Some of the many uses of scalar fields: kinks, lumps, and geometric constraints

Abstract

This perspective deals with real scalar fields in two-dimensional spacetime. We focus on models described by one and two real scalar fields, paying closer attention to kinks and lumps, which are localized structures of current interest in high energy physics and in other areas of nonlinear science. We briefly review some of the main results presented in the literature and then focus on some new issues concerning the compact and long-range behavior of solutions and the presence of geometric constraints, suggesting how they can be used in applications in other areas of nonlinear science.

Figures (3)

• Figure 1: The potentials (top) for the $\phi^4$ (solid, red) and inverted $\phi^4$ (dashed, red) models, and the corresponding solutions (middle) for kink (solid, blue) and lump (dashed, blue), and the energy densities (bottom) for kink (solid, green) and lump (dashed, green).
• Figure 2: The potentials \ref{['potco']} (top) and \ref{['potlr']} (bottom), depicted in terms of $\phi$ for $n=1$ (thicker curve), and then $n=3, 10,$ and $20$.
• Figure 3: The geometrically restricted kink (top, solid blue) and lump (bottom, dashed blue), depicted in terms of $x$ for $\alpha=0.1$ (thicker curve) and then $\alpha=0.2, 0.4$, and $0.8$.