The motion of a collection of vertical strings subject to horizontal linear vibrations
in the plane can be described by a system of first order nonlinear conservations laws. This system
-that we call the Chaplygin-Born-Infeld (CBI) system- is related to Magnetohydrodynamics and
more specifically to its shallow water version. Then, each vibrating string can be interpreted as a
magnetic line. The CBI system is also related to the Born-Infeld theory for the electromagnetic field,
a nonlinear correction to the classical Maxwell’s equations.
¶ Due to the linearity of vibrations, there is a priori no mechanism to prevent the strings to cross
each other, at least for sufficiently large initial impulse. These crossings generate concentration sin-
gularities in the CBI system. A numerical scheme is introduced to maintain order preserving strings
beyond singularities. This order preserving scheme is shown to be convergent to a distinguished
limit, which can be interpreted, through maximal monotone operator theory, as a vanishing viscosity
limit of the CBI system. Finally, models of pressureless gas with sticky particles are revisited and a
new formulation is provided.