Classical field-theory soliton

A classical field-theory soliton is a smooth, localized finite-energy solution that behaves as a persistent object. Stability may follow from a topological charge or from a balance among energy terms with different scaling behavior.

A topological charge is an integer or other discrete invariant determined by the homotopy class or characteristic class of a field configuration. Continuous finite-energy deformations cannot change it without crossing a singular or forbidden configuration.

For negative frequency $E$, the one-dimensional focusing nonlinear Schrödinger equation has the localized stationary profileGalilean boosts generate moving profiles with a translational collective coordinate and constant velocity.

A collective-coordinate approximation substitutes a finite-parameter soliton ansatz into the field action and integrates over space. Its Euler-Lagrange equations approximate the slow motion of the soliton parameters.

Derrick's scaling argument tests a static field configuration by rescaling space. A soliton can be stationary only if energy terms with opposite scaling powers balance at the original scale.

Under Derrick scaling, an energy term containing $m$ spatial derivatives and homogeneous of degree $m$ in those derivatives scales as ${\lambda }^{m-D}$ in $D$ spatial dimensions.

Stationarity of a static solution under Derrick scaling requires $dE(\lambda )/d\lambda {|}_{\lambda =1}=0$. This necessary relation among the separately scaling energy terms is the Derrick virial identity.

A scalar-field kink is a one-dimensional finite-energy static solution approaching different vacua as $x\to -\infty$ and $x\to +\infty$. Its boundary values define a topological sector.

Kinks in a phi-six model commonly arise from a nonnegative degree-six potential with three degenerate vacua. For the unit-vacuum normalizationthe kink joining $0$ to $1$ isand has mass $1/4$. A rescaled model with $U(\varphi )={\varphi }^2({\varphi }^2-4{)}^2$ has the elementary kinkand its mass is $4\sqrt{2}$ in the normalization $E=\int [({\varphi }^{\prime }{)}^2/2+U]dx$.

If a proposed kink must cross a third degenerate vacuum, its first integral makes both the field derivative and potential vanish there. ODE uniqueness prevents it from crossing at finite distance, so the two elementary kink sectors can be joined only at infinite separation.

A gauge-theory soliton is a finite-energy localized classical field configuration stabilized by topology or a balance among differently scaling energy terms.

A Bogomolny-Prasad-Sommerfield monopole is a finite-energy Yang-Mills-Higgs configuration satisfying $B_i=\pm D_i\Phi$. Its magnetic charge is the degree of the normalized Higgs field at spatial infinity.

A spherically symmetric SU(2) monopole can be written with the internal Higgs direction aligned with the spatial radial direction and the gauge field built from the invariant tensor ${ϵ}_{aij}$.

The Abelian Higgs model couples a complex scalar field to a $U(1)$ gauge field and permits magnetic vortex solitons in two spatial dimensions.

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

The vortex number is the total multiplicity of the zeros of the Higgs field, equivalently its phase winding at infinity or its magnetic flux divided by $2\pi$ in the standard normalization.

At critical coupling, the Abelian Higgs vortex moduli space is the space of gauge-equivalence classes of static $N$-vortex solutions. Vortex positions provide complex coordinates, and the field-theory kinetic energy induces an $L^2$ Riemannian metric whose geodesics approximate slow vortex motion.

A head-on geodesic through the smooth coincidence point of the two-vortex moduli space emerges along the orthogonal axis. Two identical critically coupled vortices therefore scatter through $90^{\circ }$ in the slow-motion approximation.

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

If $u$ solves the Taubes equation for a metric $g$ and $\tilde{u}$ solves it for $e^{u}g$, then $u+\tilde{u}$ solves it for $g$. The vortex divisors add, so the total vortex number is $N+\tilde{N}$.

Integrating the Taubes equation on a compact surface of area $A$ gives $A\ge 4\pi N$. A nontrivial vortex with nonzero Higgs field requires the strict inequality $A>4\pi N$.

At the integrable curvature scale on the Poincare disc, the vortex Taubes equation reduces to the Liouville equation. Holomorphic self-maps of the disc then generate explicit vortex solutions.

For the disc metric $8|dz{|}^2/(1-|z{|}^2{)}^2$ and a holomorphic map $g:D\to D$, the Higgs magnitudesolves the critically coupled vortex equations. Choosing $g(z)=z^{N+1}$ gives a radial vortex of winding $N$.

A Yang-Mills instanton is a finite-action Euclidean gauge field with self-dual or anti-self-dual curvature. On ${\mathbb{R}}^4$, its asymptotic pure gauge defines a map $S_{\infty }^3\to SU(2)$ whose degree is the instanton number.

For $k=-(8{\pi }^2{)}^{-1}\int \mathrm{Tr}(F\wedge F)$ and positive Euclidean action, decomposition into self-dual and anti-self-dual curvature givesEquality holds for a self-dual or anti-self-dual connection.

The anti-self-dual Yang-Mills equations require the curvature of a connection on an oriented Euclidean four-manifold to satisfy $F=-\ast F$. In complex coordinates $(w,z)$ they can be written

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ and $A_{\tau }=0$, one anti-self-duality convention becomes

For an $SU(2)$ connection under a standard trace convention,Its sign depends on orientation and on whether the instanton is self-dual or anti-self-dual.

For a finite-action $SU(2)$ connection on ${\mathbb{R}}^4$ that approaches $A=-dg{g}^{-1}$ at infinity,under the convention $k=(8{\pi }^2{)}^{-1}\int \mathrm{Tr}(F\wedge F)$. This integer is the degree of a map between oriented manifolds $g:S_{\infty }^3\to SU(2)\cong S^3$; orientation and trace conventions may reverse its sign.

If a four-dimensional Euclidean connection is independent of one coordinate and that connection component is renamed $\Phi$, the self-dual Yang-Mills equation reduces to the three-dimensional Bogomolny-Prasad-Sommerfield monopole equation $B_i=\pm D_i\Phi$.

The Skyrme model is a nonlinear sigma model for an $SU(2)$-valued field supplemented by a four-derivative term that evades Derrick collapse. Its topological degree is baryon number, and finite-energy solitons are Skyrmions.

A Skyrmion is a finite-energy topological soliton of the Skyrme model. Its baryon number is the degree of the compactified spatial map $S^3\to SU(2)\simeq S^3$.

The rational map approximation writes the angular dependence of a Skyrmion using a rational map $R:S^2\to S^2$ and determines a radial profile variationally. The degree of $R$ equals the baryon number, while zeros of its Wronskian mark directions of vanishing angular baryon density.