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 quasi-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 the form of curved worldlines.The results of our studies of the scattering of oscillons suggest that the radiation of the signum-Gordon model has a fractal-like nature.

Abstract:We study various properties of a perturbed signum-Gordon model, which has been obtained through the dimensional reduction of the called 'first BPS submodel of the Skyrme model'. This study is motivated by the observation that the first BPS submodel of the Skyrme model may be partially responsible for the good qualities of the rational map ansatz approximation to the solutions of the Skyrme model. We investigate the existence, stability and various properties of oscillons and other time-dependent states in this perturbed signum-Gordon model.

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.PACS numbers: I. INTRODUCTIONCompact oscillons [1] in the signum-Gordon model [2] are rather unusual field configurations. They do not match well any of two groups formed by (quasi-) periodic excitations of scalar fields -oscillons and breathers. Whereas the oscillons [3-8] are quasi-periodic and slowly radiating excitations observed in non-integrable field theories, the breathers [9-12] are exact and infinitely long living solutions that are present in many integrable field theories. Roughly speaking, the presence or lack of emission of the radiation from such periodic structures indicates if the model may or may not be integrable. The signum-Gordon oscillon is an exact, compact, perfectly periodic and finite energy solution. It does not radiate at all, however, when perturbed it emits some radiation [13]. Since the signum-Gordon model is a non-integrable field theory, then the existence of infinitely long living exact oscillons is a rather unusual fact. Such compact oscillons may have wider applications in theories with approximate scalling symmetry [14]. It is possible due to the fact that the signum-Gordon model emerges from many field theories with non-differentaiable potentials in the limit of small amplitude excitations [13][14][15][16]. Such more general models can support topological solitons and defects [17]. An interesting question about the oscillons is whether the oscillons may arise in the collision process of kinks, skyrmions and other topological or non-topologigal objects. Recently we have looked at the process of scattering of the signum-Gordon oscillons and emission of a radiation which, as reported in [18], is dominated by a huge number of smaller oscillons. Motivated by this study we have looked in more detail at the process of a collapse of the signum-Gordon shock waves which turns out to be an efficient process of production of oscillons. This issue is a main subject of the present paper. We suspect that similar phenomena may be observed * CNPq Scholarship holder -Brazil arXiv:1909.11137v1 [hep-th]