Multisoliton complex systems with explicit superpotential interactions

Abstract

Abstract We consider scalar field theories in $1+1$ dimensions with $N$ fields $\varphi_1, \dots \varphi_N$ which interact through a potential
$V=V(\varphi_1, \dots \varphi_N)$, which is defined in terms of an explicit superpotential $W$. We construct $W$ for any $N$ in terms of a known superpotential $w$ for a single-scalar model, such as that for the sine-Gordon equation or the $\varphi^4$ model, leading to an expression for $V$ which has multiple minima that supports solitons. Static solitons which minimize the total energy in each soliton sector appear as solutions of first-order Bogomolny equations, which have a gradient structure. These are identical in form to equations which arise in the context of synchronization phenomena in complex systems, with the space and time variables interchanged. The sine-Gordon superpotential, for example, leads to an explicit periodic superpotential $W$ for $N$ scalar fields, with associated Bogomolny equations that are equivalent to the well-known Kuramoto equations which describe the synchronization of identical phase oscillators on the unit circle. The known asymptotic properties of the Kuramoto system, for both positive and negative coupling constants, ensure that finite-energy solitons exist for any given set of intermediate values imposed at the origin. Besides the models derived from the sine-Gordon equation, we investigate $\varphi^4$ and $\varphi^6$ models with $N$ scalar fields and show numerically that solitons again exist over a wide range of parameters. We also derive general properties of the elementary meson excitations of the system, in particular we show that meson-soliton bound states exist over a restricted range of mass parameters with respect to an exact solution of the $\varphi^6$ system for $N=3$.