nonlinear partial differential equation, applicable to classical and quantum mechanics. Classical equation The classical field equation (in dimensionless form) is: for the complex field ψ(x,t). This equation arises from the Hamiltonian with the Poisson brackets Unlike its linear counterpart, it never describes the time evolution of a quantum state. The case with negative κ is called focusing and allows for bright soliton solutions (localized in space, and having spatial attenuation towards infinity) as well as breather solutions. It can be solved exactly by use of the inverse scattering transform, as shown by (see below). The other case, with κ positive, is the defocusing NLS which has dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude). Quantum mechanics To get the quantized version, simply replace the Poisson brackets by commutators and normal order the Hamiltonian The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by Chen-Ning Yang. Quantum correlation functions also were evaluated by Korepin in 1993. The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields. Solving the equation The nonlinear Schrödinger equation is integrable in 1d: solved it with the inverse scattering transform. The corresponding linear system of equations is known as the Zakharov–Shabat system: where The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system: 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: which leaves the system invariant. Here, φ is another invertible matrix solution (different from ϕ) of the Zakharov–Shabat system with spectral parameter Ω: 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 and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution. Galilean invariance 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