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Exact BPS double-kinks in generalized $φ^4$, $φ^6$ and sine-Gordon models

R. Casana, E. da Hora, F. C. Simas

TL;DR

The paper addresses the construction of exact BPS double-kink solutions in (1+1)D scalar field theories with a generalized kinetic term $f(\phi)$. By formulating a BPS framework with $V(\phi)=W_\phi^2/(2f)$ and introducing a coordinate $y$ via $dy/dx=1/f(\phi)$, the authors derive analytic double-kink solutions for three models ($\phi^4$, $\phi^6$, and sine-Gordon) using a common choice $f(x)=\left(\dfrac{2n+1}{x}\right)^{2n}$. The resulting solutions exhibit a central plateau, exponential tails, and a characteristic two-lump energy density; the lumps can be symmetric or asymmetric depending on the model, with explicit lump positions and peak energies determined. These exact results extend BPS soliton literature in generalized kinematics and pave the way for studying double-kink collisions and resonance phenomena with full analytic control.

Abstract

We consider a $(1+1)$-dimensional theory with a single real scalar field $φ$ whose kinematics is modified by a generalizing function $f(φ)$. After briefly reviewing its Bogomol'nyi-Prasad-Sommerfield (BPS) structure, we focus on a particular $f(φ)$ to obtain analytic BPS double-kink solutions in three different models governed by the $φ^4$, $φ^6$, and sine-Gordon superpotentials. In all cases, the resulting double-kinks approach the boundaries by following an exponential decay, with the generalizing function controlling its dependence on $x$ and mass. We also calculate the BPS bound explicitly and study how the double kinks behave near the origin. The energy distribution of the novel BPS states engenders symmetric two-lump profiles for the $φ^4$ and sine-Gordon superpotentials. Whereas, for the $φ^6$ superpotential, the BPS energy profiles form asymmetric two-lumps.

Exact BPS double-kinks in generalized $φ^4$, $φ^6$ and sine-Gordon models

TL;DR

The paper addresses the construction of exact BPS double-kink solutions in (1+1)D scalar field theories with a generalized kinetic term . By formulating a BPS framework with and introducing a coordinate via , the authors derive analytic double-kink solutions for three models (, , and sine-Gordon) using a common choice . The resulting solutions exhibit a central plateau, exponential tails, and a characteristic two-lump energy density; the lumps can be symmetric or asymmetric depending on the model, with explicit lump positions and peak energies determined. These exact results extend BPS soliton literature in generalized kinematics and pave the way for studying double-kink collisions and resonance phenomena with full analytic control.

Abstract

We consider a -dimensional theory with a single real scalar field whose kinematics is modified by a generalizing function . After briefly reviewing its Bogomol'nyi-Prasad-Sommerfield (BPS) structure, we focus on a particular to obtain analytic BPS double-kink solutions in three different models governed by the , , and sine-Gordon superpotentials. In all cases, the resulting double-kinks approach the boundaries by following an exponential decay, with the generalizing function controlling its dependence on and mass. We also calculate the BPS bound explicitly and study how the double kinks behave near the origin. The energy distribution of the novel BPS states engenders symmetric two-lump profiles for the and sine-Gordon superpotentials. Whereas, for the superpotential, the BPS energy profiles form asymmetric two-lumps.

Paper Structure

This paper contains 7 sections, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model and its BPS framework
  3. Analytical BPS double-kinks
  4. The $\phi ^{4}$ case
  5. The $\phi^{6}$ case
  6. The sine-Gordon case
  7. Summary and perspectives

Figures (3)

  • Figure 1: BPS double-kink solutions $\phi_{k_n}(x)$ (left) and their energy densities $\varepsilon _{BPS}$ (right) given by Eqs. (\ref{['sphi4']}) and (\ref{['ephi4']}), respectively. Here, $n=1$ (red line), $n=2$ (blue line), and $n=3$ (orange line). The black line represents $n=0$, i.e. the canonical single-kink solution.
  • Figure 2: BPS solutions $\phi_{k_n}(x)$ (left) and their energy densities $\varepsilon _{BPS}$ (right) given by Eqs. (\ref{['sphi6']}) and (\ref{['ephi6']}), respectively. Conventions as in Fig. \ref{['fig11xx']}.
  • Figure 3: BPS solutions $\phi_{k_n}(x)$ (left) and their energy densities $\varepsilon _{BPS}$ (right) given by Eqs. (\ref{['sphisg']}) and (\ref{['ephisg']}), respectively. Conventions as in Fig. \ref{['fig11xx']}.