Non-relativistic limit

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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 Maxwell-Klein-Gordon system (in the Coulomb gauge), where the limiting equation is a 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 ).