%0 Journal Article %J Physica D 238 (2009) 55-66 %D 2009 %T Initial value problem of the Whitham equations for the Camassa-Holm equation %A Tamara Grava %A Virgil U. Pierce %A Fei-Ran Tian %X We 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. %B Physica D 238 (2009) 55-66 %I Elsevier %G en_US %U http://hdl.handle.net/1963/3429 %1 906 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-01-13T13:01:16Z\\nNo. of bitstreams: 1\\n0805.2558v1.pdf: 498872 bytes, checksum: 4d721f99d6ae9840be2332f3cc6a4118 (MD5) %R 10.1016/j.physd.2008.08.016 %0 Report %D 2006 %T Large Parameter Behavior of Equilibrium Measures %A Tamara Grava %A Fei-Ran Tian %X We 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). %B Commun. Math. Sci. 4 (2006) 551-573 %G en_US %U http://hdl.handle.net/1963/1789 %1 2755 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-03-30T13:50:45Z\\nNo. of bitstreams: 1\\n92FM-2005.pdf: 291341 bytes, checksum: 132e37e5c4fc64315d52903eed85753f (MD5)