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Polar coordinate (r, θ) for y, and the unit vectors (e r , e θ ) in (3.10).
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We consider constructing the relativistic system of collective coordinates of
a field theory soliton on the basis of a simple principle: The collective
coordinates must be introduced into the static solution in such a way that the
equation of motion of the collective coordinates ensures that of the original
field theory. As an illustration, we appl...
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... Thus, we undertake the exploration of the interaction chiefly via direct numerical simulations of the relevant field equation. An Appendix complements our computations with analytical considerations based on the variational technique known as the "collective coordinate approximation," which is widely used in various field-theoretic problems (see, e.g., [7,[25][26][27]30,[43][44][45][46][47][48][49][50]). We note, in passing, that other approaches have been used to interrogate long-range interactions, such as evaluating the field's potential energy at the center of mass of two superimposed (anti)kinks [51,52]. ...
... It is relevant to mention here that the ansatz in Eqs. (27) and (28) is suitable not only for direct numerical simulations, but also for collective coordinate approximations of the PDE dynamics [7,[25][26][27]30,[43][44][45][46][47][48][49][50]. In particular, one can use Eqs. ...
We present a computational analysis of the long-range interactions of solitary waves in higher-order field theories. Our vehicle of choice is the φ8 field theory, although we explore similar issues in example φ10 and φ12 models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard φ4 model. Upon establishing the power-law asymptotics of the model’s solutions’ approach towards one of the steady states, we make the case that such asymptotics require particular care in setting up multisoliton initial conditions. A naive implementation of additive or multiplicative ansätze gives rise to highly pronounced radiation effects and eventually leads to the illusion of a repulsive interaction between a kink and an antikink in such higher-order field theories. We propose and compare several methods for how to “distill” the initial data into suitable ansätze, and we show how these approaches capture the attractive nature of interactions between the topological solitons in the presence of power-law tails (long-range interactions). This development paves the way for a systematic examination of solitary wave interactions in higher-order field theories and raises some intriguing questions regarding potential experimental observations of such interactions. As an Appendix, we present an analysis of kink-antikink interactions in the example models via the method of collective coordinates.
... Using the ansatz in Eq. (45), and defining K + s 1 = K s 1 (x + X(t) − x 0 ) and K − s 2 = K s 2 (x − X(t) + x 0 ), we calculate the coefficient functions in Eq. (34) as follows: ...
We present a computational analysis of the long-range interactions of solitary waves in higher-order field theories. Our vehicle of choice is the $\varphi^8$ field theory, although we explore similar issues in example $\varphi^{10}$ and $\varphi^{12}$ models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard $\varphi^4$ model. Upon establishing the power-law asymptotics of the model's solutions' approach towards one of the steady states, we make the case that such asymptotics require particular care in setting up multi-soliton initial conditions. A naive implementation of additive or multiplicative ans\"atze gives rise to highly pronounced radiation effects and eventually produces the illusion of a repulsive interaction between a kink and an antikink in such higher-order field theories. We propose and compare several methods for how to "distill" the initial data into suitable ans\"atze, and we show how these approaches capture the attractive nature of interactions between the topological solitons in the presence of power-law tails (long-range interactions). This development paves the way for a systematic examination of solitary wave interactions in higher-order field theories and raises some intriguing questions regarding potential experimental observations of such interactions. As an Appendix, we present an analysis of kink-antikink interactions in the example models via the method of collective coordinates.
... There has been some recent interest in spinning Skyrmions, namely [33] and [34], which investigate an extension of the collective coordinate quantisation procedure. The related question of isospin was examined in [14] where the authors considered the deformation introduced by isospinning Skyrmions. ...
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We study kink scattering processes in the (1+1)-dimensional φ6 model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical
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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 $\Pi_3(\mathbb{C}P^1)=\mathbb{Z}$ then for $N=1$ the term becomes integer. On the other hand for $N>1$ it became perturbative because $\Pi_3(\mathbb{C}P^N)$ is trivial. The prefactor of the Hopf term (anyon angle) $\Theta$ is not quantized and its value depends on the physical system. The corresponding fermionic models can fix value of the angle $\Theta$ 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 $\Theta=n\pi, n\in \mathbb{Z}$. We quantize the solutions and calculate several mass spectra for $N=2$. Finally we discuss generalization for $N\geqq 3$.
We calculate the leading relativistic correction to the rigid body approximation used by Adkins and Nappi (and Witten) for the collective coordinate quantization of a Skyrmion with baryon number one. We derive a field theory solution of a spinning Skyrmion, expanding the field configuration in powers of the angular velocity, and then obtain the relativistic corrections to the decay constant and various static properties of nucleons. which amount to 5% – 20%.
We give a systematic method to calculate higher derivative corrections to
low-energy effective theories of solitons, which are in general non-linear
sigma models on the moduli spaces of the solitons. By applying it to the
effective theory of a single BPS non-Abelian vortex in U(N) gauge theory with N
fundamental Higgs fields, we obtain four derivative corrections to the
effective sigma model on the moduli space C \times CP^{N-1}. We compare them
with the Nambu-Goto action and the Faddeev-Skyrme model. We also show that
Yang-Mills instantons/monopoles trapped inside a non-Abelian vortex
membrane/string are not modified in the presence of higher derivative terms.
