In this paper, we present infinitely many conserved densities satisfying particular conservation law $F_{t}=(2uF)_{x}$ for the generalized Riemann equations at $N=2,3,4$. $N=2$ case, also construct corresponding to new laws containing an arbitrary smooth function. virtue of reductions and/or changes variables, related are obtained two component Hunter-Saxton equation, Gurevich-Zybin equation an...

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in unit sphere. This problem related to solvability Monge-Ampere type equations subject certain boundary conditions depending value $p$. The special case $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, give some sufficient for general $p\neq1$.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains Cn. study certain degenerate complex Monge-Ampere equations, modelled on Kahler-Ricci flow evolving algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions Cauchy-Dirichlet boundary data, we show that envelope subsolutions is semi-concave time and continuous ...

In this paper we show that stability for holomorphic vector bundles are equivalent to the existence of solutions to certain system of Monge Amp ere equations parametrized by a parameter k. We solve this fully nonlinear elliptic system by singular perturbation technique and show that the vanishing of obstructions for the perturbation is given precisely by the stability condition. This can be int...

The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Ampère...