2013/07/31 by M. D. Groves, E. Wahlén · 1 citation
Mathematics · Physics and Astronomy · #math.AP #nlin.PS
paper · pdf · doi:10.1017/s0308210515000116
published as Proceedings of the Royal Society of Edinburgh, Section: A Mathematics 2015, 145, 791-883 · Corrected version. To appear in Proceedings of the Royal Society of Edinburgh: Section A
arxiv created 2014/05/09 · arxiv updated 2015/09/25
We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimiser of the wave energy \mathcal H subject to the constraint \mathcal I=2μ, where \mathcal I is the wave momentum and 0< μ≪ 1. Since \mathcal H and \mathcal I are both conserved quantities a standard argument asserts the stability of the set Dμ of minimisers: solutions starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg-deVries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as μ\downarrow 0