Nonlinear Schrödinger equation

In theoretical physics, the nonlinear Schrödinger equation (NLS) is a nonlinear version of Schrödinger's equation in two dimensions. It appears in optics and the theory of water waves, and it can also be considered as a second quantized bosonic theory. It is an example of an integrable model.

The equation

The nonlinear Schrödinger equation is the partial differential equation

$i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi$

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 and hence it can be solved with the inverse scattering transform, which takes the present equation and produces a linear system of equations, known as the Zakharov-Shabat system.

Some of the solutions can be expressed in analytic form. These include travelling waves and solitons.

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

The nonlinear Schrödinger equation in fiber optics

In optics, the function $\psi$ 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.

See also

• Phi to the fourth for a related model in quantum field theory.

References

• Muthiah Annamalai, Study of Split-step method to solve NLS, Term-paper for NonLinear optics course, May 2006.

• 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.