Bio: Dmitry E. Pelinovsky is a professor of applied mathematics at McMaster University.

He received MSc in 1993 at Nizhny Novgorod State University (Russia), PhD in 1997 at Monash University (Australia), and postdoctoral training in University of Cape Town (South Africa) and University of Toronto (Canada). His research includes analysis and applications of nonlinear partial differential equations and lattice equations for propagation of nonlinear waves and coherent structures. He is the Editor-in-Chief for Studies in Applied Mathematics (since 2020), the Deputy Editor-in-Chief for Physica D (since 2021), and a member of the editorial board of Physical Review E (since 2021). He is an author of the books "Stability of nonlinear waves in Hamiltonian dynamical systems" (AMS, 2025) and "Localization in periodic potentials: from Schrodinger operators to the Gross-Pitaevskii equation" (Cambridge University Press, 2011).

Abstract: We study irrotational gravity waves based on conformal transformations of Euler's equations. Stokes waves are traveling waves with either smooth or peaked periodic profiles. We give a rigorous proof that the zero eigenvalue bifurcation in the stability problem for co-periodic perturbations occurs at each extremal point of the energy function versus the steepness parameter, provided that the wave speed is not extremal at the same steepness. The normal form for the unstable eigenvalues is extended for subharmonic perturbations to explain all numerically observed instabilities of water waves with increasing steepness towards the limiting peaked profile.