Solitons emerge in various non-linear systems as stable localized configurations, behaving in many ways like particles, from non-linear optics and condensed matter to nuclear physics, cosmology and supersymmetric theories This book provides an introduction to integrable and non-integrable scalar field models with topological and non-topological soliton solutions Focusing on both topological and non-topological solitons, it brings together debates around solitary waves and construction of soliton solutions in various models and provides a discussion of solitons using simple model examples These include the Kortenweg-de-Vries system, sine-Gordon model, kinks and oscillons, and skyrmions and hopfions The classical field theory of scalar field in various spatial dimensions is used throughout the book in presentation of related concepts, both at the technical and conceptual level Providing a comprehensive introduction to the description and construction of solitons, this book is ideal for researchers and graduate students in mathematics and theoretical physics

Topological and Non-Topological Solitons in Scalar Field Theories

Kink-antikink collisions and multi-bounce resonance windows in higher-order field theories

We present TerraPN, a novel method that learns the surface properties (traction, bumpiness, deformability, etc.) of complex outdoor terrains directly from robot-terrain interactions through self-supervised learning, and uses it for autonomous robot navigation. Our method uses RGB images of terrain surfaces and the robot's velocities as inputs, and the IMU vibrations and odometry errors experienced by the robot as labels for self-supervision. Our method computes a surface cost map that differentiates smooth, high-traction surfaces (low navigation costs) from bumpy, slippery, deformable surfaces (high navigation costs). We compute the cost map by non-uniformly sampling patches from the input RGB image by detecting boundaries between surfaces resulting in low inference times (47.27% lower) compared to uniform sampling and existing segmentation methods. We present a novel navigation algorithm that accounts for a surface's cost, computes cost-based acceleration limits for the robot, and dynamically feasible, collision-free trajectories. TerraPN's surface cost prediction can be trained in ∼ 25 minutes for five different surfaces, compared to several hours for previous learning-based segmentation methods. In terms of navigation, our method outperforms previous works in terms of success rates (up to 35.84% higher), vibration cost of the trajectories (up to 21.52% lower), and slowing the robot on bumpy, deformable surfaces (up to 46.76% slower) in different scenarios. 

/pdf/terrapn-unstructured-terrain-navigation-using-online-self-2zpqibuj.pdf

TerraPN: Unstructured Terrain Navigation using Online Self-Supervised Learning

For kink–antikink scattering within the φ4 non-linear field theory in one space and one time dimension resonance type configurations emerge when the relative velocity between kink and antikink falls below a critical value. It has been conjectured that the vibrational excitation of the kink would be the source for these resonances because (simplified) collective coordinate calculations, that emphasized on this excitation, qualitatively reproduced those resonances. Surprisingly a numerical study in the ϕ6 field theory also exhibited such resonances even though it does not contain the vibrational excitation. To explore this contradiction we start from the working hypothesis that in either model any collective coordinate ansatz which includes a degree of freedom similar to the vibrational excitation leads to resonances, regardless of whether or not this mode emerges as a solution to the (linearized) field equations. To this end we compare numerical results in the φ4 and ϕ6 models that arise from the full set of partial differential equations to those from the ordinary differential equations for a collective coordinate ansatz. An inaccuracy in literature formulas for the collective coordinate approach in the φ4 model requires to revisit those calculations.

Kink-Antikink Scattering in φ^4 and φ^6 Models

We perform full two-dimensional (2D) numerical relaxations of isospinning soliton solutions in the baby Skyrme model in which the global $O(3)$ symmetry is broken by the 2D analogue of the pion mass term in the Skyrme model. In our calculations we explicitly allow the isospinning solitons to deform and to break the symmetries of the static configurations. We find that stable isospinning baby Skyrme solutions can be constructed numerically for all angular frequencies $\ensuremath{\omega}\ensuremath{\le}\mathrm{min}﻿(\ensuremath{\mu},1)$, where $\ensuremath{\mu}$ is the mass parameter of the model. Stable, rotationally symmetric baby Skyrmion solutions for higher angular velocities are simply an artefact of the hedgehog approximation. Isospinning multisoliton solutions of topological charge $B$ turn out to be unstable to break up into their $B$ charge-1 constituents at some critical breakup frequency value. Furthermore, we find that for $\ensuremath{\mu}$ sufficiently large the rotational symmetry of charge-2 baby Skyrmions becomes broken at a critical angular frequency $\ensuremath{\omega}$.

/pdf/isospinning-baby-skyrmion-solutions-2i469r6nik.pdf

Isospinning baby Skyrmion solutions

We study kink-antikink collisions in a model which interpolates smoothly between the completely integrable sine-Gordon theory, the $\phi^4$ model, and a $\phi^6$-like model with three degenerate vacua. We find a rich variety of behaviours, including integrability breaking, resonance windows with increasingly irregular patterns, and new types of windows near the $\phi^6$-like regime. False vacua, extra kink modes and kink fragmentation play important roles in the explanations of these phenomena. Our numerical studies are backed up by detailed analytical considerations.

/pdf/resonance-structures-in-kink-antikink-collisions-in-a-4uemiv5kra.pdf

Resonance structures in kink-antikink collisions in a deformed sine-Gordon model

We discuss how internal rotation with fixed angular frequency can affect the solitons in the baby Skyrme model in which the global O(3) symmetry is broken to the SO(2). Two particular choices of the potential term are considered, the "old" potential and the "new" double vacuum potential, We do not impose any assumptions about the symmetry on the fields. Our results confirm existence of two types of instabilities determined by the relation between the mass parameter of the potential and the angular frequency.

/pdf/isorotating-baby-skyrmions-3y58wtvikv.pdf

Isorotating Baby Skyrmions

We present an overview of multisoliton chains arising in various non-integrable field theories and discuss different mechanisms which may lead to the occurrence of such axially-symmetric classical solutions. We explain the pattern of interactions between different solitons, in particular Q-balls, Skyrmions, and monopoles, and show how chains of interacting non-BPS solitons may form in a dynamic equilibrium between repulsive and attractive forces.

Chains of Interacting Solitons

We introduce a system composed of two ($2+1$)-dimensional baby-Skyrmion models (BSMs) set on parallel planes and linearly coupled by tunneling of fields. This system can be realized in a dual-layer ferromagnetic medium. Unlike dual-core models previously studied in nonlinear optics and Bose-Einstein condensates, here the symmetry-breaking bifurcation (SBB) in solitons (baby Skyrmions) occurs with the increase of the intercore coupling ($\ensuremath{\kappa}$), rather than with its decrease, due to the fact that, even in the uncoupled system, neither core may be empty. Prior to the onset of the symmetry breaking between the two components of the solitons, they gradually separate in the opposite directions, due to the increase of $\ensuremath{\kappa}$, which is explained in an analytical form by means of an effective interaction potential. Such evolution scenarios are produced for originally symmetric states with topological charges in the two cores, ${B}^{(1)}={B}^{(2)}=1$, 2, 3, 4. The evolution of mixed states, of the $({B}^{(1)},{B}^{(2)})=(1,2)$ and (2,4) types, with the variation of $\ensuremath{\kappa}$ is studied too.

Spontaneous symmetry breaking in dual-core baby-Skyrmion systems

We study effects of backreaction of the fermionic modes localized by the baby Skyrmion in the (2+1)-dimensional Skyrme model. It is shown that there is a tower of fermionic modes of two different types, localized by the soliton, however there is only one fermionic level, which flows from positive to negative value as coupling increases. Considering the strong coupling regime we observe that the coupling of the bosonic field to the fermions may strongly deform the Skyrmion, in particular the regions of negative topological charge density appear.

/pdf/soliton-solutions-of-the-fermion-skyrmion-system-in-2-1-4j4yexmpit.pdf

Soliton solutions of the fermion-Skyrmion system in (2+1) dimensions.

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