Kinks from dynamical systems: domain walls in a deformedO(N) linear sigma model

Abstract

It is shown how a integrable mechanical system provides all the localized static solutions of a deformation of the linear O(N )-sigma model in two space-time dimensions. The proof is based on the Hamilton-Jacobi separability of the mechanical analogue system that follows when time-independent field configurations are being considered. In particular, we describe the properties of the different kinds of kinks in such a way that a hierarchical structure of solitary wave manifolds emerges for distinct N .

Adiabatic motion of two-component BPS kinks

“…Computation of the integrals (18) has been performed by changing variables to y = exp[2 √ 2(x+a)] in such a way that the quadratures reduce to rational definite integrals in y. The transition from one to two lumps is seen in formula (19) in the trading of arctan by i arctanh that happens at b = ±1.…”

Orbit-based deformation procedure for two-field models

“…But beside this, one is assuming that the deformation leads to well-behaved, smooth potentials. Therefore, we should ask for the first order derivatives of the deformed superpotential W(φ 1 , φ 2 ) to be continue or, alternatively, for its second order cross-derivatives to be identical ∂ 12 W = ∂ 21 W. Imposing this condition on the systems (18) and (19) leads to a very complicated constraint which suggest no obvious choice of the deformation functions f 1 and f 2 . For this reason, in order to make progress we need to introduce some assumptions to simplify the situation to a tractable case.…”

Stability of kink defects in a deformed O(3) linear sigma model

“…Defining non-dimensional space-time coordinates, fields and physical parameters as in [1], the dynamics of the model is determined by the action functional…”

“…We also use all the conventions in [1] and focus on the maximally asymmetric case, σ The mechanical system associated with the search for the solitary wave solutions of the model is completely integrable in the sense of Liouville and all the kink solutions can be found analytically. Here we shall not pursue this line of research; it has been fully developed in [1].…”

“…In Reference [1] we investigated the solitary waves that arise in a deformed O(3) linear Sigma model. These non-linear waves appear as kink defects when the system is considered in a (1 + 1)-dimensional space-time.…”