  We prove nonlinear stability of line soliton solutions of the KP-II equation
with respect to transverse perturbations that are exponentially localized as
$x\to\infty$. We find that the amplitude of the line soliton converges to that
of the line soliton at initial time whereas jumps of the local phase shift of
the crest propagate in a finite speed toward $y=\pm\infty$. The local amplitude
and the phase shift of the crest of the line solitons are described by a system
of 1D wave equations with diffraction terms.
