Nonlinear Schrödinger equation

Nonlinear Schrödinger equation
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Nonlinear_Schrödinger_equation".

In theoretical physics, the nonlinear Schrödinger equation (NLS) is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state. It is an example of an integrable model.

In quantum mechanics, it is a special case of the nonlinear Schrödinger field, and when canonically quantized, it describes bosonic point particles with delta-function interactions — the particles either repel or attract when they are at the same point. The nonlinear Schrödinger equation is integrable when the particles move in one dimension of space. In the limit of infinite strength repulsion, the nonlinear Schrödinger equation bosons are equivalent to one dimensional free fermions.

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1 The equation
2 Quantum version
3 Solving the equation
4 Galilean invariance
5 The nonlinear Schrödinger equation in fiber optics
6 The nonlinear Schrödinger equation in water waves
7 Gauge equivalent counterpart
8 See also
9 References

The equation

The nonlinear Schrödinger equation is the partial differential equation

$i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi$

for the complex field ψ.

This equation arises from the Hamiltonian

$H=\int dx \left[{1\over 2}|\partial_x\psi|^2+{\kappa \over 2}|\psi|^4\right]$

with the Poisson brackets

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

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

Quantum version

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

$[\psi(x),\psi(y)]=[\psi^*(x),\psi^*(y)]=0\,$

$[\psi^*(x),\psi(y)]=-\delta(x-y)\,$

and normal order the Hamiltonian

$H=\int dx \left[{1\over 2}\partial_x\psi^\dagger\partial_x\psi+{\kappa \over 2}\psi^\dagger\psi^\dagger\psi\psi\right].$

Solving the equation

The nonlinear Schrödinger equation is integrable: It can be solved with the inverse scattering transform. The corresponding linear system of equations is known as the Zakharov–Shabat system:

$\begin{align} \phi_x &= J\phi\Lambda+U\phi \\ \phi_t &= 2J\phi\Lambda^2+2U\phi\Lambda+(JU^2-JU_x)\phi, \end{align}$

where

$\Lambda = \begin{pmatrix} \lambda_1&0\\ 0&\lambda_2 \end{pmatrix} , \qquad J = i\sigma_z = \begin{pmatrix} i&0\\ 0&-i \end{pmatrix} , \quad\text{and}\quad U = i \begin{pmatrix} 0&q\\ r&0 \end{pmatrix}.$

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

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

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:

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

which leaves the system invariant.

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

$\begin{align} \varphi_x &= J\varphi\Omega+U\varphi \\ \varphi_t &= 2J\varphi\Omega^2+2U\varphi\Omega+(JU^2-JU_x)\varphi. \end{align}$

Starting from the trivial solution $U=0\,$ and iterating, one obtains the solutions with n solitons.

Computational solutions are found using a variety of methods, like the split-step method.

Galilean invariance

The nonlinear Schrödinger equation is Galilean invariant in the following sense:

Given a solution $\psi(x,t)\,$ a new solution can be obtained by replacing $x\,$ with $x+vt\,$ everywhere in $\psi(x,t)\,$ and by appending a phase factor of $e^{-iv(x+\frac{1}{2}vt)}:\,$

$\psi(x,t) \mapsto \psi_{[v]}(x,t)=\psi(x+vt,t)\; e^{-iv(x+\frac{1}{2}vt)}.$

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, stimulated Raman scattering, etc.

The nonlinear Schrödinger equation in water waves

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. 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.

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. Note, that 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.

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:

$\eta = a(x_0,t_0)\; \cos \left( k_0\, x_0 - \omega_0\, t_0 - \theta(x_0,t_0) \right),$

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

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

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:

$x = k_0 \left[ x_0 - \Omega'(k_0)\; t_0 \right] \qquad \text{ and } \qquad t = k_0^2 \left[ -\Omega''(k_0) \right]\; t_0.$

So ( x, t ) are a transformed coordinate system moving with the group velocity Ω'( k₀ ) of the carrier waves, The curvature Ω"( k₀ ) is always negative for water waves under the action of gravity.

For waves on the water surface of deep water, the important coefficients for the nonlinear Schrödinger equation are:

$\kappa = - 2 k_0^2, \qquad \Omega(k_0) = \sqrt{g k_0} = \omega_0, \qquad \text{so} \quad \Omega'(k_0) = \frac{1}{2} \frac{\omega_0}{k_0} \quad \text{and} \quad \Omega''(k_0) = -\frac{1}{4} \frac{\omega_0}{k_0^2},$

with g the acceleration by gravity.

Gauge equivalent counterpart

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

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

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

References

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.
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. doi:10.1007/BF00913182. Originally in: Zhurnal Prildadnoi Mekhanik i Tekhnicheskoi Fiziki 9 (2): 86–94, 1968.

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