Studies on Some Aspects of Light Beam Propagation Through ...

12 Optical Solitons : An Introduction
exactly unless for certain idealized conditions. Approximate methods have
been used to study general nonlinear partial differential equations. Variational
method is one among them. Here wc will describe the variational
method used in this thesis for studying the different nonlinear systems.
The method first introduced by \Vhitham.22 It centres upon the use of an
average Lagrangian density and the inevitable introduction of trial functions.
\Vhitham points out that the variational approach is not a separate
method and that it permits the development of quite general results but
naturally the accuracy of any description must depend upon a judicious
choice of trial functions. The variational method was applied to the NLS
equation first by Anderson. 23 By means of a Gaussian trial function and
a Ritz optimization procedure, approximate solutions arc obtained for the
evolution of pulse width, pulse amplitude and frequency chirp during propagation.
The result so obtained was in good comparison with that from
inverse scattering and numerical simulations. Since then the method has
been used successively by various authors. 24 - 27
The method proceeds by first writing the Lagrangian associated with
the system under study. A suitable trial solution is assumed using which
a reduced Lagrangian is obtained. A set of evolution equations called
Euler-Lagrange equation is obtained by finding the variation of the various
unknown parameters in the ansatz with respect to the reduced Lagrangian.
Lagrangian formulation
To illustrate the variational method, let us consider the nonlinear Schr-
6dinger equation of the form
(1.21)
where u is the field, z is the propagation direction, x and y are the two
transverse dimensions and s is a parameter signifying the strength of the
nonlinearity. The NLS can be restated as a variational problem by casting
it in the form of the Euler-Lagrangc equation
8 ( 8L) 8 ( 8L) 8 ( 8L) 8L
fJz fJXx + fJy 8Xx + 8z 8X - oX = 0,
lI
where X = 11, or 'U,* and the Lagrangian L is given by
(1.22)
i (* *) 1 12 1 2 sI ,1
L = "2 11Uz - U 11z + U;r: + U y 1 -"2 ul . (1.23)