The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of KdV with linear dispersion full water wave problem. conjectured existence highest, cusped, traveling-wave solution, and his conjecture was recently verified in periodic case by Ehrnstr\"om Wahl\'en. In present paper we prove it for solitary waves. Like case, proof based on global bifurcation...

We show that Whitham type equations u_t + u u_x -L = 0, where L is a general Fourier multiplier operator of order \alpha \in [-1,1], \alpha\neq allow for small solutions to be extended beyond their expected existence time. The result valid range quadratic dispersive with inhomogeneous symbols in the given by \alpha, and should extendable other same relative strength.

The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We...