124 research outputs found
On Kaup-Kupershchmidt–type equations and their soliton solutions
We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the
MKdV equations gauge equivalent to the KKE. Next we outline the symmetry and the spectral properties of the relevant Lax operator. Using the dressing Zakharov-Shabat method we demonstrate that the MKdV and KKE have two types of onesoliton solutions and briefly comment on their properties
On the Caudrey-Beals-Coifman System and the Gauge Group Action
The generalized Zakharov-Shabat systems with complex-valued Cartan elements
and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their
gauge equivalent are studies. This includes: the properties of fundamental
analytical solutions (FAS) for the gauge-equivalent to CBC systems and the
minimal set of scattering data; the description of the class of nonlinear
evolutionary equations solvable by the inverse scattering method and the
recursion operator, related to such systems; the hierarchies of Hamiltonian
structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings
(Submitted to J. Nonlin. Math. Phys.
Superintegrability in the Manev Problem and its Real Form Dynamics
We report here the existence of Ermanno-Bernoulli type invariants for the
Manev model dynamics which may be viewed upon as remnants of Laplace-Runge-Lenz
vector whose conservation is characteristic of the Kepler model. If the orbits
are bounded these invariants exist only when a certain rationality condition is
met and thus we have superintegrability only on a subset of initial values. We
analyze real form dynamics of the Manev model and derive that it is always
superintegrable. We also discuss the symmetry algebras of the Manev model and
its real Hamiltonian form.Comment: 12 pages, LaTeX, In: Prof. G. Manev's Legacy in Contemporary
Astronomy, Theoretical and Gravitational Physics, V. Gerdjikov, M. Tsvetkov
(Eds), Heron Press, Sofia 2005, pp. 155-16
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