166
K. S. Sreelatha
by them. If the equations of motion of a system are known, one would like to solve
them so that the dynamical variables at any time can be given as a function of initial variables and time. A dynamical system modelled using nonlinear equations
of motion becomes dependent on the configuration. This type of dynamical systems are encountered in a large number of disciplines such as physical sciences,
chemical sciences, engineering sciences, biological sciences and medical sciences.
Based on the behaviour of solutions, nonlinear dynamical systems are classified
into two: integrable systems and nonintegrable systems or chaotic systems. Systems
whose dynamical equations which can be solved exactly using some methods are
called integrable systems. Those for which no exact solution can be found are called
chaotic systems. Chaotic systems can be studied only using numerical methods. The
present article gives a review of my doctoral research work in integrable systems and
some applications of optical solitons—mainly the modelling of nonlinear waveguides for the propagation of optical solitons. A small description of chaos theory and
its importance is also presented.
2 Integrable Systems
One of the most fundamental, important and fascinating problems in the field of
nonlinear dynamical system is to develop a general criterion which decides the integrability. The idea of integrable system becomes important after the description given
by nonlinear evolution equations whose solutions represent the propagation of waves
with a permanent profile known as solitons. Nonlinear dynamical systems are generally nonintegrable. But some systems of practical interest are integrable. Nonlinear
integrable systems were discovered from eighteenth century onwards, but no one at
that time has the real understanding of their characteristics and solutions. The first
step towards the explanation of the relationship between the analytical structure of a
system and its integrability was given by the Russian mathematician Kovalevskaya
[1]. Her work focussed on the study of the motion of a rigid body with a fixed point
from an analysis of the singularities of the solutions. Then, Korteveg and de Vries
(KdV) made an important contribution to water wave problems and discovered a
nonlinear model equation for the unidirectional propagation of long surface waves
through uniform rectangular channel and the equation is now known as KdV equation. Now there exist several integrable partial differential equations (pdes) which
can be derived using physically meaningful asymptotic techniques from a very large
class of pdes.
Every integrable system has a number of special properties that hold only for
integrable equations. The concept of solitons and inverse scattering technique (IST)
method to find exact solutions for some nonlinear partial differential equations (pde)
had far reaching influence and applications in various branches of mathematics,
physics and engineering [2]. It has been established that many nonlinear wave equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. Among these, well-known equations are Korteweg the
K. S. Sreelatha
by them. If the equations of motion of a system are known, one would like to solve
them so that the dynamical variables at any time can be given as a function of initial variables and time. A dynamical system modelled using nonlinear equations
of motion becomes dependent on the configuration. This type of dynamical systems are encountered in a large number of disciplines such as physical sciences,
chemical sciences, engineering sciences, biological sciences and medical sciences.
Based on the behaviour of solutions, nonlinear dynamical systems are classified
into two: integrable systems and nonintegrable systems or chaotic systems. Systems
whose dynamical equations which can be solved exactly using some methods are
called integrable systems. Those for which no exact solution can be found are called
chaotic systems. Chaotic systems can be studied only using numerical methods. The
present article gives a review of my doctoral research work in integrable systems and
some applications of optical solitons—mainly the modelling of nonlinear waveguides for the propagation of optical solitons. A small description of chaos theory and
its importance is also presented.
2 Integrable Systems
One of the most fundamental, important and fascinating problems in the field of
nonlinear dynamical system is to develop a general criterion which decides the integrability. The idea of integrable system becomes important after the description given
by nonlinear evolution equations whose solutions represent the propagation of waves
with a permanent profile known as solitons. Nonlinear dynamical systems are generally nonintegrable. But some systems of practical interest are integrable. Nonlinear
integrable systems were discovered from eighteenth century onwards, but no one at
that time has the real understanding of their characteristics and solutions. The first
step towards the explanation of the relationship between the analytical structure of a
system and its integrability was given by the Russian mathematician Kovalevskaya
[1]. Her work focussed on the study of the motion of a rigid body with a fixed point
from an analysis of the singularities of the solutions. Then, Korteveg and de Vries
(KdV) made an important contribution to water wave problems and discovered a
nonlinear model equation for the unidirectional propagation of long surface waves
through uniform rectangular channel and the equation is now known as KdV equation. Now there exist several integrable partial differential equations (pdes) which
can be derived using physically meaningful asymptotic techniques from a very large
class of pdes.
Every integrable system has a number of special properties that hold only for
integrable equations. The concept of solitons and inverse scattering technique (IST)
method to find exact solutions for some nonlinear partial differential equations (pde)
had far reaching influence and applications in various branches of mathematics,
physics and engineering [2]. It has been established that many nonlinear wave equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. Among these, well-known equations are Korteweg the