Zakharov–Shabat system directly and employs the followingDarboux transformation:

which leaves the system invariant.

Here,φ is another invertible matrix solution (different fromϕ) of the Zakharov–Shabat system with spectral parameter Ω:

Starting from the trivial solution U = 0 and iterating, one obtains the solutions with nsolitons. This can be achieved via direct numerical simulation using, for example, thesplit-step method.^[13]This method has been implemented on both CPU and GPU.^[14][15]

Inoptics, the nonlinear Schrödinger equation occurs in theManakov 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 theκ term represents the nonlinearity. The equation models many nonlinearity effects in a fiber, including but not limited toself-phase modulation,four-wave mixing,second-harmonic generation,stimulated Raman scattering,optical solitons,ultrashort pulses, etc.

Forwater waves, the nonlinear Schrödinger equation describes the evolution of theenvelope ofmodulated wave groups. In a paper in 1968,Vladimir E. Zakharov describes theHamiltonian structure of water waves. In the same paper Zakharov shows that, for slowly modulated wave groups, the waveamplitude satisfies the nonlinear Schrödinger equation, approximately.^[16] The value of the nonlinearity parameterк depends on the relative water depth. For deep water, with the water depth large compared to thewave length of the water waves,к is negative andenvelopesolitons may occur. Additionally, the group velocity of these envelope solitons could be increased by an acceleration induced by an external time-dependent water flow.^[17]

For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameterк is positive andwave groups withenvelope solitons do not exist. In shallow watersurface-elevation solitons orwaves 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 ofrogue waves.^[18]

Thecomplex fieldψ, as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves. Consider a slowly modulatedcarrier wave with water surfaceelevationη of the form:

${\displaystyle \eta =a(x_0,t_0)\cos \left[k_0{x}_0-{\omega }_0{t}_0-\theta (x_0,t_0)\right],}$

wherea(x₀,t₀) andθ(x₀,t₀) are the slowly modulated amplitude andphase. Furtherω₀ andk₀ are the (constant)angular frequency andwavenumber of the carrier waves, which have to satisfy thedispersion relationω₀ = Ω(k₀). Then

${\displaystyle \psi =a\exp \left(i\theta \right).}$

So itsmodulus |ψ| is the wave amplitudea, and itsargument arg(ψ) is the phaseθ.

The relation between the physical coordinates (x₀,t₀) and the (x, t) coordinates, as used in thenonlinear Schrödinger equation given above, is given by:

${\displaystyle x=k_0\left[x_0-{\mathrm{\Omega }}^{\prime }(k_0)t_0\right],t=k_0^2\left[-{\mathrm{\Omega }}^{″}(k_0)\right]t_0}$

Thus (x, t) is a transformed coordinate system moving with thegroup velocity Ω'(k₀) of the carrier waves,The dispersion-relationcurvature Ω"(k₀) – representinggroup 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:

${\displaystyle \kappa =-2k_0^2,\mathrm{\Omega }(k_0)=\sqrt{g{k}_0}={\omega }_0}$  so ${\displaystyle {\mathrm{\Omega }}^{\prime }(k_0)=\frac{1}{2}\frac{{\omega }_0}{k_0},{\mathrm{\Omega }}^{″}(k_0)=-\frac{1}{4}\frac{{\omega }_0}{k_0^2},}$

whereg is theacceleration due to gravity at the Earth's surface.

In the original (x₀,t₀) coordinates the nonlinear Schrödinger equation for water waves reads:^[19]

${\displaystyle i{\mathrm{\partial }}_{t_0}A+i{\mathrm{\Omega }}^{\prime }(k_0){\mathrm{\partial }}_{x_0}A+\frac{1}{2}{\mathrm{\Omega }}^{″}(k_0){\mathrm{\partial }}_{x_0{x}_0}A-\nu |A{|}^{2}A=0,}$

with${\displaystyle A={\psi }^{\ast }}$ (i.e. thecomplex conjugate of${\displaystyle \psi }$) and${\displaystyle \nu =\kappa k_0^2{\mathrm{\Omega }}^{″}(k_0).}$ So${\displaystyle \nu =\frac{1}{2}{\omega }_0{k}_0^2}$ for deep water waves.

Hasimoto (1972) showed that the work ofda Rios (1906) onvortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently,Salman (2013) used this correspondence to show thatbreather solutions can also arise for a vortex filament.

The nonlinear Schrödinger equation isGalilean invariant in the following sense:

Given a solutionψ(x, t) a new solution can be obtained by replacingx withx +vt everywhere in ψ(x, t) and by appending a phase factor of${\displaystyle e^{-iv(x+vt/2)}}$:

${\displaystyle \psi (x,t)\mapsto {\psi }_{[v]}(x,t)=\psi (x+vt,t)e^{-iv(x+vt/2)}.}$

NLSE (1) is gauge equivalent to the following isotropicLandau–Lifshitz equation (LLE) orHeisenberg ferromagnet equation

${\displaystyle {\overrightarrow{S}}_t=\overrightarrow{S}\wedge {\overrightarrow{S}}_{xx}.}$

Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like theIshimori equation and so on.

The NLSE is equivalent to thecurvature of a particular${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$-connection on${\displaystyle {\mathbb{R}}^2}$ being equal to zero.^[20]

Explicitly, with coordinates${\displaystyle (x,t)}$ on${\displaystyle {\mathbb{R}}^2}$, the connection components${\displaystyle A_{\mu }}$ are given bywhere the${\displaystyle {\sigma }_i}$ are thePauli matrices.Then the zero-curvature equation$${\displaystyle {\mathrm{\partial }}_t{A}_x-{\mathrm{\partial }}_x{A}_t+[A_x,A_t]=0}$$

is equivalent to the NLSE${\displaystyle i{\phi }_t+{\phi }_{xx}+2|\phi {|}^2\phi =0}$. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined${\displaystyle F_{\mu \nu }=[{\mathrm{\partial }}_{\mu }-A_{\mu },{\mathrm{\partial }}_{\nu }-A_{\nu }]}$.

The pair of matrices${\displaystyle A_x}$ and${\displaystyle A_t}$ are also known as aLax pair for the NLSE, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.

• AKNS system
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• "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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