On integrability of one third-order nonlinear evolution equation

We study one third-order nonlinear evolution equation, recently introduced by Chou and Qu in a problem of plane curve motions, and find its transformation to the modified Korteweg - de Vries equation, its zero-curvature representation with an essential parameter, and its second-order recursion operator.Comment: 10 page

On Transformations of the Rabelo Equations

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On integrability of the Yao-Zeng two-component short-pulse equation

We show how the Yao-Zeng system of coupled short-pulse equations is related to the original short-pulse equation and obtain the correct zero-curvature representation of the Yao-Zeng system via this relationship.Comment: 5 page

On a Whitham-Type Equation

The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly

On two aspects of the Painleve analysis

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.Comment: 8 page

On bosonic limits of two recent supersymmetric extensions of the Harry Dym hierarchy

Two generalized Harry Dym equations, recently found by Brunelli, Das and Popowicz in the bosonic limit of new supersymmetric extensions of the Harry Dym hierarchy [J. Math. Phys. 44:4756--4767 (2003)], are transformed into previously known integrable systems: one--into a pair of decoupled KdV equations, the other one--into a pair of coupled mKdV equations from a bi-Hamiltonian hierarchy of Kupershmidt.Comment: 7 page