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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.

Write the scalar field energy as $E=T+V$, whereUnder the Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$, a change of variables givesA finite-energy solution of the Euler-Lagrange equation is stationary under this admissible variation. The Derrick virial identity is therefore

The static field equation is ${\varphi }^{\prime \prime }=U^{\prime }(\varphi )$, so integration givesThe polynomial before $C$ is nonnegative and vanishes, so requiring the minimum to be zero fixes $C=0$. Hence the vacuum manifold iswhich has three elements. A finite-energy scalar-field kink can join only adjacent vacua: a solution cannot cross the intermediate vacuum at finite $x$ because its first integral would have $\varphi ={\varphi }^{\prime }=0$ there and the Picard-Lindelof theorem would make it constant. There are therefore four oriented topological sectors,comprising two increasing kinks and their two antikinks. Symmetry under $\varphi \mapsto -\varphi$ and spatial reflection generates all four from one profile.

For the $0\to 1$ sector, completing the square gives the Bogomolny boundEquality holds for the Bogomolny equationWith $y={\varphi }^2$, this becomes the logistic differential equation $y^{\prime }=2y(1-y)$. Translation invariance supplies an arbitrary center $x_0$, and the explicit kink in a phi-six model isIt tends to $0$ and $1$ at the two spatial ends and saturates the bound. Its mass is consequently

A classical field-theory soliton is a smooth, spatially localized, finite-energy solution which retains its identity under time evolution and is stable against small perturbations, commonly because of a topological charge or a balance between energy terms with different scaling behavior.

For a static field writewhereUnder the Derrick scaling ${\varphi }_{\lambda }(\mathbf{x})=\varphi (\lambda \mathbf{x})$, a change of variables givesA static solution must be stationary under this variation, so the Derrick virial identity isAll three energies are nonnegative. For $D\ge 4$, every coefficient is nonpositive and the coefficient of the strictly positive $E_2$ of any nonconstant field is negative. The identity is impossible. Thus, when the quartic-gradient term is available,For $D=2$ or $3$, the $E_4$ term has the opposite sign to at least one other term and the Derrick theorem does not rule out a soliton. If $\kappa =0$ from the outset, the identity reduces to $(2-D)E_2-D{E}_0=0$, recovering the stronger standard obstruction for $D\ge 2$.