Few-optical-cycle solitons: Modified Korteweg–de Vries sine-Gordon equation versus other non–slowly-varying-envelope-approximation models

We put forward through both analytical and numerical methods the advantage of using non–slowly-varying envelope-approximation model equations for describing the propagation of few-optical-cycle pulses in transparent media. It is proven that the dynamical model based on the generic modified Korteweg–de Vries sine-Gordon equation retrieves the results reported so far in the literature, and so demonstrating its remarkable mathematical capabilities in describing the physics of few-cycle-pulse optical solitons.

Figure 1

(Color online) Compression of a Gaussian input pulse to a FCP soliton, as described by the mKdV-sG equation with parameters $c_1=50$, $c_2=0.5$, and $c_3=1$.

Figure 2

(Color online) Top: the intensity ${\left|u\right|}^2$ of the input FCP [light blue (gray) curve] and its evolution at $z=0.2$ according to the focusing mKdV equation [dark blue (black) curve] for $c_2=+1/3$. Bottom: the linear dispersion (for $c_2=0$) of the same input at the same propagation distance [light blue (gray)], and its nonlinear dispersion according to the defocusing mKdV equation [dark blue (black) curve] for $c_2=-1/3$. The input is a breather of the focusing mKdV equation, with angular frequency of four and inverse pulse duration of two.

Figure 3

(Color online) Propagation of a FCP soliton in the frame of the mKdV-sG model (arbitrary units, $c_1=c_2=1$, $c_3=0.0001$). The input is an adequately rescaled sG breather.

Figure 4

(Color online) Pulse compression described by mKdV-sG equation (12) with defocusing mKdV part. Input is a pulse with hyperbolic secant envelope solution to the SVEA limit of mKdV-sG, with pulse length ${\tau }_0=25$ and angular frequency $\omega =0.5$. Parameters are $c_1=c_2=1$, $c_3=-0.5$.

Figure 5

(Color online) FCP soliton propagation according to mKdV-sG equation (12) with defocusing mKdV part. From the sech-shaped input pulse with length ${\tau }_0=8$ and angular frequency $\omega =0.5$, two FCP solitons, one larger than the other, plus dispersing waves, are formed. Parameters are $c_1=c_2=1$, and $c_3=-0.5$.