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Using a renormalization-inspired perturbation expansion we show that oscillons in a generic field theory in (1+1)-dimensions arise as dressed Q-balls of a universal (up to the leading nonlinear order) complex field theory. This theory reveals a close similarity to the integrable complex sine-Gordon model which possesses exact multi-$Q$-balls. We show that excited oscillons, with characteristic modulations of their amplitude, are two-oscillons bound states generated from a two $Q$-ball solution., Comment: 5 pages, 3 figures

We show that the decay of strongly excited oscillons in a single vacuum model reveals a chaotic, fractal-like pattern very much like one found in kink-antikink collision in the $\phi^4$ model. This structure can be attributed to the resonant energy transfer mechanism triggered by the modulations of amplitudes of constituent oscillons which form the excited oscillon. We also find evidence that such modulations arise as a motion of two quasi-breathers inside the constituent oscillon., Comment: 5 pages, 8 figures

We show that large scale oscillons, i.e., quasi-periodic, long living particle like solutions, may exist in massless theories, too. Their existence is explained using an effective (smeared) mass threshold which takes into account nonlinear (finite) perturbations., Comment: 6 pages, 13 figures, 2 movies

We find that an oscillon can possess a characteristic double oscillation structure even though it results in a decay of a sphaleron which does not have any positive energy vibrational mode. We show that dynamics of such an oscillon can still be captured by collective coordinates provided by the sphaleron. Namely, its unstable mode and its scaling deformation i.e., Derrick mode., Comment: 10 pages, 12 figures

We construct a simple field theory in which a sphaleron, i.e., a saddle-point particle-like solution, forms a semi-BPS state with a background defect that is an impurity. This means that there is no static force between the sphaleron and the impurity. Therefore such a sphaleron-impurity system is very much like usual BPS multi-solitons, however, still possessing an unstable direction allowing for its decay. We study dynamics of the sphaleron in such a system., Comment: 13 pages, 12 figures, references added and improved discussion

We analyze the perturbative Relativistic Moduli Space approach, where the amplitudes of the Derrick modes are promoted to collective coordinates. In particular, we analyse the possibility to calculate the critical velocity, i.e., the initial velocity of kinks at which single bounce scattering changes into a multi-bounce or annihilation collision, in the resulting Collective Coordinate Model (CCM). We find that for a growing number of modes the critical velocity of the CCM approaches the full field theory value. This is verified in the case of the $\phi^4$ model, where we reach a $99\%$ accuracy. We also see such a convergence for a wide range of models belonging to the family of the double sine-Gordon and Christ-Lee theories, especially in those cases where the kinks do not reveal a too well pronounced half-kink inner structure., Comment: 12 pages

Oscillons in a simple, 1-dimensional scalar field theory with a cubic potential are discussed. The theory has a classical sphaleron, whose decay generates a version of the oscillon. A good approximation to the small-amplitude oscillon is constructed explicitly using the asymptotic expansion of Fodor et al., but for larger amplitudes a better approximation uses the discrete, unstable and stable deformation modes of the sphaleron., Comment: 26 pages, 18 figures

We show that constrained $CP^1$ instantons, combined with the Relativistic Moduli Space approach, can accurately describe kink-antikink collisions in the $\phi^4$ model., Comment: 8 pages, 5 figures

We find that for various solitonic processes the corresponding canonical moduli space can have a boundary which is accessible in a finite time evolution. We show that such a boundary is not a failure of the moduli space approach but has a physical meaning. In our example, it corresponds to the complete annihilation of a colliding kink and antikink after a finite time. We further find that, close to the boundary, the solutions have an approximate self-similar form., Comment: 8 pages, 8 figures

We show that thick spectral walls exist in antikink-kink collisions in the $\phi^6$ model. In this model, they are triggered by the so-called $delocalized$ $modes$ which do not exist in the single-soliton sector but emerge in antikink-kink ($\bar{K}K$) collisions. Therefore, spectral walls are a rather common phenomenon that should occur in many solitonic collisions involving non-symmetric kinks as, e.g., in the $\phi^8$ or higher power models., Comment: 6 pages, 4 figures

Antikink-kink ( $\bar{\rm K} $$ {\rm K}$) collisions in the $\phi^6$ model exhibit resonant scattering although the $\phi^6$ kinks do not support any bound states to which energy could be transferred. In Phys. Rev. Lett. 107 (2011) 091602 it was conjectured that, instead, energy is transferred to a collective bound mode of the full $\bar{\rm K} $$ {\rm K}$ configuration. Here we present further strong evidence for this conjecture. Further, we construct a collective coordinate model (CCM) for $\bar{\rm K} $$ {\rm K}$ scattering based on this collective bound mode trapped between the $\bar{\rm K} $$ {\rm K}$ pair which allows us to reproduce the full dynamics of $\bar{\rm K} $$ {\rm K}$ scattering with striking accuracy. We also study kink-antikink (${\rm K} $$ \bar{\rm K}$) scattering and its description by a CCM. In this case a significant role of radiation is discovered., Comment: 17 pages, 17 figures

The moduli space approximation to kink dynamics permits a relativistic generalization if the Derrick scaling parameter is used as a collective coordinate. We develop a perturbative approach to the resulting relativistic moduli space by expanding the Derrick scaling parameter about unity and treating the higher-order Derrick modes as new degrees of freedom. This approach allows us to resolve (coordinate) singularities order-by-order, and systematically incorporates relativistic corrections {\it perturbatively} in kink scattering. It gives an excellent description of kink-antikink collisions in $\phi^4$ field theory already at first order, and at higher order, reproduces the fractal structure in the formation of the final state with an error of only $4\%$., Comment: v4: Minor revisions. This version accepted for publication in Physical Review D

We show that in some kink-antikink (KAK) collisions sphalerons, i.e., unstable static solutions - rather than the asymptotic free soliton states - can be the source of the internal degrees of freedom (normal modes) which trigger the resonance phenomenon responsible for the fractal structure in the final state formation., Comment: 12 pages, 6 figures

The fractal velocity pattern in symmetric kink-antikink collisions in $\phi^4$ theory is shown to emerge from a dynamical model with two effective moduli, the kink-antikink separation and the internal shape mode amplitude. The shape mode usefully approximates Lorentz contractions of the kink and antikink, and the previously problematic null-vector in the shape mode amplitude at zero separation is regularized., Comment: the metric and potential of the effective model are now included

We show that spectral walls are common phenomena in the dynamics of kinks in (1+1) dimensions. They occur in models based on two or more scalar fields with a nonempty Bogomol'nyi-Prasam-Sommerfield (BPS) sector, hosting two zero modes, where they are one of the main factors governing the soliton dynamics. We also show that spectral walls appear as singularities of the dynamical vibrational moduli space., Comment: improved, references added

We discover a new class of topological solitons. These solitons can exist in a space of infinite volume like, e.g., $\mathbb{R}^n$, but they cannot be placed in any finite volume, because the resulting formal solutions have infinite energy. These objects are, therefore, interpreted as totally incompressible solitons. As a first, particular example we consider (1+1) dimensional kinks in theories with a nonstandard kinetic term or, equivalently, in models with the so-called runaway (or vacummless) potentials. But incompressible solitons exist also in higher dimensions. As specific examples in (3+1) dimensions we study Skyrmions in the dielectric extensions both of the minimal and the BPS Skyrme models. In the the latter case, the skyrmionic matter describes a completely incompressible topological perfect fluid., Comment: reference added

Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $\phi^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to be extended from real to imaginary values, although the field remains real. The previously discussed null-vector problem for the shape modes of $\phi^4$ kinks is resolved by a better coordinate choice. In sine-Gordon theory, moduli spaces can be constructed using exact solutions at the critical energy separating scattering and breather (or wobble) solutions; here, energy conservation relates the metric and potential. The reduced dynamics on these moduli spaces accurately reproduces properties of the exact solutions over a range of energies., Comment: presentation improved, new plots added

We study kink-antikink scattering in a one-parameter variant of the $\phi^4$ theory where the model parameter controls the static intersoliton force. We interpolate between the limit of no static force (BPS limit) and the regime where the static interaction is small (non-BPS). This allows us to study the impact of the strength of the intersoliton static force on the soliton dynamics. In particular, we analyze how the transition of a bound mode through the mass threshold affects the soliton dynamics in a generic process, i.e., when a static intersoliton force shows up. We show that the thin, precisely localized spectral wall which forms in the limit of no static force, broadens in a well-defined manner when a static force is included, giving rise to what we will call a thick spectral wall. This phenomenon just requires that a discrete mode crosses into the continuum at some intermediate stage of the dynamics and, therefore, should be observable in many soliton-antisoliton collisions., Comment: version accepted in Phys. Rev. E

Kink-antikink scattering in the $\phi^4$ model is investigated in the limit when the static inter-soliton force vanishes. We observe the formation of spectral walls and, further, identify a new phenomenon, the vacuum wall, whose existence gives rise to a bouncing structure for the annihilating solitons. Furthermore, we discover higher order spectral walls, i.e., spectral walls which form when higher harmonics enter the continuous spectrum. These higher order spectral walls not only deform the soliton trajectories, they also can be observed easily as very intense radiation bursts.

We find a family of (half) self-dual impurity models such that the self-dual (BPS) sector is exactly solvable, for any spatial distribution of the impurity, both in the topologically trivial case and for kink (or antikink) configurations. This allows us to derive the metric on the corresponding one-dimensional moduli space in an analytical form. Also the generalized translational symmetry is found in an exact form. This symmetry provides a motion on moduli space which transforms one BPS solution into another. Finally, we analyse exactly how vibrational properties (spectral modes) of the BPS solutions depend on the actual position on moduli space. These results are obtained both for the nontrivial topological sector (kinks or antikinks) as well as for the topologically trivial sector, where the motion on moduli space represents a kink-antikink annihilation process., Comment: sec. 6.1 corrected

During defect-antidefect scattering, bound modes frequently disappear into the continuous spectrum before the defects themselves collide. This leads to a structural, nonperturbative change in the spectrum of small excitations. Sometimes the effect can be seen as a hard wall from which the defect can bounce off. We show the existence of these spectral walls and study their properties in the $\phi^4$ model with BPS preserving impurity, where the spectral wall phenomenon can be isolated because the static force between the antikink and the impurity vanishes. We conclude that such spectral walls should surround all solitons possessing internal modes., Comment: latex, 5 pages, 3 figures; v2: version published in PRL

We present a field theoretical model which allows for domain walls that do not get stuck on impurities. For this purpose, we consider a generalized chiral magnet domain wall action with an impurity which couples both to a potential term as well as to the Dzyaloshinskii-Moriya interaction energy. We show that in the so-called critical coupling limit the interaction energy between two types of domain walls and the impurity vanishes. Therefore, at low velocity, such domain walls pass freely through the impurity with almost no loss of energy., Comment: LaTex, 15 pages, 6 figures

The $\phi^4$ model is coupled to an impurity in a way that preserves one-half of the BPS property. This means that the antikink-impurity bound state is still a BPS solution, i.e., a zero-pressure solution saturating the topological energy bound. The kink-impurity bound state, on the other hand, does not saturate the bound, in general. We found that, although the impurity breaks translational invariance, it is, in some sense, restored in the BPS sector where the energy of the antikink-impurity solution does not depend on their mutual distance. This is reflected in the existence of a generalised translational symmetry and a zero mode. We also investigate scattering processes. In particular, we compare the antikink-impurity interaction close to the BPS regime, which presents a rather smooth, elastic like nature, with other scattering processes. However, even in this case, after exciting a sufficiently large linear mode on the incoming antikink, we can depart from the close-to-BPS regime. This results, for example, in a backward scattering., Comment: 49 pages, 30 figures, LaTex file; v2: minor changes: typos corrected, some comments and references added

We introduce an idea of experimental verification of the counterintuitive negative radiation pressure effect in some classical field theories by means of buckled graphene. In this effect, a monochromatic plane wave interacting with topological solutions pulls these solutions towards the source of radiation. Using extensive molecular dynamics simulations, we investigate the traveling wave-induced motion of kinks in buckled graphene nanoribbons. It is shown that depending on the driving source frequency, amplitude and direction, the kink behavior varies from attraction to repulsion (the negative and positive radiation pressure effects, respectively). Some preliminary explanations are proposed based on the analogy to certain field theory models. Our findings open the way to a new approach to motion control on the nanoscale.

The Bogomolnyi-Prasad-Sommerfield (BPS) baby Skyrme model coupled to gravity is considered. We show that in an asymptotically flat space-time the model still possesses the BPS property, i.e., admits a BPS reduction to first order Bogomolnyi equations, which guarantees that the corresponding proper energy is a linear function of the topological charge. We also find the mass-radius relation as well as the maximal mass and radius. All these results are obtained in an analytical manner, which implies the complete solvability of this selfgravitating matter system. If a cosmological constant is added, then the BPS property is lost. In de Sitter (dS) space-time both extremal and non-extremal solutions are found, where the former correspond to finite positive pressure solutions of the flat space-time model. For the asymptotic anti-de Sitter (AdS) case, extremal solutions do not exist as there are no negative pressure BPS baby Skyrmions in flat space-time. Non-extremal solutions with AdS asymptotics do exist and may be constructed numerically. The impact of the negative cosmological constant on the mass-radius relation is studied. We also found two potentials for which exact multi-soliton solutions in the external AdS space can be obtained. Finally, we elaborate on the implications of these findings for certain three-dimensional models of holographic QCD., Comment: Latex file, 34 pages, 13 figures, v2: conclusions on the stability of AdS Skyrmions weakened, results and remaining conclusions unchanged

Radial vibrations of charge one hedgehog Skyrmions in the full Skyrme model are analysed. We investigate how the properties of the lowest resonance modes (quasi normal modes) - their frequencies and widths - depend on the form of the potential (value of the pion mass as well as the addition of further potentials) and on the inclusion of the sextic term. Then we consider the inverse problem, where certain values for the frequencies and widths are imposed, and the field theoretic Skyrme model potential giving rise to them is reconstructed. This latter method allows to reproduce the physical Roper resonances, as well as further physical properties of nucleons, with high precision., Comment: LaTex, 24 pages, 18 figures

We study radial vibrations of spherically symmetric skyrmions in the BPS Skyrme model. Concretely, we numerically solve the linearised field equations for small fluctuations in a skyrmion background, both for linearly stable oscillations and for (unstable) resonances. This is complemented by numerical solutions of the full nonlinear system, which confirm all the results of the linear analysis. In all cases, the resulting fundamental excitation provides a rather accurate value for the Roper resonance, supporting the hypothesis that the BPS Skyrme model already gives a reasonable approximate description of this resonance. Further, for many potentials additional higher resonances appear, again in agreement with known experimental results., Comment: Latex, 41 pages, 22 pdf figures; v2: minor changes

We demonstrate that the gauged BPS baby Skyrme model with a double vacuum potential allows for phase transitions from a non-solitonic to a solitonic phase, where the latter corresponds to a ferromagnetic liquid. Such a transition can be generated by increasing the external pressure $P$ or by turning on an external magnetic field $H$. As a consequence, the topological phase where gauged BPS baby skyrmions exist, is a higher density phase. For smaller densities, obtained for smaller values of $P$ and $H$, a phase without solitons is reached. We find the critical line in the $P,H$ parameter space. Furthermore, in the soliton phase, we find the equation of state for the baby skyrmion matter $V=V(P,H)$ at zero temperature, where $V$ is the "volume", i.e., area of the solitons., Comment: Latex file, 16 pages, 13 figures

The magnetothermodynamics of skyrmion type matter described by the gauged BPS baby Skyrme model at zero temperature is investigated. We prove that the BPS property of the model is preserved also for boundary conditions corresponding to an asymptotically constant magnetic field. The BPS bound and the corresponding BPS equations saturating the bound are found. Further, we show that one may introduce pressure in the gauged model by a redefinition of the superpotential. Interestingly, this is related to non-extremal type solutions in the so-called fake supersymmetry method. Finally, we compute the equation of state of magnetized BSP baby skyrmions inserted into an external constant magnetic field $H$ and under external pressure $P$, i.e., $V=V(P,H)$, where $V$ is the "volume" (area) occupied by the skyrmions. We show that the BPS baby skyrmions form a ferromagnetic medium., Comment: Latex, 39 pages, 14 figures. v2: New results and references added, physical interpretation partly changed

We present a numerical study of the process of the kink-antikink collisions in the coupled one-dimensional two-component $\phi^4$ model. Our results reveal two different soliton solutions which represent double kink configuration and kink-non-topological soliton (lump) bound state. Collision of these solitons leads to very reach resonance structure which is related to reversible energy exchange between the kinks, non-topological solitons and the internal vibrational modes. Various channels of the collisions are discussed, it is shown there is a new type of self-similar fractal structure which appears in the collisions of the relativistic kinks, there the width of the resonance windows increases with the increase of the impact velocity. An analytical approximation scheme is discussed in the limit of the perturbative coupling between the sectors. Considering the spectrum of linear fluctuations around the solitons we found that the double kink configuration is unstable if the coupling constant between the sectors is negative., Comment: 21 pages, 19 figures

We present a numerical study of the process of production of kink-antikink pairs in the collision of particle-like states in the one-dimensional $\phi^4$ model. It is shown that there are 3 steps in the process, the first step is to excite the oscillon intermediate state in the particle collision, the second step is a resonance excitation of the oscillon by the incoming perturbations, and finally, the soliton-antisoliton pair can be created from the resonantly excited oscillon. It is shown that the process depends fractally on the amplitude of the perturbations and the wave number of the perturbation. We also present the effective collective coordinate model for this process., Comment: 4 pages, 4 figures, revtex4

A restriction of the baby Skyrme model consisting of the quartic and potential terms only is investigated in detail for a wide range of potentials. Further, its properties are compared with those of the corresponding full baby Skyrme models. We find that topological (charge) as well as geometrical (nucleus/shell shape) features of baby skyrmions are captured already by the soliton solutions of the restricted model. Further, we find a coincidence between the compact or non-compact nature of solitons in the restricted model, on the one hand, and the existence or non-existence of multi-skyrmions in the full baby Skyrme model, on the other hand., Comment: latex, 18 pages, 2 figures; some typos corrected

The strongly coupled limit of the Skyrme-Faddeev-Niemi model (i.e., without quadratic kinetic term) with a potential is considered on the spacetime S^3 x R. For one-vacuum potentials two types of exact Hopf solitons are obtained. Depending on the value of the Hopf index, we find compact or non-compact hopfions. The compact hopfions saturate a Bogomolny bound and lead to a fractional energy-charge formula E \sim |Q|^{1/2}, whereas the non-compact solitons do not saturate the bound and give E \sim |Q|. In the case of potentials with two vacua compact shell-like hopfions are derived. Some remarks on the influence of the potential on topological solutions in the full Skyrme-Faddeev-Niemi model or in (3+1) Minkowski space are also made., Comment: Latex, 25 pages, 9 figures, extensive revision; a section on the comparison with the full model added