Shouxin Chen

Existence of multiple vortices in supersymmetric gauge field theory

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group G=U(1)×SU(N) and with N Higgs scalar fields in the fundamental representation of G. Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume |Ω|, the existence of a unique multiple vortex solution representing n1,…,nN, respectively, prescribed vortices arising in the N species of the Higgs fields is established under the explicitly stated necessary and sufficient condition where e and g are the U(1) electromagnetic and SU(N) chromatic coupling constants, v measures the energy scale of broken symmetry and is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed n-vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.

Exact kink solitons in Skyrme crystals

We present an explicit integration of the kink soliton equation obtained in a recent interesting study of the classical Skyrme model where the field configurations are of a generalized hedgehog form which is of a domain-wall type. We also show that in such a reduced one-dimensional setting the first-order and second-order equations are equivalent. Consequently, in such a context, all finite-energy solitons are Bogomolnyi-Prasad-Sommerfield type and precisely known.

Domain wall equations, Hessian of superpotential, and Bogomol'nyi bounds

Abstract An important question concerning the classical solutions of the equations of motion arising in quantum field theories at the BPS critical coupling is whether all finite-energy solutions are necessarily BPS. In this paper we present a study of this basic question in the context of the domain wall equations whose potential is induced from a superpotential so that the ground states are the critical points of the superpotential. We prove that the definiteness of the Hessian of the superpotential suffices to ensure that all finite-energy domain-wall solutions are BPS. We give several examples to show that such a BPS property may fail such that non-BPS solutions exist when the Hessian of the superpotential is indefinite.

Friedmann's equations in all dimensions and Chebyshev's theorem

This short but systematic work demonstrates a link between Chebyshevs theorem and the explicit integration in cosmological time

Existence of steady-state solutions in a nonlinear photonic lattice model

t

Explicit integration of Friedmann’s equation with nonlinear equations of state

and conformal time

Exact kink solitons in a monopole confinement problem

\eta

Friedmann-Lemaitre Cosmologies via Roulettes and Other Analytic Methods

of the Friedmann equations in all dimensions and with an arbitrary cosmological constant

Phase transition solutions in geometrically constrained magnetic domain wall models

\Lambda

Existence of vortices in nonlinear optics

. More precisely, it is shown that for spatially flat universes an explicit integration in