Forces in the systems of two opposite sign and three identical charges coupled to the dynamical scalar field of the signum-Gordon model are investigated. Three-body force is present, and the exact formula for it is found. Flipping the sign of one of
the two charges changes not only the sign but also the magnitude of the force. Both effects are due to nonlinearity of the field equation.
Force between static point particles coupled to a classical ultra-massive scalar field is calculated. The field potential is proportional to the modulus of the field. It turns out that the force exactly vanishes when the distance between the particle
s exceeds certain finite value. Moreover, each isolated particle is surrounded by a compact cloud of the scalar field that completely screens its scalar charge.
Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with nonlinear term of the form sign(phi) are presented. They are constructed from cubic polynomials in the scale invariant variable t/r. One class of solu
tions describes a process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.
We present a new class of oscillons in the (1+1)-dimensional signum-Gordon model. The oscillons periodically move to and fro in the space. They have finite total energy, finite size, and are strictly periodic in time. The corresponding solutions of t
he scalar field equation are explicitly constructed from the second order polynomials in the time and position coordinates.
We present explicit solutions of the signum-Gordon scalar field equation which have finite energy and are periodic in time. Such oscillons have a strictly finite size. They do not emit radiation.
