Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
169
The amplitude of the wave given in Eq. (4) depends on the strength of the perturbation,
i e as the value of f (t) increases, the amplitude decreases, which is as expected.
Thus, we conclude from this study that existence of soliton solution in a nonlinear
dynamical system is sensitive to perturbations [3].
3 Propagation of Solitons Through 2D Lattice
Research on propagation of solitons through discrete lattices dates back to the early
days of soliton theory. The relevant nonlinear equations which model these lattices
are very difficult to solve analytically. Generally, one looks for possible pulse soliton
solutions in the continuum or longwavelength approximation. It has been found that
by considering the weak nonlinear case, it is possible to reduce a large number of
one-dimensional nonlinear systems to integrable ones. This nonlinear approximation
has two assumptions:
(1) the amplitude of the wave is small but finite, and
(2) the wave is a long wave or a modulation of a monochromatic wave.
From recent literatures, it is found that the reductive perturbation method (RPM) is
very useful for carrying out weak nonlinear approximation since it takes into account
a competition between nonlinearity and dispersion in a systematic manner. The perturbation method has been developed and formulated in a general way by Taniuti
and his collaborators. This method was first established for the reduction of a fairly
general nonlinear system to a single tractable nonlinear equation. By using RPM,
it is easy to reduce nonlinear systems to soliton equations. In the present work, we
studied the wave propagation through a 2D lattice for three specific cases : quadratic
nonlinearity, cubic nonlinearity and both of these together [4]. In each case, these
equations reduce to three different nonlinear equations. We studied the integrability
of these equations using Painleve method and also compared the solutions of these
equations.
3.1 Deduction of the Nonlinear Equations for Wave
Propagation Through 2D Lattice
We considered a nonlinear lattice with nonuniform mass distribution. The force
between two adjacent particles is given by
F = k(( + αα
2
+ ββ
3
+ · · · )
(5)
where is the elongation of the spring and k is the spring constant. For a particle of
mass m i and displacement a i , the equation of motion can be given as
169
The amplitude of the wave given in Eq. (4) depends on the strength of the perturbation,
i e as the value of f (t) increases, the amplitude decreases, which is as expected.
Thus, we conclude from this study that existence of soliton solution in a nonlinear
dynamical system is sensitive to perturbations [3].
3 Propagation of Solitons Through 2D Lattice
Research on propagation of solitons through discrete lattices dates back to the early
days of soliton theory. The relevant nonlinear equations which model these lattices
are very difficult to solve analytically. Generally, one looks for possible pulse soliton
solutions in the continuum or longwavelength approximation. It has been found that
by considering the weak nonlinear case, it is possible to reduce a large number of
one-dimensional nonlinear systems to integrable ones. This nonlinear approximation
has two assumptions:
(1) the amplitude of the wave is small but finite, and
(2) the wave is a long wave or a modulation of a monochromatic wave.
From recent literatures, it is found that the reductive perturbation method (RPM) is
very useful for carrying out weak nonlinear approximation since it takes into account
a competition between nonlinearity and dispersion in a systematic manner. The perturbation method has been developed and formulated in a general way by Taniuti
and his collaborators. This method was first established for the reduction of a fairly
general nonlinear system to a single tractable nonlinear equation. By using RPM,
it is easy to reduce nonlinear systems to soliton equations. In the present work, we
studied the wave propagation through a 2D lattice for three specific cases : quadratic
nonlinearity, cubic nonlinearity and both of these together [4]. In each case, these
equations reduce to three different nonlinear equations. We studied the integrability
of these equations using Painleve method and also compared the solutions of these
equations.
3.1 Deduction of the Nonlinear Equations for Wave
Propagation Through 2D Lattice
We considered a nonlinear lattice with nonuniform mass distribution. The force
between two adjacent particles is given by
F = k(( + αα
2
+ ββ
3
+ · · · )
(5)
where is the elongation of the spring and k is the spring constant. For a particle of
mass m i and displacement a i , the equation of motion can be given as