00963nas a2200133 4500008004300000245008100043210006900124260001300193520052900206100001800735700002200753700001800775856003600793 2009 en_Ud 00aInitial value problem of the Whitham equations for the Camassa-Holm equation0 aInitial value problem of the Whitham equations for the CamassaHo bElsevier3 aWe study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp.1 aGrava, Tamara1 aPierce, Virgil U.1 aTian, Fei-Ran uhttp://hdl.handle.net/1963/342900940nas a2200109 4500008004300000245005300043210005300096520060900149100001800758700001800776856003600794 2006 en_Ud 00aLarge Parameter Behavior of Equilibrium Measures0 aLarge Parameter Behavior of Equilibrium Measures3 aWe study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(x) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x).1 aGrava, Tamara1 aTian, Fei-Ran uhttp://hdl.handle.net/1963/1789