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Scattering of compact oscillons

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Head-on collisions of signum-Gordon oscillons can scatter with almost no radiation, and what radiation remains is a self-similar cascade of smaller oscillons.

desk verdict Solid numerical mapping of two-oscillon scattering with a genuinely new exact construction; the fractal-like radiation claim is an honest but under-supported conjecture. read the letter →

arxiv 1909.01992 v2 pith:7II5BFEN submitted 2019-09-04 hep-th

F. M. Hahne , P. Klimas , J. S. Streibel , W. J. Zakrzewski This is my paper · ORCID
classification hep-th
keywords signum-Gordonmodelcompactoscillonsoscillonscatteringfractalradiationself-similarcascadeshockwavesscalingsymmetryno-radiationwindows
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies head-on collisions of exact compact oscillons in the (1+1)-dimensional signum-Gordon model, a non-integrable scalar field theory with a V-shaped potential. It claims that two incoming oscillons generally turn into two outgoing quasi-oscillons plus a jet-like cascade of smaller oscillon-like structures, and that for particular high velocities and relative phases this cascade nearly vanishes, so the scattering resembles what one expects from an integrable model. The paper further claims that the radiation is dominated by self-similar oscillon structures at arbitrarily small scales, suggesting a fractal-like nature of the model's radiation. If true, this gives a concrete non-integrable model with radiation-free scattering windows and a cascade mechanism that repeats at all scales.

What carries the argument

The argument is carried by the exact scaling symmetry of the signum-Gordon equation, $\varphi^{(\lambda)}(t,x)=\lambda^2\varphi(t/\lambda,x/\lambda)$, together with Lorentz covariance and the notion of an oscillon's phase $\alpha$. The scaling symmetry guarantees that exact oscillon solutions exist with arbitrarily small size and energy (energy scales as $\lambda^3$), so any perturbed oscillon can shed energy by emitting smaller oscillons, and a scattering diagram can in principle repeat itself at every scale; this underpins the fractal conjecture. The phase $\alpha$ determines the shape of each incoming oscillon at the moment its support touches the other, and the paper shows that the no-radiation choices lie on curves in $(\alpha,V)$ space, with the special phase shift $\Delta\alpha=1/2$ relating configurations by a global sign flip in the $v=0$ case. Comparison with the exact shock-wave solutions whose zeros lie on hyperbolas $x_k(t)=\pm\sqrt{t^2-4a_k}$ is what identifies the diamond-shaped structures seen in high-velocity collisions.

What would settle it

Run the same symmetric collision (for instance V=0.93, v=0, alpha=0.414) at grid spacings differing by a factor of two and measure both the radiated-energy fraction and the number of resolved oscillon structures in the central cascade; if the near-zero radiation window shifts or the small-scale structures appear or disappear, the fractal and clean-scattering claims as stated fail.

Watch

Extended reading notes

Core claim

The core claim, stated on the paper's own terms, is that the radiation emitted by the signum-Gordon model is dominated by oscillons and has a fractal-like nature. In symmetric collisions of two exact compact oscillons, the interaction temporarily shrinks the support to a minimal size and then produces two main outgoing quasi-oscillons; the intervening radiation appears as cascades of smaller oscillons, and repeated blow-ups of the spacetime diagram show similar oscillating structures at successively smaller scales. For velocities roughly above V~0.7 the outgoing quasi-oscillons are regular, and the cascade can be made essentially to vanish by fine-tuning the initial phase alpha (and the border velocity parameter v), yielding almost radiation-free scattering in a non-integrable model. The paper also identifies intermediate shock-wave-like structures that decay into the oscillon cascade. Thus the paper asserts that the signum-Gordon model possesses both clean scattering windows and a self-similar, scale-repeating radiation mechanism.

Load-bearing premise

The load-bearing premise is that the finite-difference fourth-order time-integration grid faithfully resolves the continuum evolution at the smallest scales, a premise the paper itself flags as delicate because halving or doubling the number of grid points can make tiny structures appear or disappear and the low-velocity no-radiation void could be an artefact.

Editorial extensions

If this is right

  • If the fractal picture is right, then the signum-Gordon model's radiation is not a continuous spray of small waves but a discrete hierarchy: outgoing energy is carried by nested generations of smaller compact oscillons.
  • The near-zero-radiation windows mean a non-integrable model can exhibit integrable-like scattering; two exact oscillons can pass through each other leaving almost nothing behind when V and alpha are tuned.
  • The decay of the intermediate shock-wave-like structures into oscillon cascades gives a concrete production channel for quasi-oscillons, and connects the diamond-shaped wave seen in the simulations to the known exact shock solutions.
  • Because the exact generalized oscillons with non-uniformly moving borders match the outgoing objects, collisions generically convert one class of exact compact oscillons into a broader class with curved, accelerating borders.
  • The radiation-free conditions being achievable with nonzero border velocity v (with four equivalent initial configurations instead of two) widens the family of initial states that scatter cleanly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One consequence the authors leave implicit: if the cascade is truly self-similar, counting collisions or oscillon-like structures in successive blow-ups should follow a power law (for example, numbers growing by a fixed factor per zoom), which could be tested in the existing high-resolution data.
  • The same scaling argument predicts that the emitted energy spectrum should be scale-free; measurements of the radiated energy fraction at several zoom levels could distinguish true self-similarity from a finite-resolution artefact.
  • The near-absence of radiation at fine-tuned phases resembles resonance conditions in integrable scattering; a natural extension is to search for an effective conserved quantity that selects those phases, using the zero-line matching conditions as a starting point.
  • If the low-velocity no-radiation void is physical rather than numerical, it suggests a velocity threshold near V~0.7 below which the outgoing oscillons reabsorb the radiation, a behaviour the paper notes but does not resolve.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies collisions of exact compact oscillons in the (1+1)-dimensional signum-Gordon model. After constructing traveling oscillons by Lorentz-boosting known generalized oscillons and parametrizing symmetric/antisymmetric two-oscillon initial data by a phase α, the authors perform extensive numerical scattering simulations as functions of boost V, border speed v, and phase. They report that collisions generically produce two outgoing quasi-oscillons plus jets/cascades of smaller oscillon-like objects; that for fine-tuned high-velocity initial data the cascades nearly vanish, giving clean scattering windows reminiscent of integrable models; that intermediate stages sometimes contain shock-wave-like structures matching exact shock-wave zero hyperbolas; that outgoing oscillons generically have curved (accelerating) borders, for which an exact class with uniformly accelerated borders is constructed; and that high-resolution zooms suggest a fractal-like self-similar structure of the radiation. The paper is candid about numerical limitations in Appendices A and B.

Significance. If the numerical claims hold, the paper offers a striking phenomenology: exact compact oscillons in a non-integrable model can scatter almost elastically, the radiation is dominated by oscillon-like objects at multiple scales, and the model's exact scaling symmetry provides a natural mechanism for self-similar cascades. The analytic constructions—Lorentz-boosted oscillons, endpoint formulas x0(α), shock-wave matching, and accelerated-border oscillons—are internally consistent and constitute a useful contribution independent of the numerics. The paper's honest acknowledgment of resolution sensitivity is a strength, but the headline fractal claim is not yet quantitatively established; as presented it is a conjecture supported mainly by visual zooms.

major comments (3)
  1. [Section III F and Appendix B] The fractal-like-radiation claim rests on the smallest resolved structures in Fig. 38, yet Appendix B states that changing the number of grid points by a factor of two can make tiny structures appear or disappear and that only larger structures are stable. No grid-convergence study, no quantitative scaling exponent, and no comparison of blow-ups at two independent resolutions is provided; the dilation symmetry (III.53) proves the existence of exact oscillons at every scale but not that the numerically generated cascade is scale-invariant. Please provide a convergence test (e.g., compare N=2^19 and N=2^20 for the same run, quantify L1 differences and the counts/amplitudes of nested oscillons in the zoomed region), or explicitly state in the abstract and conclusions that the fractal nature is an unresolved conjecture.
  2. [Section III C 3 and Section III D (Figs. 24, 30, 32, 33, 34)] The no-radiation windows and the low-V no-radiation void are produced from lower-resolution scans with N=2^12, without error bars, without a stated definition of Erad, and without a description of how EL and ER are assigned to outgoing oscillons. The text itself says in Section III C 3 that outgoing-oscillon energies are less reliable for small V and in Section III D that the low-V void "could well be a numerical artefact." Because the existence of clean scattering windows is a central claim, please recompute representative strips of the parameter maps at N=2^15 or higher, show that Erad/E converges, and specify the radiation-extraction procedure.
  3. [Appendix A] The numerical method is standard RK4 applied to a second-order system with a discontinuous sgn(φ) term, but no convergence-order test, energy conservation monitor, or treatment of the sign discontinuity is reported. Given that the model has compact supports and shock-like structures, the continuum limit of the discretization is not automatic. Please add at least one quantitative convergence check (e.g., a scattering run at N=2^15 and N=2^16 with fixed Δt/Δx, reporting pointwise differences and energy drift over the simulation time) and discuss how the sgn(0)=0 convention is implemented in the discrete update.
minor comments (5)
  1. [Throughout] Numerous typos and grammatical slips remain, e.g., "quase-integravel" in the abstract, "intarval" near Eq. (II.7), "potentails" in the Introduction, "deatil" in Section III B, "decaysed" in Section IV, "in stability" and "the details of the details" in Appendix B. A careful language edit is needed.
  2. [Figure 33 caption] The caption says the plot is made "while holding fixed the right input oscillon's phase αR = 0," but the text says αL is fixed and αR is varied; the caption should read αL = 0.
  3. [Sections III C 3 and III D] The quantity Erad/E is never defined by an equation. Please state how the radiation energy is computed and how the outgoing oscillon energies EL and ER are separated from radiation.
  4. [Figure 38] The amplitude scales in the blow-ups (down to about 3×10^-5) should be stated in the caption together with the grid resolution used, so the reader can judge the smallest scales against the discretization error.
  5. [Section III C 1, Eq. (III.14)] The derivation of ts and Δxmin assumes the border moves freely until it hits the future light cone; the paper should state that this is a kinematic estimate and quantify the agreement in Fig. 19, where no deviations between the dots and the curves are discussed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: scattering outcomes are generated by numerical evolution of exact solutions; fits and self-citations are explicit and not load-bearing.

full rationale

The paper's central results are numerical scattering outcomes starting from exact compact oscillon solutions of the signum-Gordon equation, and those outcomes are not encoded in the inputs by construction. The exact oscillons used as initial data come from prior work [25,26,27], and they are independent exact solutions, not objects defined in terms of the scattering results. The no-radiation windows are found by scanning the velocity and phase parameters (e.g., Figs. 24, 30, 32, 33), not imposed; the special role of phases differing by 1/2 for v = 0 is explained by the derived relation that such a shift changes the field by a sign, which is a symmetry property, not a circular input. The shock-wave comparison in Section III C 2 fits the free parameter a0 and time shift T0 to numerical zero positions and then checks the remaining zeros against the exact shock-wave recurrence; this is an explicit hypothesis test with two fitted parameters and additional non-fitted zeros, not a prediction fabricated from the conclusion. The fractal-like radiation claim is explicitly labeled a conjecture in Section III F ('This statement still has a status of a conjecture'), and its supporting facts are the exact dilation symmetry (III.53), which is an independent mathematical property of the model, and visual blow-ups of nested numerical structures; the dilation symmetry guarantees existence of exact oscillons at arbitrarily small scales but is not used as the definition of the observed radiation. Self-citations to [25,26,27,35,38] provide exact solutions and prior numerics; none of these citations assumes the target claim that scattering radiation is fractal-like. The numerical-resolution caveat in Appendix B ('changing the number of points by a factor of two resulted in the appearance or even disappearance of some tiny structures') is a correctness risk about small-scale reliability, not a circularity, because the claim is not derived from the finite-difference scheme nor from a fitted parameter. Accordingly, no step in the paper's derivation chain reduces, by its own equations or by self-citation, to its own inputs.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis depends on exact prior solutions and symmetries, not on new fitted constants. The only explicit fits are the shock-wave comparison parameters a0 and T0; the no-radiation phases are found by scanning and are listed as free parameters because they are not derived analytically. No new physical entities are introduced.

free parameters (3)
  • a0 (shock-wave zero scale) = 0.00541
    Chosen to match the first ten zero trajectories of the numerical wave to the exact shock-wave solution in Fig.21.
  • T0 (time offset of shock-wave formation) = 0.282
    Chosen jointly with a0 to align the analytic zero hyperbolas with the numerical diamond wave in Fig.21.
  • No-radiation phase pairs (per V and v) = e.g., alpha=0.414 and 0.914 for V=0.93, v=0
    Found numerically by scanning; not derived analytically, and central to the claim that radiation can vanish for fine-tuned phases.
assumptions (4)
  • domain assumption The Euler-Lagrange equation with sgn(phi), with sgn(0)=0, defines a well-posed initial value problem on which the RK4 discretization converges.
    The numerical evolution assumes this; the paper itself notes numerical sensitivity for small oscillons (Appendix B).
  • standard math Partial solutions (I.3) can be patched across sign boundaries into global continuous solutions; delta contributions from step-function boundaries are ignored because the sum is continuous (II.10).
    Used throughout to construct exact oscillons; standard method in signum-Gordon literature.
  • standard math Lorentz covariance and scaling symmetry (III.53) hold and are used to build traveling oscillons and to argue for fractal-like radiation.
    These are exact symmetries of the signum-Gordon equation, not new.
  • domain assumption The estimate of the collision time ts assumes the oscillon borders move freely with constant velocity until they hit the future light cone of the collision event (III.14), (III.19).
    This assumption is used to derive ts and dx_min and is validated only by agreement with numerical data (Fig.19).

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Cite this review

Pith. "Pith review of Scattering of compact oscillons." pith.science (2026). https://pith.science/paper/7II5BFEN

@misc{pith2026190901992,
  author       = {Pith},
  title        = {Pith review of: Scattering of compact oscillons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7II5BFEN}},
  note         = {Machine review of arXiv:1909.01992}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 1909.01992 by the authors.

Figure 1
Figure 1. FIG. 1: The world sheet of the generalized exact oscillon seen in its own rest frame [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Function [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The profile function [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (35 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The generalized exact oscillon with the velocity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Initial field configuration (a) Ψ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Symmetric initial configuration of two compact exact oscillons parametrized by [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Another possible symmetric initial configuration of two compact oscillons parametrized by [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: (a) Oscillon [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Symmetric initial configurations of the oscillons Ψ [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Functions [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Symmetric configurations with [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Generic configurations with [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: The dependence of the scattering of oscillons (antisymmetric configuration) on their initial speed [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: The scattering of oscillons (antisymmetric configuration) as a function of their phase [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: The scattering of oscillons (antisymmetric configuration) in dependence on parameter [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Worldsheets of two incoming oscillons for [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: The scatterings of oscillons (symmetric configuration) as a function of their initial speed [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: The scattering process for initial symmetric configurations containing oscillons with speed [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: The cases of different velocities [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: The exact shock wave solution for [PITH_FULL_IMAGE:figures/full_fig_p025_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Scattering of two exact oscillons for symmetric field configuration. The incident oscillons move [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22: The scatterings of the symmetric initial configuration with [PITH_FULL_IMAGE:figures/full_fig_p027_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23: Initial profile of the signum-Gordon field and worldsheets of the incoming oscillons for [PITH_FULL_IMAGE:figures/full_fig_p027_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24: Fraction of the total energy of initial configuration carried out by radiation for [PITH_FULL_IMAGE:figures/full_fig_p028_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25: Worldsheets of two incoming generalized oscillons with [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26: The scattering process for the initial symmetric configuration containing oscillons with speed [PITH_FULL_IMAGE:figures/full_fig_p029_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27: The scattering process for initial symmetric configuration containing oscillons with speed [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28: The scattering processes for the initial symmetric configuration containing oscillons with speed [PITH_FULL_IMAGE:figures/full_fig_p031_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29: Initial configurations that minimize the radiation for [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30: Fraction of the total energy of the initial configuration carried out by the radiation for [PITH_FULL_IMAGE:figures/full_fig_p031_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31: The scattering process for initial configuration containing oscillons with speed [PITH_FULL_IMAGE:figures/full_fig_p032_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32: Fraction of the total energy of initial configuration carried out by the radiation as a function of [PITH_FULL_IMAGE:figures/full_fig_p033_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33: Fraction of the total energy of the initial configuration carried out by the radiation as a function [PITH_FULL_IMAGE:figures/full_fig_p034_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34: Balance of energy after the interaction (in percentage of incident energy) for each oscillon, where [PITH_FULL_IMAGE:figures/full_fig_p034_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35: Scattering process for [PITH_FULL_IMAGE:figures/full_fig_p035_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36: Borders [PITH_FULL_IMAGE:figures/full_fig_p037_36.png]
Figure 37
Figure 37. Figure 37: FIG. 37: Generalized exact oscillon with uniformly accelerated border for different values of the boost [PITH_FULL_IMAGE:figures/full_fig_p039_37.png]
Figure 38
Figure 38. Figure 38: FIG. 38: Scattering process of two oscillons: (b) blow-up of the rectangular region in (a); (c) blow-up of the [PITH_FULL_IMAGE:figures/full_fig_p042_38.png]

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Works this paper leans on

59 extracted references · 58 canonical work pages · cited by 1 Pith paper

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    For better transparency we present first only the cases with v = 0

    Overview of the numerical results In this section we present a general overview of the numerical results obtained in the scattering of solitons described by symmetric initial configurations parametrized by the boost velocity V and initial phase α. For better transparency we present first only the cases with v = 0. They are sufficient to provide us a basic fee...

    work page

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    Symmetric configurations A numerical study of the scattering process shows that the process depends on many parameters likev1,v2, the relative velocity of the oscillons, their initial distance, time shift and reflections. In order to simplify the set of parameters we have decided to restrict our considerations to symmetric and anti-symmetric initial configur...

    work page

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    Phase of the oscillon Unlike for the case of compact kinks, the scattering process of two oscillons depends on the initial distance between them. This observation follows from the fact that the shape of the oscillons changes with time and the outcome of the scattering process depends strongly 3 on the shapes of oscillons at the moment when their supports ...

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    To obtain it is sufficient to restrict considerations to v≥ 0 because any non-travelling oscillon satisfies relation φ(t,x ;−v) = φ(t, 1−x)

    Determination of the endpoints x(α) and ˜x(α) of a oscillon We start with the determination of the function x0(α) which describes the position of the right endpoint of the oscillon. To obtain it is sufficient to restrict considerations to v≥ 0 because any non-travelling oscillon satisfies relation φ(t,x ;−v) = φ(t, 1−x). So we shall consider here the boosts ...

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    ( A) (−1, 0) for t′∈ (− 1 2, 0) ( B) (1, 0) for t′∈ (0, 1

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    ( C) (−1, 1) for t′∈ ( 1 2, 1) ( D) (1,−1) for t′∈ (1, 3

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    There are two different cases dependent on the absolute value of the boost velocity V =|u|

    ( E) (−1, 2) for t′∈ ( 3 2, 2) ( F ) . There are two different cases dependent on the absolute value of the boost velocity V =|u|. They are separated by the critical case for which the velocity has value Vc≡ 1 2 +v. The straight line y(t′,α ) crosses maxima of τ(t′) i.e. τ(t′ max) = 1 2 at t′ max ={− 1 2, 1 2} for u = +V and at t′ max ={ 1 2, 1} in the cas...

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    In the case of oscillons with|u| < van additional caution is necessary

    Remarks on the initial configurations Having determined the expressions for x0(α) and ˜x0(α) we can now construct arbitrary initial configurations containg generalized exact oscillons which begin to collide at t = 0. In the case of oscillons with|u| < van additional caution is necessary. In Fig.11 we present the worldsheets of two generalized exact oscillon...

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    First we note that for small velocities the numerical solution is very irregular

    High velocities - formation of shock waves Next we discuss in more detail some of our numerical results. First we note that for small velocities the numerical solution is very irregular. Looking at figures Fig.17(a)-(c) we clearly see a formation of a strongly perturbed oscillo...

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    Namely, some symmetric initial configurations evolve in such a way that the resulting field contains only the main quasi- oscillons i.e

    Vanishing of the radiation In our numerical studies we have also spotted a very interesting case. Namely, some symmetric initial configurations evolve in such a way that the resulting field contains only the main quasi- oscillons i.e. the amount of radiation generated in this pr...

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    Moreover, it can also be checked that both initial configurations characterized by phases α and α + 1 2 differ only by the sign i.e

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    Oscillons with v⁄= 0 In this section we present some of our results on the scattering of generalized exact oscillons i.e. oscillons which depend on v – the parameter controlling the swaying motion of the oscillon endpoints. In Fig.25 we show the wordsheets of two such oscillon...

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