Two-component breather solution of the nonlinear Klein-Gordon equation

Abstract

The generalized perturbative reduction method is used to find the two-component vector breather solution of the nonlinear Klein-Gordon equation. It is shown that the nonlinear pulse oscillates with the sum and difference of frequencies and wave numbers in the region of the carrier wave frequency and wave number. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are obtained. In the particular case, the vector breather coincides with the vector 0π pulse of self-induced transparency.

Small amplitude breather of the nonlinear Klein-Gordon equation

“…obtained by expanding in Taylor series the right side of the sine-Gordon equation for U ≪ 1. In [16] the generalized pertubative reduction method developed in [17][18][19] is used. The solution obtained in [16] is…”

“…In [16] the generalized pertubative reduction method developed in [17][18][19] is used. The solution obtained in [16] is…”

“…Solution (26) is similar in form to solutions (15), (23) obtained in this paper. The essence of the perturbation reduction method used in [16][17][18][19] is the search for a solution to a nonlinear partial differential equation in the form of an amplitude modulated plane wave: where ε is some small parameter,…”

“…However, as shown in [18], the solution of equation (26), taken in the form of the amplitude modulated plane wave, turns out to be unstable. From our point of view, the method proposed in this paper for obtaining the approximate solution of the nonlinear Klein−Gordon equation in the form of smallamplitude breather has an advantage over the perturbation reduction method used in [16][17][18][19] due to its simplicity.…”