Question

Compitability condition for the form of nonlinear schrodinger equation

Answer 1

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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 and planar waveguides[2] and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime.[3] Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water;[2] the Langmuir waves in hot plasmas;[2] the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere;[4] the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains;[5] 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.[2] Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state (except hypothetically, as in some early attempts in the 1970s, to explain the quantum measurement process[6]). The 1D NLSE is an example of an integrable model.