## 1 Introduction  
In recent years considerable interest has developed in a nonperturbative, dynamical description of transport phenomena in condensed matter physics inspired by the physics of strong electromagnetic fields [1]. Particular attention was devoted to graphene (see, e.g., [2, 3]). In this case there is an obvious similarity with the dynamical Schwinger effect in QED, the creation of electron-positron pairs from the vacuum in strong electromagnetic fields [4, 5, 6]. In this context the nonperturbative kinetic approach has proven successful. It is based on the transition to a quasiparticle representation in the presence of an external, quasiclassical electric field facilitated by a time dependent Bogoliubov transformation [7, 8, 9, 10]. It would be natural to adopt these methods to specific problems in condensed matter physics and, in particular, to the physics of graphene. Such an adaptation is performed in the present work. The application of these methods allows for advancement to nonperturbative investigations of nonlinear effects in graphene in the presence of strong external electric fields. We want to give a detailed outline of the contents of the present work. In Section 2, Section 2.1 the basic kinetic equation (KE) for the simplest dynamical model of graphene [2, 3, 11, 12, 13, 14] (a single layer graphene sheet with two Dirac points of the Brillouin zone and absence of the standard scattering mechanism of carriers) is obtained using nonperturbative techniques for the case of a spatially homogeneous, time-dependent external electric field of arbitrary polarization in the graphene plane. The transition to the quasiparticle representation is obtained with the help of a unitary transformation expressed in explicit form [15]. All subsequent consideration is essentially nonperturbative. The process of electron-hole (\(e\)-\(h\)) pair creation in a strong electric field can be considered as a specific field-induced phase transition in a system with broken symmetry [1, 10, 16]. In Section 2.2 some features of this process are considered in graphene. Section 3 is devoted to the connection of observables such as the quasiparticle number and current densities with the kinetic theory. Here we discuss also the energy conservation law for a system in an external electric field. Here, in particular, an order parameter is introduced which describes the polarization properties of graphene. It is shown that after switching off the external field pulse the order parameter survives and oscillates with momentum dependent amplitude. In other words, the evolution of the order parameter is defined by the entire prehistory of the graphene evolution during the application of the external field. In particular, this effect becomes apparent in the damped oscillations of the residual polarization current on the background of a constant residual conduction current (Section 4). Here it is also shown that the polarization current dominates over the conduction current. This dominance turns out also in calculations of the currents in the framework of the standard perturbation theory. For example, in the Appendix A, we reproduce the well-known results for the polarization and conduction currents in the leading orders of the expansion with respect to \(E/E_{0}\ll 1\), where \(E_{0}\) is the characteristic field (1). Section 5 contains results of the numerical calculations of the distribution functions of the carriers for electric fields of different magnitude and spectral composition models both for linear and elliptic polarizations. This fact is an important hint that the similar situation is valid also in \(D=3+1\) QED, where analogous calculations can be very complicated [17]. In Section 6 we outline the effect of manipulating a weak signal with a current by means of generating active properties of graphene with the help of another (basic) field. Finally, in Section 7, by analogy with Section 2, we derive the KE in the \(D=2+1\) tight binding model of the nearest neighbor interaction [3, 11, 12].