Fei-Ran Tian

Singularities in Hele-Shaw flows

We study, analytically and numerically, singularities in an interface flow driven by surface tension and a sink in a two-dimensional Hele--Shaw cell. Our analysis proves that singularity formation is inevitable in generic situations. Our numerical simulations suggest that the Hele--Shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out. A corner is formed right at the tip of the interface that touches the sink.

A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation

We study the solution of the focusing nonlinear Schrodinger equation in the semiclassical limit. Numerical solutions are presented for four different kinds of initial data, of which three are analytic and one is nonanalytic. We verify numerically the weak convergence of the oscillatory solution by examining the strong convergence of the spatial average of the solution.

ON THE WHITHAM EQUATIONS FOR THE DEFOCUSING COMPLEX MODIFIED KDV EQUATION

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasi-linear hyperbolic equations and describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV equation are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solutions of the Whitham equations when the initial values are given by a step-like function. We also compare the results with those of the defocusing nonlinear Schrodinger (NLS) equation. For the NLS equation, the Whitham equations are strictly hyperbolic and genuinely nonlinear. We show that the weak hyperbolicity of the mKdV–Whitham equations is responsible for some new structure in the dispersive shocks which has not been found in the NLS case.

On the Initial Value Problem of the Whitham Averaged System

The initial value problem in question is concerning the Whitham averaged system:

Singularities in Hele--Shaw Flows Driven by a Multipole

{{\beta }_{{it}}} + {{\lambda }_{i}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right){{\beta }_{{ix}}} = 0i = 1,2,3

On the breaking of the single phase Whitham solution

(1) where

Oscillations of the zero dispersion limit of the korteweg‐de vries equation

{{\lambda }_{1}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right) = 2\left( {{{\beta }_{1}} + {{\beta }_{2}} + {{\beta }_{3}}} \right) + 4\left( {{{\beta }_{1}} - {{\beta }_{2}}} \right)\frac{{K\left( s \right)}}{{E\left( s \right)}}