Graphane -- material for hydrogen storage, breathers and kinks

In this paper we study the graphane. The Frenkel-Kontorova model on hexagonal lattice was used. We studied the case of one H atom above the C atom in the plane of graphane (we used the approximation of the hexagonal lattice in the plane). Continuous limit of the Lagrange-Euler equations is found from the Hamiltonian for $H$ atoms motion, they enabled us to study kink and breather excitations of $H$ atoms in the $H$ plane above the $C$ plane. We have found that there are three cases in the one $H$ atom motion The case $1$, when the $H$ atom is at the position which is below the position at which it is desorbed. Then the motion of this $H$ atom at time $t_{h}$ is described. The case $2$, when the $H$ atom is at the position of the suppressed atom $H$ in the direction to the $C$ (nearer) atom. This $H$ atom will be desorbed from the graphane going through the minimum of the potential energy and then through the point of desorption. Its motion of at time $t_{h}$ is described. The case $3$, when the $H$ atom is near the position of small oscillations near the potential energy minimum. The position of the atom $H$ at the time $t_{0}$ is the position to which the atom $H$ was excited with external force. The lattice of $H$ atoms in graphane may be excited as described by the kink solution of the Sine-Gordon equation. The kink has its velocity $U$, $U^{2} < 1$, and in time $T$ and in $X^{'}$ coordinate direction localization. The Sine-Gordon equation has the breather solution in the $X^{'}$ direction. There $\omega$ is the frequency of the breather, $T_{0}$ and $X^{'}_{0}$ are in time $T$ and in $X^{'}$ direction localization.