Self-induced transparency with level degeneracy

Traveling wave solutions of nonlinear partial differential equations

“…The solution for an n-tuple cosine equation would provide an approximate solution for a nonlinear wave equation the inhomogeneous term of which may be approximated by a partial sum of a Fourier cosine series, a result entirely analogous to the previously obtained result for a Fourier sine series approximation [2]. Further, it is shown that each TW-solution of the single cosine equation immediately leads to a TW-solution of a double sine-cosine equation, a special case of which is a double cosine equation.…”

“…When E vanishes, this first-integral reduces to a solution obtained previously for the SG-equation [2]. The damped and forced SG-equation to be considered may be written as…”

“…While the SG-equation has been studied using the present procedure, [1], it has been shown that an alternate approach, based on a special transformation, leads to more detailed results [2]. Thus, both approaches will be utilized.…”

“…With specified coefficients A i and T i = sin(u/i), equation (48) is an n-tuple SG-equation of particular interest in nonlinear optics, providing the equations which govern the propagation of ultrashort light pulses when the effects of level degeneracy must be included (Q-transition). The solution of this problem was discussed in a recent paper [2] for the more general case of arbitrary coefficients Ai thus providing, in addition, an approximate solution for an arbitrary KG-equation, the inhomogeneous term of which may be approximated by a partial sum of a Fourier sine series. Solutions (by quadrature) for the other trigonometric functions will be given in this section.…”

“…This procedure reduces the problem to finding the solution of an ordinary differential equation of order one less than the order of the partial differential equation and, while this of course means the potential loss of some solutions, the solutions obtained often yield direct information about the structure of a solution so that more general solutions may be obtained by extrapolation. The method has been found to be quite useful in finding solutions of the Klein-Gordon (KG)-equation [1], including the special case which has become known as the Sine-Gordon (SG)-equation [2], [3].…”