Nonlinear evolution of a pulse with a linear frequency...

4. Envelope of a Chirped Pulse

[11] The second-order correction to the complex amplitude F₂ satisfies the inhomogeneous equation

and the boundary conditions partial

F₂/

r|_r=0 =0 and F₂

0 at r

. The condition of solvability of this problem implies the equation for the pulse envelope U(q,s)

(9)

The coefficients of the equation are given by

The dependence of the coefficients of (9) on variable s manifests the influence of the longitudinal irregularity of the waveguide channel on evolution of the pulse envelope and, in particular, on its linear frequency modulation.

[12] Solution of (9) in a general case can be performed by the method described in the books by Molotkov [2003] and Molotkov et al. [1999]. As a first step, it is necessary to solve a simplified model problem, which will lead to the conclusions on a qualitative behavior of the solution. Then this solution can be used as a structural basis of the ansatz for the solution of the complete problem.

[13] To simplify the problem (9), we impose additional restrictions on the longitudinal inhomogeneity of the waveguide. We write the sought function U in the form

extracting explicitly an exponential multiplier with the envelope phase and, similarly to Bisyarin and Molotkov [2002], the multiplier that characterizes amplitude variations as functions of the longitudinal coordinate. Let us suppose that the u(q,s) function defined in such a way depends on the variable

alone, which can be achieved if the coefficients of (9) are related by

It is these relations that present additional assumptions on the longitudinal irregularity of the waveguide. Under these conditions the u(x) function obeys the second Painlevé equation

(10)

This equation is well studied from the point of view of existence of moving critical points in the solutions [see Golubev, 1950; Kudryashov, 2004, and references therein]; a relation with linear integral equations of definite kinds has been investigated in detail by Ablowitz et al. [1978, 1980]. Gibbon et al. [1985] has proved that it is possible to construct N -soliton solutions of nonlinear differential equations in partial derivatives if they possess the Painlevé property. Reduction of the second Painlevé equation to a linear integral equation and the proof of the existence of bounded solutions were given by Ablowitz and Segur [1977].

Figure 1

[14] In our work, the numerical solution of (10) was performed by the Runge-Kutta method. Figure 1 shows the graph of this solution (curve 1). For the sake of a comparison, Figure 1 presents a sech soliton with the same amplitude (curve 2) and the Airy function (curve 3). The soliton of the nonlinear Schrödinger equation is the solution of a similar problem [Bisyarin and Molotkov, 2002] without frequency modulation. The Airy function is presented for comparison because linearization of (10) is the Airy equation. The plotted solution has a localized character, that is, it tends to zero at |x|

0. Comparison with the sech soliton leads to the conclusion that a linear frequency modulation of the high- frequency carrier of the pulse gives rise to formation of an oscillating tail of the envelope and leads to a greater steepness of the leading edge of the pulse.