Nonlinear Schrödinger equation

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

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

${\displaystyle {\begin{aligned}\phi _{x}&=J\phi \Lambda +U\phi \\\phi _{t}&=2J\phi \Lambda ^{2}+2U\phi \Lambda +\left(JU^{2}-JU_{x}\right)\phi ,\end{aligned}}}$

where

${\displaystyle \Lambda ={\begin{pmatrix}\lambda _{1}&0\\0&\lambda _{2}\end{pmatrix}},\quad J=i\sigma _{z}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}},\quad U=i{\begin{pmatrix}0&q\\r&0\end{pmatrix}}.}$

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

${\displaystyle \phi _{xt}=\phi _{tx}\quad \Rightarrow \quad U_{t}=-JU_{xx}+2JU^{2}U\quad \Leftrightarrow \quad {\begin{cases}iq_{t}=q_{xx}+2qrq\\ir_{t}=-r_{xx}-2qrr.\end{cases}}}$

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:

${\displaystyle {\begin{aligned}\phi \to \phi [1]&=\phi \Lambda -\sigma \phi \\U\to U[1]&=U+[J,\sigma ]\\\sigma &=\varphi \Omega \varphi ^{-1}\end{aligned}}}$

which leaves the system invariant.

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

${\displaystyle {\begin{aligned}\varphi _{x}&=J\varphi \Omega +U\varphi \\\varphi _{t}&=2J\varphi \Omega ^{2}+2U\varphi \Omega +\left(JU^{2}-JU_{x}\right)\varphi .\end{aligned}}}$

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 ${\displaystyle e^{-iv(x+vt/2)}\,}$:

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

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.

A hyperbolic secant (sech) envelope soliton for surface waves on deep water.
Blue line: water waves.
Red line: envelope soliton.

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. Additionally, the group velocity of these envelope solitons could be increased by an acceleration induced by an 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:

${\displaystyle \eta =a(x_{0},t_{0})\;\cos \left[k_{0}\,x_{0}-\omega _{0}\,t_{0}-\theta (x_{0},t_{0})\right],}$

where a(x₀, t₀) and θ(x₀, t₀) are the slowly modulated amplitude and phase. Further ω₀ and k₀ are the (constant) angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ω₀ = Ω(k₀). Then

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

So its modulus |ψ| is the wave amplitude a, and its argument arg(ψ) is the phase θ.

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

${\displaystyle x=k_{0}\left[x_{0}-\Omega '(k_{0})\;t_{0}\right],\quad t=k_{0}^{2}\left[-\Omega ''(k_{0})\right]\;t_{0}}$

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

${\displaystyle \kappa =-2k_{0}^{2},\quad \Omega (k_{0})={\sqrt {gk_{0}}}=\omega _{0}\,\!}$   so   ${\displaystyle \Omega '(k_{0})={\frac {1}{2}}{\frac {\omega _{0}}{k_{0}}},\quad \Omega ''(k_{0})=-{\frac {1}{4}}{\frac {\omega _{0}}{k_{0}^{2}}},\,\!}$

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

In the original (x₀, t₀) coordinates the nonlinear Schrödinger equation for water waves reads:[18]

${\displaystyle i\,\partial _{t_{0}}A+i\,\Omega '(k_{0})\,\partial _{x_{0}}A+{\tfrac {1}{2}}\Omega ''(k_{0})\,\partial _{x_{0}x_{0}}A-\nu \,|A|^{2}\,A=0,}$

with ${\displaystyle A=\psi ^{*}}$ (i.e. the complex conjugate of ${\displaystyle \psi }$) and ${\displaystyle \nu =\kappa \,k_{0}^{2}\,\Omega ''(k_{0}).}$ So ${\displaystyle \nu ={\tfrac {1}{2}}\omega _{0}k_{0}^{2}}$ for deep water waves.

NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation

${\displaystyle {\vec {S}}_{t}={\vec {S}}\wedge {\vec {S}}_{xx}.\qquad }$

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

The NLSE is equivalent to the curvature of a particular ${\displaystyle {\mathfrak {su}}(2)}$-connection on ${\displaystyle \mathbb {R} ^{2}}$ being equal to zero.[19]

Explicitly, with coordinates ${\displaystyle (x,t)}$ on ${\displaystyle \mathbb {R} ^{2}}$, the connection components ${\displaystyle A_{\mu }}$ are given by

$${\displaystyle A_{x}={\begin{pmatrix}i\lambda &i\varphi ^{*}\\i\varphi &-i\lambda \end{pmatrix}}}$$

$${\displaystyle A_{t}={\begin{pmatrix}2i\lambda ^{2}-i|\varphi |^{2}&2i\lambda \varphi ^{*}+\varphi _{x}^{*}\\2i\lambda \varphi -\varphi _{x}&-2i\lambda ^{2}+i|\varphi |^{2}\end{pmatrix}}}$$

where the ${\displaystyle \sigma _{i}}$ are the Pauli matrices. Then the zero-curvature equation

$${\displaystyle \partial _{t}A_{x}-\partial _{x}A_{t}+[A_{x},A_{t}]=0}$$

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

The pair of matrices ${\displaystyle A_{x}}$ and ${\displaystyle A_{t}}$ are also known as a Lax pair for the NLSE, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.

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.

• AKNS system
• Eckhaus equation
• Quartic interaction for a related model in quantum field theory
• Soliton (optics)
• Logarithmic Schrödinger equation

• "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