Related papers: The well-posedness of the Cauchy problem for self-…

Numerical simulations of the Cauchy problem for self-interacting massive vector fields often face instabilities and apparent pathologies. We explicitly demonstrate that these issues, previously reported in the literature, are actually due…

We study the well-posedness of the initial value (Cauchy) problem of vacuum Einstein-aether theory. The latter is a Lorentz-violating gravitational theory consisting of General Relativity with a dynamical timelike 'aether' vector field,…

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real…

We introduce a class of "Lipschitz horizontal" vector fields in homogeneous groups, for which we show equivalent descriptions, e.g. in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness…

Previous studies have identified potential instabilities in self-interacting vector theories associated with the breakdown of the well-posedness of the initial-value problem. However, these conclusions are restricted to Abelian vector…

Scalar-tensor theories with first-derivative self interactions, known as $k$-essence, may provide interesting phenomenology on cosmological scales. On smaller scales, however, initial value evolutions (which are crucial for predicting the…

We show that self-interacting vector field theories exhibit unphysical behaviour even when they are not coupled to any external field. This means any theory featuring such vectors is in danger of being unphysical, an alarming prospect for…

We investigate some well-posedness issues for the initial value problem (IVP) associated to the system \begin{equation} \{ \begin{array} [c]{l} 2i\partial_{t}u+q\partial_{x}^{2}u+i\gamma\partial_{x}^{3}u=F_{1}(u,w)\\…

We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem…

In this paper, we consider the initial value problem for the Einstein-Vlasov-Scalar field equations in temporal gauge, where the initial data are prescribed on two characteristic smooth intersecting hypersurfaces. From a suitable choice of…

Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…

A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…

For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…

We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on…

As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…

The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes…

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for…

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the…

We study the vacuum Cauchy problem for K-essence, i.e. cosmologically relevant scalar-tensor theories that involve first-order derivative self-interactions, and which pass all existing gravitational wave bounds. We restrict to spherical…