2018/07/30 by Mathew A. Johnson, Johnson, Mathew A., J. Douglas Wright +1
Mathematics · #76B45 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:76B45
paper · pdf · doi:10.48550/arxiv.1807.11469
arxiv created 2018/07/30 · arxiv updated 2018/07/31
We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur & Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between 0 and 1/3.