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Register a new userCompact oscillons in the signum-Gordon model
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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.
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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 the scalar field equation are explicitly constructed from the second order polynomials in the time and position coordinates.
The regularized signum-Gordon potential has a smooth minimum and is linear in the modulus of the field value for higher amplitudes. The Q-ball solutions in this model are investigated. Their existence for charges large enough is demonstrated. In three dimensions numerical solutions are presented and the absolute stability of large Q-balls is proved. It is also shown, that the solutions of the regularized model approach uniformly the solution of the unregularized signum-Gordon model. From the stability of Q-balls in the regularized model follows the stability of the solutions in the original theory.
We study various aspects of the scattering of generalized compact oscillons in the signum-Gordon model in (1+1) dimensions. Using covariance of the model we construct traveling oscillons and study their interactions and the dependence of these interactions on the oscillons initial velocities and their relative phases. The scattering processes transform the two incoming oscillons into two outgoing ones and lead to the generation of extra oscillons which appear in the form of jet-like cascades. Such cascades vanish for some values of free parameters and the scattering processes, even though our model is non-integrable, resemble typical scattering processes normally observed for integrable or quase-integrable models. Occasionally, in the intermediate stage of the process, we have seen the emission of shock waves and we have noticed that, in general, outgoing oscillons have been more involved in their emission than the initial ones i.e. they have a border in form of curved world-lines. The results of our studies of the scattering of oscillons suggest that the radiation of the signum-Gordon model has a fractal-like nature.
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 solutions describes a process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.
The signum-Gordon model in 1+1 dimensions possesses the exact shockwave solution with discontinuity of the field at the light cone and infinite gradient energy. The energy of a regular part of the wave inside the light cone is finite and it grows linearly with time. The initial data for such waves contain a field configuration which is null in the space and has time derivative proportional to the Dirac delta. We study regularized initial data that lead to shock-like waves with finite gradient energy. We found that such waves exist in the finite time intervals and finally they decay and produce a cascade of oscillon-like structures. A pattern of the decay is very similar to the one observed in process of scattering of compact oscillons.
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