Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
167
de-Vries (KdV), modified KdV, nonlinear Schrödinger(NLS), and sine Gordon (SG)
nonlinear dynamical systems. These are completely integrable nonlinear dynamical
equations.
The most remarkable property of integrable systems is the existence of special type
of solutions called solitons. They ae localized waves that travel without much change
in shape. The word soliton refers to solitary travelling waves which preserve their
identities even after a collision. Solitons can be found everywhere; in the sky as density waves in spiral galaxies, as red spots in the atmosphere of Jupiter, in the ocean
as waves bombarding oilwells and also in smaller natural and laboratory systems
such as plasmas, molecular systems, laser pulses propagating in solids, superfluid
He, superconducting Josephson junction magnetic system, structural phase transitions, polymers, fluid flows, elementary particles and in liquid crystals. Apart from
the ubiquitous existence, the importance of solitary waves lies in their interesting
properties as nonlinear waves.
Nonlinear evolution equations (NLEE) possessing soliton solutions shows many
special properties such as an infinite sequence of conservation laws, Lie-Backlünd
symmetries, multisoliton solutions, Backlünd transformations and reduction to ordinary differential equations of Painleve-type. Furthermore, these equations may be
obtained by considering the compatibility of two associated linear operator that can
be expressed in the Lax’s form. All these suggest that the equations are exactly solvable. Soliton bearing equations such as sine-Gordon, KdV or NLSE are familiar to
mathematicians because of the complete integrability of their Hamiltonian systems
from which they derive. But when realistic applications in various fields such as condensed matter physics and engineering are considered, we have to include various
perturbations which leads to problems beyond those of pure integrable systems. This
was the motivation of the my Ph.D. thesis work. We studied the integrability of some
perturbed nonlinear partial differential equations using Lax method and Painleve
analysis.
2.1 Perturbed Nonlinear Schrödinger Equation
Among the important class of nonlinear integrable systems, the nonlinear Schrödinger
equation (NLSE) plays a significant role due to the presence of the special type of
stable solitary wave solutions, called envelope solitons. In optics, these solitons are
expected to be suitable information carriers in optical fiber communication systems
and are known as optical solitons. Solitons themselves can form a nonlinear superposition without exchange of energies; i.e. the interaction of solitons in any integrable
system is elastic. This is because the associated equations possess an infinite number
of conserved quantities. The simplest form of NLSE is
βU xx + γ U |U |
2
= iU t
(1)
167
de-Vries (KdV), modified KdV, nonlinear Schrödinger(NLS), and sine Gordon (SG)
nonlinear dynamical systems. These are completely integrable nonlinear dynamical
equations.
The most remarkable property of integrable systems is the existence of special type
of solutions called solitons. They ae localized waves that travel without much change
in shape. The word soliton refers to solitary travelling waves which preserve their
identities even after a collision. Solitons can be found everywhere; in the sky as density waves in spiral galaxies, as red spots in the atmosphere of Jupiter, in the ocean
as waves bombarding oilwells and also in smaller natural and laboratory systems
such as plasmas, molecular systems, laser pulses propagating in solids, superfluid
He, superconducting Josephson junction magnetic system, structural phase transitions, polymers, fluid flows, elementary particles and in liquid crystals. Apart from
the ubiquitous existence, the importance of solitary waves lies in their interesting
properties as nonlinear waves.
Nonlinear evolution equations (NLEE) possessing soliton solutions shows many
special properties such as an infinite sequence of conservation laws, Lie-Backlünd
symmetries, multisoliton solutions, Backlünd transformations and reduction to ordinary differential equations of Painleve-type. Furthermore, these equations may be
obtained by considering the compatibility of two associated linear operator that can
be expressed in the Lax’s form. All these suggest that the equations are exactly solvable. Soliton bearing equations such as sine-Gordon, KdV or NLSE are familiar to
mathematicians because of the complete integrability of their Hamiltonian systems
from which they derive. But when realistic applications in various fields such as condensed matter physics and engineering are considered, we have to include various
perturbations which leads to problems beyond those of pure integrable systems. This
was the motivation of the my Ph.D. thesis work. We studied the integrability of some
perturbed nonlinear partial differential equations using Lax method and Painleve
analysis.
2.1 Perturbed Nonlinear Schrödinger Equation
Among the important class of nonlinear integrable systems, the nonlinear Schrödinger
equation (NLSE) plays a significant role due to the presence of the special type of
stable solitary wave solutions, called envelope solitons. In optics, these solitons are
expected to be suitable information carriers in optical fiber communication systems
and are known as optical solitons. Solitons themselves can form a nonlinear superposition without exchange of energies; i.e. the interaction of solitons in any integrable
system is elastic. This is because the associated equations possess an infinite number
of conserved quantities. The simplest form of NLSE is
βU xx + γ U |U |
2
= iU t
(1)