Non-relativistic limit: Difference between revisions

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


==Non-relativistic limit of NLKG==
By inserting a parameter <math>c</math> (the speed of light), one can rewrite the [[NKLG|nonlinear Klein-Gordon equation]] as
<center><math>u_{tt}/c^2 - \Delta u + c^2 u + f(u) = 0</math>.</center>
One can then ask for what happens in the non-relativistic limit <math>c \rightarrow \infty</math> (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, <math>u</math> should be localized near the double hyperboloid
<center><math>t  = \pm c \sqrt{c^2 +  x^2}</math>.</center>
In the non-relativistic limit this becomes two paraboloids
<center><math>t  = \pm (c^2 +  x^2/2)</math></center>
and so one expects <math>u</math> to resolve as
<center><math> u = exp(i c^2 t) v_+ + exp(-i c^2 t) v_- </math></center>
<center><math> u_t = ic^2 exp(ic^2 t) v_+ - ic^2 exp(ic^2 t) v_- </math></center>
where <math>v_+</math>, <math>v_-</math> solve some suitable NLS.
A special case arises if one assumes <math>(u_t - ic^2 u)</math> to be small at time zero (say <math>o(c)</math> in some Sobolev norm). Then one expects <math>v_-</math> to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], [[Bibliography#Nj1990|Nj1990]], [[Bibliography#Ts1984|Ts1984]], [MacNaOz-p], [Na-p]. In more general situations one expects <math>v_+</math> and <math>v_-</math> to evolve by a coupled NLS; see [[Bibliography#MasNa2002|MasNa2002]].
Heuristically, the frequency <math>\ll c</math> portion of the evolution should evolve in a Schrodinger-type manner, while the frequency <math>\gg c</math> 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 <br /> Schrodinger-Poisson system
<center><math>i v^+_t + \Delta v/2 = u v^+ </math></center>
<center><math>i v^-_t - \Delta v/2 = u v^- </math></center>
<center><math>\Delta u = - |v^+|^2 + |v^-|^2</math></center>
under reasonable <math>H^1</math> hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields  <math>f</math> , <math>A</math>, <math>A_0</math> and the Schrodinger-Poisson fields u, v^+, v^- are
<center><math>A_0 \sim u </math></center>
<center><math>f  \sim exp(ic^2 t) v^+ + exp(-ic^2 t) v^- </math></center>
<center><math>f _t \sim i M exp(ic^2)v^+ - i M exp(-ic^2 t) v^-</math></center>
where <math>M = sqrt{c^4 - c^2 \Delta}</math> (a variant of <math>c^2</math>).
[[Category:wave]]
[[Category:limits]]
[[Category:limits]]

Revision as of 21:51, 30 July 2006

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

Non-relativistic limit of NLKG

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

.

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

.

In the non-relativistic limit this becomes two paraboloids

and so one expects to resolve as

where , solve some suitable NLS.

A special case arises if one assumes to be small at time zero (say in some Sobolev norm). Then one expects 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 and to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency portion of the evolution should evolve in a Schrodinger-type manner, while the frequency 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

under reasonable hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields , , and the Schrodinger-Poisson fields u, v^+, v^- are

where (a variant of ).