In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation in two dimensions. It can be considered as a classical equation, or a second quantized bosonic theory. It is an example of an integrable model .
Classically, we have a complex field ψ satisfying the partial differential equation
It is described by the Hamiltonian
![H=\int dx \left[{1\over 2}|\partial_x\psi|^2+{\kappa \over 2}|\psi|^4\right]](a/../upload/math/7c911a29de7398645238d2bcb6d94ff8.png)
with the Poisson brackets
- {ψ(x),ψ(y)} = {ψ * (x),ψ * (y)} = 0
- {ψ * (x),ψ(y)} = iδ(x - y)
To get the quantized version, simply replace the Poisson brackets by commutators
- [ψ(x),ψ(y)] = [ψ * (x),ψ * (y)] = 0
- [ψ * (x),ψ(y)] = - δ(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]](a/../upload/math/aa9f5122ce7a3f7a278012529f75fc19.png)
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