Non-relativistic limit

The non-relativistic limit of a relativistic equation (which thus involves the speed of light 'c') denotes the limit when $c\to \infty$. It is the opposite of the vanishing dispersion limit.

Non-relativistic limit of NLKG

By inserting a parameter $c$ (the speed of light), one can rewrite the nonlinear Klein-Gordon equation as

$u_{tt}/c^{2}-\Delta u+c^{2}u+f(u)=0$.

One can then ask for what happens in the non-relativistic limit $c\rightarrow \infty$ (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, $u$ should be localized near the double hyperboloid

$t=\pm c{\sqrt {c^{2}+x^{2}}}$.

In the non-relativistic limit this becomes two paraboloids

$t=\pm (c^{2}+x^{2}/2)$

and so one expects $u$ to resolve as

$u=exp(ic^{2}t)v_{+}+exp(-ic^{2}t)v_{-}$ $u_{t}=ic^{2}exp(ic^{2}t)v_{+}-ic^{2}exp(ic^{2}t)v_{-}$

where $v_{+}$, $v_{-}$ solve some suitable NLS.

A special case arises if one assumes $(u_{t}-ic^{2}u)$ to be small at time zero (say $o(c)$ in some Sobolev norm). Then one expects $v_{-}$ to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], Nj1990, Ts1984, [MacNaOz-p], [Na-p]. In more general situations one expects $v_{+}$ and $v_{-}$ to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency $\ll c$ portion of the evolution should evolve in a Schrodinger-type manner, while the frequency $\gg c$ portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).

A similar non-relativistic limit result holds for the [#mkg Maxwell-Klein-Gordon] system (in the Coulomb gauge), where the limiting equation is the coupled
Schrodinger-Poisson system

$iv_{t}^{+}+\Delta v/2=uv^{+}$ $iv_{t}^{-}-\Delta v/2=uv^{-}$ $\Delta u=-|v^{+}|^{2}+|v^{-}|^{2}$

under reasonable $H^{1}$ hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields $f$ , $A$, $A_{0}$ and the Schrodinger-Poisson fields u, v^+, v^- are

$A_{0}\sim u$ $f\sim exp(ic^{2}t)v^{+}+exp(-ic^{2}t)v^{-}$ $f_{t}\sim iMexp(ic^{2})v^{+}-iMexp(-ic^{2}t)v^{-}$

where $M=sqrt{c^{4}-c^{2}\Delta }$ (a variant of $c^{2}$).