%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
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