We show that the Painlevé equations PIII −PV I can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrödinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three different branches, each described by an infinite chain of equations. The Painlevé equations are obtained by closing the chain periodically at the lowe...

The Euler’s equations describing the dynamics of capillarygravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussinesq type of equation is derived involving the small-amplitude and longwavelength parameters. A recently introduced sixth-order Bouss...

In this paper, the Exp-function method is used to obtain generalized travelling wave solutions with free parameters of the MKdV-sine-Gordon and Boussinesq-double sine-Gordon equations. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides an effective mathematical tool for nonlinear evolution equations arising in mathematical physics.

The improved Boussinesq equation is studied in this paper. Control properties for this equation posed on a bounded interval are first considered. When the control acts through the Dirichlet boundary condition the linearized system is proved to be approximately but not spectrally controllable. In a second part, the equation is posed on the one-dimensional torus and distributed moving controls ar...

Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. Various families of their travelling wave solutions are generated, which include periodic wave, solitary wave, periodic-like wave, solitonlike wave, Jacobi elliptic function periodic wave, combined non-degenerative Jacobi elliptic function-like wave, Weierstrass ellip...