910 research outputs found
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
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
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