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Nonlinear Schrödinger Equation

A nonlinear Schrödinger equation adds a nonlinear amplitude term to the linear Schrödinger equation. A power-law defocusing normalization is

ipartial_tu+Deltau=|u|^(p-1)u,

where u is complex-valued and p>1. Its energy has the nonnegative density ||del u||^2/2+|u|^(p+1)/(p+1). The scaling-critical Sobolev exponent in dimension d is s_c=d/2-2/(p-1). Energy supercriticality means s_c>1.

See also

Schrödinger Equation, Sobolev Space

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References

Tao, T. Nonlinear Dispersive Equations: Local and Global Analysis. Providence, RI: Amer. Math. Soc., 2006. https://doi.org/10.1090/cbms/106.

Cite this as:

Weisstein, Eric W. "Nonlinear Schrödinger Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NonlinearSchroedingerEquation.html

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