Kink with Allowance for the Energy Dissipation and Pumping

Abstract

In the present work, the (1+1)-dimensional nonlinear equation of motion of 4 ϕ scalar theory is examined with simultaneous allowance for the processes of energy dissipation and pumping in the system. The energy dissipation is described in the equation by a linear term, and the energy pumping is described by a squared time derivative of the field. A single-kink solution is constructed for the given problem. It is demonstrated that a scalar field expansion in the inflationary model can occur with balanced energy dissipation and pumping.The kink solution in 4 ϕ scalar theory is one of the most extensively studied solutions of nonlinear equations widely used for a description of various phenomena and processes in many branches of physics [1][2][3]. The equation of motion in this theory in the (1+1)-dimensional case has the formwhere 2 m and λ are positive constants and 2 2 tt t ϕ = ∂ ϕ ∂ . Alongside with this equation, its modifications are constantly examined constructed so that to consider the processes of energy dissipation (through attenuation or friction) or pumping in the system or dissipation and pumping simultaneously. A review of publications devoted to this problem with linear dependence of energy dissipation or pumping is presented in [4]. In a number of works (for example, see [5][6][7][8][9]) this problem was considered with allowance for the nonlinear energy dissipation or pumping.Single-kink solutions for all above-mentioned equations are self-similar and are described by expressions of the form ( ) f kx t − ω , where f is a function and k and ω are the parameters of the solution determined by the model. In the present work, Eq. (1) is modified so that at least one parameter of the solution appears independent from the model parameters. In fact, this means that under certain conditions, the processes of energy dissipation and pumping in the system will be balanced to some extent; as a result, one of the parameters of the system state described by the self-similar solution will be independent of the energy transfer in this system. Let us consider the equation of motion of 4 ϕ theory with simultaneous allowance for the energy dissipation and pumping. Let the energy dissipation be described by a linear function, and pumping be described by a quadratic function. In this case, Eq. (1) is reduced to the formwhere α and β are positive constants (the dissipation and pumping coefficients, respectively). Since we study the dependence of the solution only on the energy dissipation and pumping, without loss of generality all other coefficients can be set equal to unity.To construct a single-kink solution of Eq.(2), the direct method generalizing the Hirota method to the case of degenerated solution parameters was used [4,10]. The sought-after solution can be written as follows:Byelorussian National Technical University.