Bogomol'nyi equations for Kruglov strings
We construct the Bogomol'nyi equations for Abelian gauge--Higgs vortices in which the Maxwell gauge sector is replaced by Kruglov nonlinear electrodynamics, a power-law family that interpolates between Maxwell theory, Born--Infeld electrodynamics, and exponential electrodynamics, characterized by a dimensionless exponent $σ$. Using the stressless method, we derive a pair of first-order equations directly from the vanishing of the spatial stress tensor, without assuming the Higgs potential \textit{a priori}. For generic $σ$, the gauge and Higgs sectors are coupled through an implicit algebraic relation. We therefore introduce a constitutive map $Φ(Y;σ)$ and analyze its monotonicity and range to determine the conditions for a smooth admissible Bogomol'nyi branch. For $σ>1/2$, the constitutive map is strictly monotonic and unbounded, whereas for $0<σ<1/2$ it possesses a finite maximum; the marginal case $σ=1/2$ is bounded. These properties yield explicit bounds on the nonlinear parameter $β$ for the latter cases. We further obtain closed-form constitutive relations, BPS potentials, and gauge-field equations for six representative values of $σ$, spanning linear, quadratic, and cubic algebraic structures. The corresponding vortex profiles are then computed numerically. The resulting BPS string tension is purely topological, $μ_{\rm BPS}=2πn$, independent of both $σ$ and $β$.