TY - JOUR
T1 - Initial value problem of the Whitham equations for the Camassa-Holm equation
JF - Physica D 238 (2009) 55-66
Y1 - 2009
A1 - Tamara Grava
A1 - Virgil U. Pierce
A1 - Fei-Ran Tian
AB - 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.
PB - Elsevier
UR - http://hdl.handle.net/1963/3429
U1 - 906
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - RPRT
T1 - Large Parameter Behavior of Equilibrium Measures
Y1 - 2006
A1 - Tamara Grava
A1 - Fei-Ran Tian
AB - 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).
JF - Commun. Math. Sci. 4 (2006) 551-573
UR - http://hdl.handle.net/1963/1789
U1 - 2755
U2 - Mathematics
U3 - Mathematical Physics
ER -