We consider an autonomous Hamiltonian system $\ddot {q}+V_q(q)=0$ in ${\bf R}^2$, where the potential $V$ has a global maximum at the origin and singularities at some points $\xi_1$, $\xi_2 \in {\bf R}^2 \setminus \{0\}$. Under some compactness conditions on $V$ at infinity and assuming a strong force type condition at the singularities, we study, using variational arguments, the existence of various types of heteroclinic and homoclinic solutions of the system.

The limit Wigner measure of a WKB function satisfies a simple transport equation in phase-space and is well suited for capturing oscillations at scale of order $O(\epsilon)$, but it fails, for instance, to provide the correct amplitude on caustics where different scales appear. We define the semi-classical Wigner function of an $N$-dimensional WKB function, as a suitable formal approximation of its scaled Wigner function. The semi-classical Wigner function is an oscillatory integral that provides an $\epsilon$-dependent regularization of the limit Wigner measure, it obeys a transport-dispersive evolution law in phase space, and it is well defined even at simple caustics.

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position and time form the intersection of a plane in the space of slownesses with the slowness surface. The field and especially its singularities are strongly dependent on the varying geometry of these loops; we use a level set numerical technique to compute those real loops which essentially gives us second order accuracy. We give the static term corresponding to the mode with zero wave speed. Numerical evaluation of the solution is presented graphically followed by some concluding remarks.

The purpose of this paper is to give a pricing analysis for the American option in a jump-diffusion model by PDE arguments. Existence and uniqueness of the solution to the obstacle problem for the associated model is shown in suitable spaces. We also prove the unique existence of the solution of the corresponding free boundary problem. Furthermore, smoothness and monotonicity of the free boundary which is the optimal exercise boundary of the option are deduced.

Droplet profiles under the influence of van der Waals forces

Euro. Jnl. of Applied Maths.12, 367–393

ELI MINKOV and AMY NOVICK-COHEN