Soliton Dynamics in Time-Dependent External Fields

The so-called non-Newtonian behaviour of solitons at small times is discussed in ϕ 4 theory for time-dependent external forces F ( t ) ∼ t n , n = 0, 1, 2,.... By the adiabatic approach, ϕ( x, t ) ≈ exact kink + u ( t ), it is shown that (i) the field momentum P = - ∫ - ∞ + ∞ ϕ x ϕ t d x satisfies Newton's law d P /d t ∼ F ( t ), and (ii) the acceleration of the kink in leading order of t is proportional to t 0 , t 1 or t n +2 , depending on the initial conditions u (0) and u t (0) for the motion of the vacuum. In consequence, for n ⩾ 2 the kink's acceleration is definitively different from the Newtonian acceleration (∼ t n ), whatever the initial conditions for the vacuum are. For F ( t ) = t 2 a numerical experiment is performed, showing good agreement with the adiabatic result, even for relativistic initial velocities. Expanding the perturbation function in the eigenfunctions to the linearized ϕ 4 equation, one finds that in the two leading orders of t it does not depend on x while in the next two orders its x -dependence is given by a term proportional to the translation mode.