Related papers: Hyperbolic Monge-Amp\`ere systems with $S_1=0$
The complex Monge-Amp\`ere operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded…
We define a nondegenerate Monge-Amp\`ere structure on a 6-dimensional manifold as a pair $(\Omega,\omega)$, such that $\Omega$ is a symplectic form and $\omega$ is a 3-differential form which satisfies $\omega\wedge\Omega=0$ and which is…
We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of…
The existence and multiplicity and nonexistence of nontrivial radial convex solutions of systems of Monge-Amp\`ere equations are established with superlinearity or sublinearity assumptions for an appropriately chosen parameter. The proof of…
The real homogeneous Monge-Amp\`{e}re equation in one space and one time dimensions admits infinitely many Hamiltonian operators and is completely integrable by Magri's theorem. This remarkable property holds in arbitrary number of…
It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1}…
The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…
We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.
An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems.…
Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample…
We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this…
We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (\Omega,\omega), such that \Omega is a symplectic form and \omega is a 3-differential form which satisfies…
The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate…
We revisit the convex integration constructions for the Monge-Amp\`ere system and prove its flexibility in dimension $d=2$ and codimension $k=3$, up to $\mathcal{C}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the…
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of K\"ahler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Amp\`ere…
We consider a 3rd-order generalized Monge-Ampere equation u_yyy - u_xxy^2 + u_xxx u_xyy = 0 (which is closely related to the associativity equation in the 2-d topological field theory) and describe all integrable structures related to it…
In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.
Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is…
Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by…
In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Amp\`ere equation with weak initial data. As…