Shape Changing and Accelerating Solitons in the Integrable Variable Mass Sine-Gordon Model

The sine-Gordon model with a variable mass (VMSG) appears in many physical systems, ranging from the current through a nonuniform Josephson junction to DNA-promoter dynamics. Such models are usually nonintegrable with solutions found numerically or perturbatively. We construct a class of VMSG models, integrable at both the classical and the quantum levels with exact soliton solutions, which can accelerate and change their shape, width, and amplitude simulating realistic inhomogeneous systems at certain limits.

Figure 1

Exact soliton solutions: $s=\sin \frac{u}{2}(x,t)$ of the integrable VMSG equation with variable mass $m$. (a) $m=m_0(x^2-t^2)$ having an intriguing flattening of the soliton at the center. (b) Short time interval limit of the above soliton showing the flattening prominently. (c) $m=2m_0\cos q(x+t)$ with oscillatory behavior of the soliton. (d) Static soliton in the zone ($x\le 1.2$) with $m=\mathrm{const}$ and initially static soliton in the zone (at $x=4.8$) with variable mass: $m=m_0\exp (\rho x)\approx m(1+\rho x)$, with $\rho =0.1$ moves backward with acceleration, resembling soliton propagation in an inactive or active promoter region in a DNA chain.