  In an attempt to clarify what is the velocity of a particle in doubly special
relativity, we solve Maxwell's equations invariant under the position-space
nonlinear Lorentz transformation proposed by Kimberly, Magueijo, and Medeiros.
We show that only the amplitude of the Maxwell wave, not the phase, is affected
by the nonlinearity of the transformation. Thus, although the Maxwell wave
appears to have infinitely large energy near the Planck time, the wave velocity
is the same as the conventional light velocity. Surprisingly, the velocity of
the Maxwell wave is not the same as the maximum signal velocity determined by
the null geodesic condition, which is infinitely large near the Planck time and
monotonically decreases in time to the conventional light velocity when time
approaches infinity. This implies that, depending on the position of the
particle in question, the light cone determined by Maxwell's equations may be
inside or outside the null cone determined by the null geodesic equation, which
may lead to the causality problem.
