Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms

Abstract

We consider the propagation of ultrashort optical solitons in media described by a general Hamiltonian of multilevel atoms. Assuming that all transition frequencies of the medium are well below the typical wave frequency, i.e., only the contribution of infrared transitions is taken into account, we use a shortwave approximation and a rigorous application of the reductive perturbation formalism to derive a cumbersome coupled system of nonlinear partial differential equations describing ultrashort soliton evolution in such systems. The rather complicated set of coupled equations can be simplified to a generic double-sine-Gordon equation for a special case of identical three-level atoms, whereas for a special case of identical four-level atoms the system of coupled equations can be reduced to a generalized double-sine-Gordon equation. Numerical simulations showing the formation of robust breather-type solutions of both the standard double-sine-Gordon and of the generalized double-sine-Gordon equations from sinusoidal inputs with Gaussian envelopes are also presented.

A Fourth Order Finite Difference Method for the Good Boussinesq Equation

“…Such solutions are called solitons. Many equations admitting soliton solutions are as follows: sine Gordon and double sine Gordon equations, Schrodinger equation, and KdV, MKdV, and complex modified KdV equations; many research works have been done on these equations [1][2][3][4][5][6][7][8]. Most of the current research is directed to solve coupled nonlinear systems analytically and numerically [9][10][11][12][13][14][15][16][17][18][19][20][21].…”

Models for supercontinuum generation beyond the slowly-varying-envelope approximation

“…For the former set, the optical pulse can be considered formally as a low-frequency wave; then using the so-called long-wave approximation, the one-dimensional propagation is shown to obey a mKdV-type model [26,43]. For the latter set, it is a shortwave approximation that leads to a sG-type model [26,44]. For a more general medium, both types of transitions are present and the propagation of ultrashort pulses can be well described by the mixed mKdV-sG equation [27], which we write in a dimensionless form as…”

Few-cycle spatiotemporal optical solitons in waveguide arrays

“…[44][45][46][47]. The theoretical models beyond the SVEA rely on the modified Korteweg-de Vries (mKdV) [48], the short-pulse [49][50][51], the sine-Gordon (sG) [52], the double sG [53,54], and the mKdV-sG [55][56][57] equations; see also Refs. [58][59][60][61][62].…”