Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
173
This equation is an integro-differential equation which cannot be solved exactly.This
means that as we apply the quadratic and cubic nonlinearity together, the system
become more perturbes and may lead to chaotic situation. This work considered the
nonlinear wave propagation through a two-dimensional lattice having nonuniform
mass distribution. We used weak nonlinear approximation for quadratic nonlinearity
and cubic nonlinearity seperately and both together. Using RPM, we reduced the
equations of motion for three nonlinear equations, namely, Kadomtsev Petviashvili
(KP) equation, modified KP equation and an integro-differential equation [4]. The
integrability studies of these equations were also carried out and the details are given
in next session.
3.2 Integrability Studies of KP and mKP Equations
The integrability of the derived KP, mKP and integro-differential equations is studied using both Painleve method (P-test) and Lax method. For P-test, the presence or
absence of movable, non-characteristic, critical singular manifolds should be determined. When the system is free from movable critical manifolds, the P-property
holds suggesting P-integrability. Main steps involved in the P-test of pdes include (i)
determination of leading order behaviours, (ii) identification of powers at which arbitrary functions can enter into the Laurent series called resonances and (iii) verifying
that at the resonance values, sufficient number of arbitrary functions exist without
the introduction of movable critical manifolds. With quadratic nonlinearity only, the
KP equation holds all these properties and it is found to be integrable in P-sense. For
mKP equation with cubic nonlinearity, the resonant values do not provide sufficient
number of arbitrary functions and hence it is not integrable in the Painleve sense.
When both the nonlinearities were considered together, due to the presence of the
integral term, it is not possible to apply the P-test and hence it does not belong to
the integrable class. Hence, it has been proved that a perturbed nonlinear equation
is integrable when the perturbation is homogeneous and when the perturbation is
inhomogeneous, the equation becomes nonintegrable.
We also tried to find the lax pair for all these equations. Existence of Lax pair
for KP equation is already proved in literature and it is considered as a completely
integrable system. We could not find any Lax pair for mKP equation and also for the
integro-differential equation. The non-existence of Lax pairs clearly indicates the
nonintegrability of the corresponding system (in the Lax sense).
Kadomtsev and Petviashvili (KP) equation is the generalized KdV equation for
the two-dimensional case. These equations model slowly varying waves in dispersive
media. The equation with +ve sign (α
2
= −1) arises in the study of plasmas and also
in the modulation of long weakly nonlinear water waves which propagate in one
dimension. The equation with +ve sign (α
2
= +1) arises in acoustics and lattice
dynamics. Researches into physical, earth and life sciences have led to the discovery
of hundreds more nonlinear evolution equations. Of these, only a few are known to
have soliton solutions.
173
This equation is an integro-differential equation which cannot be solved exactly.This
means that as we apply the quadratic and cubic nonlinearity together, the system
become more perturbes and may lead to chaotic situation. This work considered the
nonlinear wave propagation through a two-dimensional lattice having nonuniform
mass distribution. We used weak nonlinear approximation for quadratic nonlinearity
and cubic nonlinearity seperately and both together. Using RPM, we reduced the
equations of motion for three nonlinear equations, namely, Kadomtsev Petviashvili
(KP) equation, modified KP equation and an integro-differential equation [4]. The
integrability studies of these equations were also carried out and the details are given
in next session.
3.2 Integrability Studies of KP and mKP Equations
The integrability of the derived KP, mKP and integro-differential equations is studied using both Painleve method (P-test) and Lax method. For P-test, the presence or
absence of movable, non-characteristic, critical singular manifolds should be determined. When the system is free from movable critical manifolds, the P-property
holds suggesting P-integrability. Main steps involved in the P-test of pdes include (i)
determination of leading order behaviours, (ii) identification of powers at which arbitrary functions can enter into the Laurent series called resonances and (iii) verifying
that at the resonance values, sufficient number of arbitrary functions exist without
the introduction of movable critical manifolds. With quadratic nonlinearity only, the
KP equation holds all these properties and it is found to be integrable in P-sense. For
mKP equation with cubic nonlinearity, the resonant values do not provide sufficient
number of arbitrary functions and hence it is not integrable in the Painleve sense.
When both the nonlinearities were considered together, due to the presence of the
integral term, it is not possible to apply the P-test and hence it does not belong to
the integrable class. Hence, it has been proved that a perturbed nonlinear equation
is integrable when the perturbation is homogeneous and when the perturbation is
inhomogeneous, the equation becomes nonintegrable.
We also tried to find the lax pair for all these equations. Existence of Lax pair
for KP equation is already proved in literature and it is considered as a completely
integrable system. We could not find any Lax pair for mKP equation and also for the
integro-differential equation. The non-existence of Lax pairs clearly indicates the
nonintegrability of the corresponding system (in the Lax sense).
Kadomtsev and Petviashvili (KP) equation is the generalized KdV equation for
the two-dimensional case. These equations model slowly varying waves in dispersive
media. The equation with +ve sign (α
2
= −1) arises in the study of plasmas and also
in the modulation of long weakly nonlinear water waves which propagate in one
dimension. The equation with +ve sign (α
2
= +1) arises in acoustics and lattice
dynamics. Researches into physical, earth and life sciences have led to the discovery
of hundreds more nonlinear evolution equations. Of these, only a few are known to
have soliton solutions.