19 research outputs found
Lagrangian Approach to Dispersionless KdV Hierarchy
We derive a Lagrangian based approach to study the compatible Hamiltonian
structure of the dispersionless KdV and supersymmetric KdV hierarchies and
claim that our treatment of the problem serves as a very useful supplement of
the so-called r-matrix method. We suggest specific ways to construct results
for conserved densities and Hamiltonian operators. The Lagrangian formulation,
via Noether's theorem, provides a method to make the relation between
symmetries and conserved quantities more precise. We have exploited this fact
to study the variational symmetries of the dispersionless KdV equation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
pplications) at http://www.emis.de/journals/SIGMA
Dynamical systems theory for nonlinear evolution equations
We observe that the fully nonlinear evolution equations of Rosenau and
Hymann, often abbreviated as equations, can be reduced to
Hamiltonian form only on a zero-energy hypersurface belonging to some potential
function associated with the equations. We treat the resulting Hamiltonian
equations by the dynamical systems theory and present a phase-space analysis of
their stable points. The results of our study demonstrate that the equations
can, in general, support both compacton and soliton solutions. For the
and cases one type of solutions can be obtained from the
other by continuously varying a parameter of the equations. This is not true
for the equation for which the parameter can take only negative
values. The equation does not have any stable point and, in the
language of mechanics, represents a particle moving with constant acceleration.Comment: 5 pages, 4 figure
On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy
A general form of the fifth-order nonlinear evolution equation is considered.
Helmholtz solution of the inverse variational problem is used to derive
conditions under which this equation admits an analytic representation. A
Lennard type recursion operator is then employed to construct a hierarchy of
Lagrangian equations. It is explicitly demonstrated that the constructed system
of equations has a Lax representation and two compatible Hamiltonian
structures. The homogeneous balance method is used to derive analytic soliton
solutions of the third- and fifth-order equations.Comment: 16 pages, 1 figur
On the supersymmetric nonlinear evolution equations
Supersymmetrization of a nonlinear evolution equation in which the bosonic
equation is independent of the fermionic variable and the system is linear in
fermionic field goes by the name B-supersymmetrization. This special type of
supersymmetrization plays a role in superstring theory. We provide
B-supersymmetric extension of a number of quasilinear and fully nonlinear
evolution equations and find that the supersymmetric system follows from the
usual action principle while the bosonic and fermionic equations are
individually non Lagrangian in the field variable. We point out that
B-supersymmetrization can also be realized using a generalized Noetherian
symmetry such that the resulting set of Lagrangian symmetries coincides with
symmetries of the bosonic field equations. This observation provides a basis to
associate the bosonic and fermionic fields with the terms of bright and dark
solitons. The interpretation sought by us has its origin in the classic work of
Bateman who introduced a reverse-time system with negative friction to bring
the linear dissipative systems within the framework of variational principle.Comment: 12 pages, no figure
Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrodinger equation
We have studied the modulational instability (MI) of the higher-order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearities in an optical context and presented an analytical expression for MI gain to show the effects of non-Kerr nonlinearities and higher-order dispersions on MI gain spectra. In our study, we demonstrate that MI can exist not only for the anomalous group-velocity dispersion (GVD) regime, but also in the normal GVD regime in the HNLS equation in the presence of non-Kerr quintic nonlinearities. The non-Kerr quintic nonlinear effect reduces the maximum value of the MI gain and bandwidth and plays a sensitive role over the Kerr nonlinearity, which leads to continuous wave breaking into a number of stable wave trains of ultrashort optical pulses that can be used to generate the stable supercontinuum white-light coherent sources