Ultrashort pulses and short-pulse equations in 2+1 dimensions

In this paper, we derive and study two versions of the short pulse equation (SPE) in $(2+1)$ dimensions. Using Maxwell's equations as a starting point, and suitable Kramers-Kronig formulas for the permittivity and permeability of the medium, which are relevant, e.g., to left-handed metamaterials and dielectric slab wave guides, we employ a multiple scales technique to obtain the relevant models. General properties of the resulting $(2+1)$-dimensional SPEs, including fundamental conservation laws, as well as the Lagrangian and Hamiltonian structure and numerical simulations for one- and two-dimensional initial data, are presented. Ultrashort one-dimensional breathers appear to be fairly robust, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing wave forms.

Figure 1

Top (four) panels: contour plots showing profiles of the evolution of a perturbed 1D breather in the $(x,t$) plane, when evolved according to Eq. (15). Snapshots correspond to $z=0$ (top left), $z=50$ (top right), $z=100$ (bottom left), and $z=150$ (bottom right). Bottom panel: the evolution of the breather for $x=0$. Parameter values are $c=0$, $s=-1/3$, and $\alpha =-2$. All depicted quantities are dimensionless.

Figure 2

Evolution of the conserved quantities of the SPE-I for the simulation shown in Fig. 1: top panel shows the Hamiltonian $H$ and bottom panels show the momenta $M_t$ (left) and $M_t$ (right). The relative error for $H$ and $M_t$ is of order ${10}^{-4}$ and ${10}^{-3}$, respectively, and $M_x$ is zero. All depicted quantities are dimensionless.

Figure 3

Contour plots showing profiles of the field $E$, in the $(x,t)$ plane, evolved as per SPE-I, Eq. (15), with localized initial data [cf. Eq. (27)]. The snapshots, from left to right, and top to bottom correspond to $z=0,3,10,20,30,38$. Parameter values are $s=-1/3$ and $\alpha =-2$. All depicted quantities are dimensionless.

Figure 4

Top (four) panels: contour plots showing profiles of the 1D breather solution of Eq. (38) in the $(x,t$) plane, when evolved as per Eq. (28). Snapshots correspond to $z=0$ (top left), $z=10$ (top right) $z=30$ (bottom left), and $z=50$ (bottom right). Bottom panel: the evolution of the breather for $x=0$. Parameter values are $c=0$, $s=-1/3$, $\alpha =-2$, and $\sigma =1$. All depicted quantities are dimensionless.

Figure 5

Evolution of the conserved quantities of SPE-II for the simulation shown in Fig. 4: the top panel shows the Hamiltonian $H$ and the bottom panels show the momenta $M_t$ (left) and $M_x$ (right). The relative error for $H$ and $M_t$ is of order ${10}^{-4}$ or less, and the value of $M_x$ is zero (and remains so throughout the simulation). All depicted quantities are dimensionless.

Figure 6

Contour plots showing profiles of the field $E$, in the $(x,t)$ plane, evolved as per SPE-II, Eq. (15), with localized initial data [cf. Eq. (27)]. The snapshots, from top to bottom and left to right, correspond to $z=0,3,10,20,30,40$. The domain for numerical computation in $x\times t$ plane is $[-20\pi ,20\pi ]\times [-20\pi ,20\pi ]$; we are zooming in here in the snapshots to show more detail. All depicted quantities are dimensionless.