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Pawel Klimas

Publications and source records attributed to Pawel Klimas.

10 recordsLinked to original sources
arXiv2026

Signum-Gordon spectral mass from nonlinear Fourier mode mixing

We investigate the emergence of a spectral mass in the signum-Gordon model, a nonlinear field theory characterized by a non-analytic, V-shaped potential where standard perturbative mass definitions are inapplicable. By analyzing the evolution of monochromatic wave trains, we identify two distinct dynamical regimes governed by the relationship between the wave's amplitude and its wavenumber. In the nonlinear regime, the model exhibits nonlinear Fourier mode mixing, where the potential's lack of analyticity acts as a source that populates higher-order harmonics. Using two complementary numerical methods -- tracking frequency distributions from initial wavenumbers and measuring spatial responses to boundary signals -- we construct comprehensive dispersion maps in energy-momentum space. Our results demonstrate that the signum-Gordon field effectively mimics a massive theory. Specifically, we show that a particular initial wave amplitude induces a spectral mass of unity, perfectly matching the behavior of the massive Klein-Gordon equation and providing a robust framework for quantifying mass in non-analytic scalar models.

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arXiv2024

Signum-Gordon shock waves in (2+1) and (3+1) dimensions

This study introduces novel, exact solutions to the scalar field Signum-Gordon equation that feature a discontinuity near the light cone. These solutions, applicable in higher spatial dimensions ($n > 1$), extend previous limitations to one dimension. Our chosen ansatz leads to an ordinary equation with exact solutions obtained for $n = 2$ and $3$ spatial dimensions. The shock wave's energy trapped within the light cone is proportional to the wave's $n$-dimensional volume and the field discontinuity at the wavefront. The investigation delves further into the behavior of shock waves when their driving force, represented by a delta function at the light cone, is disabled. Disabling this delta function disrupts energy transfer, preventing the wave's propagation as predicted by analytical calculations. We identify the region within the light cones where the field remains unaffected. Two-dimensional ($n = 2$) simulations reveal the formation of intriguing structures upon source removal. These structures include a central, stable feature resembling an oscillon and a surrounding ring that breaks down into smaller oscillations.

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arXiv2019

Gravitating compact $Q$-ball and $Q$-shell solutions in the $\mathbb{C}P^N$ nonlinear sigma model

We study compact gravitating $Q$-ball, $Q$-shell solutions in a sigma model with the target space $\mathbb{C}P^N$. Models with odd integer $N$ and suitable potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions. The energy of the solutions behaves as $E\sim |Q|^{5/6}$, where $Q$ stands for the $U(1)$ Noether charge, for both the gravitating and the black hole solutions.Notable difference from the solutions of the flat space is that upper bound of $|Q|$ appears when the coupling with gravity is stronger. The maximal value of $|Q|$ quickly reduces for larger coupling constant. It may give us a useful hint of how a star forms its shape with a certain finite number of particles.

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arXiv2016

Collective coordinate quantization and spin statistics of the solitons in the $\mathbb{C}P^N$ Skyrme-Faddeev model

The $\mathbb{C}P^N$ extended Skyrme-Faddeev model possesses planar soliton solutions. We consider quantum aspects of the solutions applying collective coordinate quantization in regime of rigid body approximation. In order to discuss statistical properties of the solutions we include an Abelian Chern-Simons term (the Hopf term) in the Lagrangian. Since $Π_3(\mathbb{C}P^1)=\mathbb{Z}$ then for $N=1$ the term becomes an integer. On the other hand for $N>1$ it became perturbative because $Π_3(\mathbb{C}P^N)$ is trivial. The prefactor of the Hopf term (anyon angle) $Θ$ is not quantized and its value depends on the physical system. The corresponding fermionic models can fix value of the angle $Θ$ for all $N$ in a way that the soliton with $N=1$ is not an anyon type whereas for $N>1$ it is always an anyon even for $Θ=nπ, n\in \mathbb{Z}$. We quantize the solutions and calculate several mass spectra for $N=2$. Finally we discuss generalization for $N\geqq 3$.

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arXiv2016

Composite BPS skyrmions from an exact isospin symmetry breaking

We study the BPS Skyrme model with potentials breaking the isospin symmetry and analyse how properties of exact solitonic solutions depend on a form of the isospin breaking potential. In the case of the strong symmetry breaking a new topologic structure is observed which enables us to decompose a BPS skyrmion into a lower dimensional defect localised on a brane (kink). We investigate some thermodynamical properties of such solitons as well as the role of the symmetry breaking potential in the resulting mean-field equation of state.

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arXiv2016

Quasi-integrable deformations of the $SU(3)$ Affine Toda Theory

We consider deformations of the $SU(3)$ Affine Toda theory (AT) and investigate the integrability properties of the deformed theories. We find that for some special deformations all conserved quantities change to being conserved only asymptotically, {\it i.e.} in the process of the scattering of two solitons these charges do vary in time, but they return, after the scattering, to the values they had prior to the scattering. This phenomenon, which we have called quasi-integrability, is related to special properties of the two-soliton solutions under space-time parity transformations. Some properties of the AT solitons are discussed, especially those involving interesting static multi-soliton solutions. We support our analytical studies with detailed numerical ones in which the time evolution has been simulated by the 4th order Runge-Kutta method. We find that for some perturbations the solitons repel and for the others they form a quasi-bound state. When we send solitons towards each other they can repel when they come close together with or without `flipping' the fields of the model. The solitons radiate very little and appear to be stable. These results support the ideas of quasi-integrability, {\it i.e.} that many effects of integrability also approximately hold for the deformed models.

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arXiv2015

Potentials and the vortex solutions in the $CP^N$ Skyrme-Faddeev model

The extended Skyrme-Faddeev model possesses vortex solutions in a (3+1) dimensional Minkowski space-time with target space $CP^N$. They have finite energy per unit of length and contain waves propagating along vortices with the speed of light. We introduce various types of the potentials which correspond with holomorphic solutions of the integrable sector and also with several numerical solutions outside of this sector. The presented solutions constitute a strong indication that the current model contains large class of solutions with much wider range of coupling constants than the previously known exact solution.

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arXiv2008

Perturbations of the signum-Gordon model

We investigate a perturbation of a scalar field model (called here the signum-Gordon model) with the potential $V(f)=|f|$. The perturbation generalizes the signum-Gordon model to the signum-Klein-Gordon model i.e. to the case $V(f)=|f|-{1/2}λf^2$, where $λ$ is a small parameter. Such a generalization breaks the scaling symmetry of the signum-Gordon model. In this paper we concentrate on solutions for self-similar initial data. Such data are particulary useful for identification of the effects caused by the term that breaks the scaling symmetry. We have found that the behaviour of the solutions is quite interesting - they escape and return periodically to the self-similar initial data.

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arXiv2006

On Shock Waves in Models with V-Shaped Potentials

The recently found shock wave solution in the scalar field model with the field potential $V(ϕ)=|ϕ|$ is generalized to the case $V(ϕ)=|ϕ|-{1/2}λϕ^2$. We find two kinds of the shock waves, which are analogous of compression and expansion waves. The dependence of the waves on the parameter $λ$ is investigated in detail.

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