Figure 1

Periodic traveling wave solutions with chirp, as functions of the propagation distance. (a) Intensity $|u{|}^2$ of solution 1 and (b) of solution 2 from Table , presented as functions of $k_{0}x+l_{0}y+m_{0}t$ and $z$. Coefficients and parameters: $\beta (z)=\cos (z)$, $\gamma (z)={\gamma }_0=0$, $ϵ=0$, $M=0.9999$, $a_0=0.1$, $b_0=1$, $e_0=0$, $k_0=l_0=m_0=1$, and ${\omega }_0=0$.Reuse & Permissions

Figure 2

Combined intensity distributions of the periodic wave solutions 1 and 4, as functions of the propagation distance, with $ϵ=1$. (a) Distribution without and (b) with chirp. Other parameters are the same as in Fig. 1.Reuse & Permissions

Figure 3

Light bullet soliton solutions without chirp. The setup and parameters are as in Fig. 1, except for $M=1$ and $a_0=0$.Reuse & Permissions

Figure 4

Light bullet soliton solutions with chirp. Setup is the same as in Fig. 3, except for $a_0=0.1$.Reuse & Permissions

Figure 5

Numerical simulation of the light bullet from Fig. 3b. Initial data from Eq. (24) are propagated according to Eq. (1) for 90 diffraction lengths along the $z$ axis. Only the dependence on $t$ is shown. The initial profile is noted by open circles. The curves to the left present intensity profiles at the left turning point, the curves to the right the profiles at the right turning point. The curves at the center are snapshots of the profiles passing approximately through the point $t=0$ (i.e., the frames closest to $t=0$ are recorded). Three sets of 15 profiles are overlapped at different $z$ points, to show that no instabilities develop.Reuse & Permissions