## Abstract

In a recent papers^{1}, we presented exact analytic solutions for a pair of solitary waves which can propagate through a medium of three-level and a medium of five-level atoms without loss and with their shapes invariant. The solitary-wave pair or matched pulses for the three-level A system is of particular interest. Denote the Rabi frequencies of the two pulses by Ω_{j} = 2d_{j} *E* _{j}/ħ, j=1,2, where d_{j} is the dipole matrix element between levels j and j+1, and Ej is the slowly varying electromagnetic field amplitude. The Rabi frequencies are functions of both space z and time t. For the solitary pulses, the Rabi frequencies depend on t and z through ξ= (t - z/v)/τ, where r is the pulse length and v is the velocity of the pulse pair. Our analytic solutions give Ω_{1}(ξ) = A_{1}f_{1}(ξ) + A_{2}f_{2}(ξ) and Ω_{2}(ξ) = B_{1}f_{1}(ξ) + B_{2}f_{2}(ξ), where A_{1}, A_{2}, B_{1} and B_{2} are real constants, and where f_{1} and f_{2} are dimensionless functions of ξ. There are two independent sets of solutions for the pair of elliptic functions f_{1} and f_{2}. The canonical solutions to which other forms of solutions can be transformed are given in Table 1 where a = (A_{l} ^{2}+B_{1} ^{2}+A_{2} ^{2}+B_{2} ^{2})/4, b = (A_{2} ^{2}+B_{2} ^{2}−A_{l} ^{2}−B_{1} ^{2})/4 ≥ 0 and A_{1}A_{2}+B_{1}B_{2}=0 The solitary-wave pair has a constant velocity v given by 1/v=1/c+4μ_{1}τ^{2}/R^{2}, where μ_{j}=π*N* d_{j}ω_{j}/ћc, *N* is the density of the atoms in the medium, ω_{j} is the laser frequency, and it is assumed that μ_{1}=μ_{2}. Notice that f_{1} and f_{2} satisfy f _{1} ^{2} + f _{2} ^{2} = 1. The constant R, its relation to the squared modulus 0 ≤ k^{2} ≤ 1 (or the squared complementary modulus k′^{2}=1−k^{2}) of f_{1} and f_{2}, and one other relationship which must be satisfied by the amplitudes and length of the pulses, are given in the last two rows of Table 1.

## Keywords

Solitary Wave Pulse Length Elliptic Function Input Pulse Rabi Frequency## References

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