  The existence of localized electromagnetic structures is discussed in the
framework of the 1-dimensional relativistic Maxwell-fluid model for a cold
plasma with immobile ions. New partially localized solutions are found with a
finite-difference algorithm designed to locate numerically exact solutions of
the Maxwell-fluid system. These solutions are called quasi-solitons and consist
of a localized electromagnetic wave trapped in a spatially extended electron
plasma wave. They are organized in families characterized by the number of
nodes $p$ of the vector potential and exist in a continuous range of parameters
in the $\omega-V$ plane, where $V$ is the velocity of propagation and $\omega$
is the vector potential angular frequency. A parametric study shows that the
familiar fully localized relativistic solitons are special members of the
families of partially localized quasi-solitons. Soliton solution branches with
$p>1$ are therefore parametrically embedded in the continuum of quasi-solitons.
On the other hand, geometric arguments and numerical simulations indicate that
$p=0$ solitons exist only in the limit of either small amplitude or vanishing
velocity. Direct numerical simulations of the Maxwell-fluid model indicate that
the $p>0$ quasi-solitons (and embedded solitons) are unstable and lead to wake
excitation, while $p=0$ quasi-solitons appear stable. This helps explain the
ubiquitous observation of structures that resemble $p=0$ solitons in numerical
simulations of laser-plasma interaction.
