Kink shape solutions of the Maxwell-Lorentz system

In the limit of high amplitude oscillating electromagnetic fields, a sequence of kink antikink shaped optical waves has been found in the Maxwell’s equations coupled to a single Lorentz oscillator and with Kerr nonlinearity. The individual kinks and antikinks result from a traveling wave assumption and their stability has been assessed by numerical simulations. For typical physical parameter values the kink width is of the order of tens of femtoseconds.

Figure 1

Phase plane flow pattern calculated numerically from the scaled Eqs. (12), using the parameters in Eq. (4) and $\tilde{v}=0.6545$.

Figure 2

Solid curve: The kink solution, $\tilde{E}$ versus $\tilde{\xi }$, calculated numerically from the scaled Eqs. (12). Dashed curve: The approximation $\tilde{E}\left(\tilde{\xi }\right)=A_{\mathit{\text{kink}}}\tanh [(\tilde{\xi }-239.2)∕w]$, $A_{\mathit{\text{kink}}}=5.7336\times {10}^{-2}$ and $w=32.32$. The parameters are given in Eq. (4) and $\tilde{v}=0.6545$.

Figure 3

The real part $\mathrm{Re}\left[n\right]$ (a) and the imaginary part $\kappa$ (b) of the index of refraction $n$ from Eq. (13) as function of the frequency ratio $\omega ∕{\omega }_0$. The normalized damping coefficient is $\gamma =0.1$.

Figure 4

Numerical solutions of Eqs. (12). (a) Oscillating waves corresponding to the nearly linear case starting from $\mathbf{(}\tilde{E}\left(0\right),\tilde{Y}\left(0\right)\mathbf{)}=(0,0.001)$. (b) Initial conditions $\mathbf{(}\tilde{E}\left(0\right),\tilde{Y}\left(0\right)\mathbf{)}=(0,0.00175)$. (c) A solution close to a train of kink-antikinks using the initial conditions $\mathbf{(}\tilde{E}\left(0\right),\tilde{Y}\left(0\right)\mathbf{)}=(0,0.00177436)$. $\tilde{v}=0.6545$. $E$ and $t$ in the figure labels refer to the scaled variables $\tilde{E}$ and $\tilde{t}$.

Figure 5

A traveling kink and antikink wave, forming a squarelike pulse, using the parameter values in Eq. (4). The initial data for the kink are $\tilde{E}(\tilde{z},0)=A_{\mathit{\text{kink}}}\tanh [(\tilde{z}-z_1)∕w]$ where $A_{\mathit{\text{kink}}}=5.7336\times {10}^{-2}$, $w=32.32$, $\tilde{v}=0.6545$, and $z_1=400$. For the antikink we used $A_{\mathit{\text{kink}}}=-5.7336\times {10}^{-2}$, $w=32.32$, $\tilde{v}=0.6545$, and $z_1=800$. $E$, $z$, and $t$ in the figure labels refer to the scaled variables $\tilde{E}$, $\tilde{z}$, and $\tilde{t}$.