9 research outputs found
Integrable systems with quadratic nonlinearity in Fourier space
The Lax pair representation in Fourier space is used to obtain a list of
integrable scalar evolutionary equations with quadratic nonlinearity. The
famous systems of this type such as KdV, intermediate long-wave equation (ILW),
Camassa-Holm and A. Degasperis systems are represented in this list. Some new
systems are obtained as well. The generalizations on two-dimensional and
discrete systems are discussed.Comment: 6 page
On the structure of the B\"acklund transformations for the relativistic lattices
The B\"acklund transformations for the relativistic lattices of the Toda type
and their discrete analogues can be obtained as the composition of two duality
transformations. The condition of invariance under this composition allows to
distinguish effectively the integrable cases. Iterations of the B\"acklund
transformations can be described in the terms of nonrelativistic lattices of
the Toda type. Several multifield generalizations are presented
Non-homogeneous systems of hydrodynamic type possessing Lax representations
We consider 1+1 - dimensional non-homogeneous systems of hydrodynamic type
that possess Lax representations with movable singularities. We present a
construction, which provides a wide class of examples of such systems with
arbitrary number of components. In the two-component case a classification is
given.Comment: 22 pages, latex, minor change
Chaplygin ball over a fixed sphere: explicit integration
We consider a nonholonomic system describing a rolling of a dynamically
non-symmetric sphere over a fixed sphere without slipping. The system
generalizes the classical nonholonomic Chaplygin sphere problem and it is shown
to be integrable for one special ratio of radii of the spheres. After a time
reparameterization the system becomes a Hamiltonian one and admits a separation
of variables and reduction to Abel--Jacobi quadratures. The separating
variables that we found appear to be a non-trivial generalization of
ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be
useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem
in theta-functions of the new time.Comment: This is an extended version of the paper to be published in Regular
and Chaotic Dynamics, Vol. 13 (2008), No. 6. Contains 20 pages and 2 figure