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

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

This equation arises from the Hamiltonian

$H=\backslash int\; dx\; \backslash left2\}|\backslash partial\_x\backslash psi|^2+\{\backslash kappa\; \backslash over\; 2\}|\backslash psi|^4\backslash right$

with the Poisson brackets

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

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

Quantum version

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

$*=*=0$

$*=-\backslash delta(x-y)$

and normal order the Hamiltonian

$H=\backslash int\; dx\; \backslash left2\}\backslash partial\_x\backslash psi^\backslash dagger\backslash partial\_x\backslash psi+\{\backslash kappa\; \backslash over\; 2\}\backslash psi^\backslash dagger\backslash psi^\backslash dagger\backslash psi\backslash psi\backslash right.$

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 non-linear Schrödinger equation occurs in the Manakov system, a model of wave propagation in fiber optics. The function $\backslash psi$ 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

• 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.

Partial differential equations Exactly solvable models