Solitons in Field Theory and Nonlinear Analysis

Magnetic impurity inspired Abelian Higgs vortices

“…See also [30,31]. Using the methods in [47,49], it is not hard to show that (6.7) is also sufficient to ensure the existence…”

Universal growth law for knot energy of Faddeev type in general dimensions

“…Although both homotopy groups are isomorphic to the set of all integers, Z, the dependence relationships between the corresponding minimum energies and topologies are drastically different, which lead to the existence of different types of solitons: point-like ones in the Skyrme theory but knot-like ones in the Faddeev theory. More precisely, let us use E and Q to collectively denote the energy and topological invariant in either the Skyrme theory or the Faddeev theory, u is any static field configuration, N is a given integer, and Such a property is also commonly seen in previously well-studied gauge field theory soliton configurations including vortices and monopoles (Bogomol'nyi 1976;Actor 1979;Jaffe & Taubes 1980;Yang 2001) and instantons (Witten 1977;Atiyah et al 1978;Actor 1979;Rajaraman 1982;Nash & Sen 1983;Freed & Uhlenbeck 1991;Yang 2001). On the other hand, however, for the Faddeev theory case, we have, instead, the sublinear asymptotics E N wjN j 3=4 ; ð1:3Þ which is analogous to the ropelength energy, crossing number relation EwN p (3/4%p%1) stated earlier but is uncommonly seen in quantum field theory.…”

Direct and inverse relationships between Riccati systems coupled with multiplicative terms

“…Then, necessary and sufficient conditions for (26) to coincide with the first differential equation of the system (17) read:…”

“…The interest to transient nonlinear effects in competing populations has been also amplified by the development of mathematical and computational techniques dedicated for the analysis of solitary waves in the middle of the twentieth century. Many nonlinear models, including the Korteweg-de-Vries equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Lax equation, have been studied in the context of the existence and the qualitative description of solitary solutions in these systems [4,26]. The main objective of this paper is to demonstrate that the existence of solitary solutions is also possible in a system of coupled Riccati equations with the multiplicative term.…”