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 $\sigma$. 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 $\mathit{\text{a}}\mathit{\text{priori}}$. For generic $\sigma$, the gauge and Higgs sectors are coupled through an implicit algebraic relation. We therefore introduce a constitutive map $\mathrm{\Phi }(Y;\sigma )$ and analyze its monotonicity and range to determine the conditions for a smooth admissible Bogomol’nyi branch. For $\sigma >1\mathrm{/}2$, the constitutive map is strictly monotonic and unbounded, whereas for $0<\sigma <1\mathrm{/}2$ it possesses a finite maximum; the marginal case $\sigma =1\mathrm{/}2$ is bounded. These properties yield explicit bounds on the nonlinear parameter $\beta$ for the latter cases. We further obtain closed-form constitutive relations, BPS potentials, and gauge-field equations for six representative values of $\sigma$, spanning linear, quadratic, and cubic algebraic structures. The corresponding vortex profiles are then computed numerically. The resulting BPS string tension is purely topological, ${\mu }_{\mathrm{B}\mathrm{P}\mathrm{S}}=2\pi n$, independent of both $\sigma$ and $\beta$.