Speaker 1 :  Mats Ehrnström  (Norwegian University of Science and Technology)
    Date & Time : November 19th (Fri.), 2021 / 10:00-10:40PM (Tokyo (JST)) 
                  (--> November 19th (Fri.), 2021 / 2:00-2:40PM (Oslo (CET)) ) 
    Title : On the bifurcation diagram of the capillary-gravity Whitham equation
    Abstract : 
         We study the bifurcation of periodic travelling waves of the capillary-gravity 
       Whitham equation. This is a nonlinear pseudo-differential equation that combines 
       the canonical shallow water nonlinearity with the exact (unidirectional) dispersion 
       for finite-depth capillary-gravity waves. Starting from the line of zero solutions, 
       we give a complete description of all small periodic solutions, unimodal as well 
       bimodal, using simple and double bifurcation via Lyapunov--Schmidt reductions. 
       Included in this study is the resonant case when one wavenumber divides another. 
       Some bifurcation formulas are studied, enabling us, in almost all cases, to continue 
       the unimodal bifurcation curves into global curves. By characterizing the range of 
       the surface tension parameter for which the integral kernel corresponding to the 
       linear dispersion operator is completely monotone (and therefore positive and convex; 
       the threshold value for this to happen turns out to be T = 4/\pi^2, not the critical 
       Bond number 1/3, we are able to say something about the nodal properties of solutions, 
       even in the presence of surface tension. Finally, we present a few general results 
       for the equation and discuss, in detail, the complete bifurcation diagram as far as 
       it is known from analytical and numerical evidence. Interestingly, we find, 
       analytically, secondary bifurcation curves connecting different branches of solutions; 
       and, numerically, that all supercritical waves preserve their basic nodal structure, 
       converging asymptotically in L^2 but not in L^\infty towards one of the two constant 
       solution curves.
         This is joint work with Mathew A. Johnson (University of Kansas), Ola Maehlen 
       (NTNU/University og Oslo) and Filippo Remonato (NTNU/Sintef Research Foundation).