Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers, planar waveguides and hot rubidium vapors and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model. In quantum mechanics, the 1D NLSE is a special case of the classical nonlinear Schrödinger field, which in turn is a classical limit of a quantum Schrödinger field. Conversely, when the classical Schrödinger field is canonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantum nonlinear Schrödinger equation″) that describes bosonic point particles with delta-function interactions—the particles either repel or attract when they are at the same point. In fact, when the number of particles is finite, this quantum field theory is equivalent to the Lieb–Liniger model. Both the quantum and the classical 1D nonlinear Schrödinger equations are integrable. Of special interest is the limit of infinite strength repulsion, in which case the Lieb–Liniger model becomes the Tonks–Girardeau gas (also called the hard-core Bose gas, or impenetrable Bose gas). In this limit, the bosons may, by a change of variables that is a continuum generalization of the Jordan–Wigner transformation, be transformed to a system one-dimensional noninteracting spinless fermions. The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) in their study of optical beams. Multi-dimensional version replaces the second spatial derivative by the Laplacian. In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.

Definition The nonlinear Schrödinger equation is a 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 ) {\displaystyle \psi (x,t)} . This equation arises from the Hamiltonian

H = ∫ d x [ 1 2 | ∂ x ψ | 2 + κ 2 | ψ | 4 ] {\displaystyle H=\int \mathrm {d} x\left[{1 \over 2}|\partial _{x}\psi |^{2}+{\kappa \over 2}|\psi |^{4}\right]}

with the Poisson brackets

{ ψ ( x ) , ψ ( y ) } = { ψ ∗ ( x ) , ψ ∗ ( y ) } = 0 {\displaystyle \{\psi (x),\psi (y)\}=\{\psi ^{*}(x),\psi ^{*}(y)\}=0\,}

{ ψ ∗ ( x ) , ψ ( y ) } = i δ ( x − y ) . {\displaystyle \{\psi ^{*}(x),\psi (y)\}=i\delta (x-y).\,}

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 Zakharov & Shabat (1972) (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

[ ψ ( x ) , ψ ( y ) ] = [ ψ ∗ ( x ) , ψ ∗ ( y ) ] = 0

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