Propagation of few-cycle pulses in a nonlinear medium and an integrable generalization of the sine-Gordon equation

The generalized sine-Gordon equation is obtained under the theoretical investigation of interaction of few-cycle pulses in a nonlinear medium modeled by a set of four-level atoms. This equation is derived without the use of the slowly varying envelope approximation and is shown to be integrable in the frameworks of the inverse scattering transformation method. Its solutions describing the propagation of the solitons and breathers and their interaction are investigated. In the case of different signs of the parameters of the equation considered, it is revealed, in particular, that the collision of solitons with opposite polarities can lead to an appearance of the short-living pulse having extraordinarily large amplitude, whose dynamics is similar to that of rogue waves. Also, the solitons of “rectangular” form and the breathers with rectangular oscillations exist in the case of the same signs of the parameters.

Figure 1

(a) Quantum levels of symmetric two-pit potential. Levels $1^{+}$ and $2^{-}$ correspond to the tunnel splitting; $3^{-}$ and $4^{+}$ correspond to the remote quantum levels; the superscripts “$+$” and “$-$” designate positive and negative parity, respectively. (b) The scheme of the allowed quantum transitions.

Figure 2

Profiles of $\partial \theta /\partial \tau$ and $u$ (thin line, the designations of the axes are in the parentheses) of one-soliton solutions with parameters $\alpha =0.5{\kappa }^{2}c$, $\beta =-0.5/c$, ${\chi }^{\left(0\right)}=0$, $k=0$, and $\mu =0.2/\left|\kappa \right|$ (a), $\mu =0.5/\left|\kappa \right|$ (b), and $\mu =0.65/\left|\kappa \right|$ (c).

Figure 3

Profiles of $\partial \theta /\partial \tau$ of two-soliton solutions with parameters $\alpha =0.5{\kappa }^{2}c$, $\beta =-0.5/c$, ${\chi }_1^{\left(0\right)}={\chi }_2^{\left(0\right)}=0$, $k_1=k_2=0$ and ${\mu }_1=0.2/\left|\kappa \right|$, ${\mu }_2=-0.3/\left|\kappa \right|$ (a) and ${\mu }_1=0.1/\left|\kappa \right|$, ${\mu }_2=-0.45/\left|\kappa \right|$ (b).

Figure 4

Profiles of $\partial \theta /\partial \tau$ of two-soliton solution with parameters $\alpha =0.5{\kappa }^{2}c$, $\beta =-0.5/c$, ${\chi }_1^{\left(0\right)}={\chi }_2^{\left(0\right)}=0$, $k_1=k_2=0$, ${\mu }_1=-0.69/\left|\kappa \right|$, ${\mu }_2=-0.7/\left|\kappa \right|$, and $z=-15\left|\kappa \right|$ (a), $z=0$ (b), and $z=15\left|\kappa \right|$ (c).

Figure 5

Profiles of $\partial \theta /\partial \tau$ and $u$ (thin line, the designations of the axes are in the parentheses) of one-soliton solutions with parameters $\alpha ={0.5\left|\kappa \right|}^{2}c$, $\beta =0.5/c$, ${\chi }^{\left(0\right)}=0$, and $\mu =0.2/\left|\kappa \right|$ (a), $\mu =1/\left|\kappa \right|$ (b), and $\mu =16/\left|\kappa \right|$ (c).

Figure 6

Profiles of $\partial \theta /\partial \tau$ of two-soliton solution with parameters $\alpha =0.5{\kappa }^{2}c$, $\beta =0.5/c$, ${\chi }_1^{\left(0\right)}={\chi }_2^{\left(0\right)}=0$, ${\mu }_1=-10/\left|\kappa \right|$, ${\mu }_2=-14/\left|\kappa \right|$, and $z=-500\left|\kappa \right|$ (a), $z=7\left|\kappa \right|$ (b), and $z=100\left|\kappa \right|$ (c).

Figure 7

Profile of $\partial \theta /\partial \tau$ of breather solution with parameters $\alpha =0.5{\kappa }^{2}c$, $\beta =0.5/c$, ${\chi }_R^{\left(0\right)}={\chi }_I^{\left(0\right)}=0$, ${\mu }_R=1/\left|\kappa \right|$, ${\mu }_I=12/\left|\kappa \right|$, $z=0$.