Sevdzhan Hakkaev

Local Well-Posedness and Orbital Stability of Solitary Wave Solutions for the Generalized Camassa–Holm Equation

ABSTRACT We establish local well-posedness in the Sobolev space H s with any for the generalized Camassa–Holm equation. Furthermore, we consider the stability and instability problem of the solitary wave solutions.

Stability of periodic travelling shallow-water waves determined by Newton's equation

We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin–Bona–Mahony and Camassa–Holm equations. To prove orbital stability, we use the abstract results of Grillakis–Shatah–Strauss and the Floquet theory for periodic eigenvalue problems.

Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability

We construct various periodic traveling wave solutions of the Ostrovsky/Hunter--Saxton/short pulse equation and its KdV regularized version. For the regularized short pulse model with small Coriolis parameter, we describe a family of periodic traveling waves which are a perturbation of appropriate KdV solitary waves. We show that these waves are spectrally stable. For the short pulse model, we construct a family of traveling peakons with corner crests. We show that the peakons are spectrally stable as well.

STABILITY OF PERIODIC TRAVELING WAVES FOR THE QUADRATIC AND CUBIC NONLINEAR SCHRÖDINGER EQUATIONS

We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schrodinger equations in one space dimension.