Related papers: Global solution to the cubic Dirac equation in two…

In this paper we prove global well-posedness and scattering for the defocusing, cubic, nonlinear wave equation on $\mathbf{R}^{1 + 3}$ with radial initial data lying in the critical Sobolev space $\dot{H}^{1/2}(\mathbf{R}^{3}) \times…

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1(\mathbb{R}^3)$. The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation.…

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small…

The Dirac equation is exactly solved for a pseudoscalar linear plus Coulomb-like potential in a two-dimensional world. This sort of potential gives rise to an effective quadratic plus inversely quadratic potential in a Sturm-Liouville…

In the present article a semilinear wave equation with scale-invariant damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension $n$ is proved using weighted $L^\infty-L^\infty$…

This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity…

We solved the one-dimensional position-dependent mass Dirac equation in the presence of the cusp potential and reported the solutions in terms of the Whittaker functions. We have derived the reflection and transmission coefficients by…

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of…

The two-body Dirac equation with general local potential is reduced to the pair of ordinary second-order differential equations for radial components of a wave function. The class of linear + Coulomb potentials with complicated spin-angular…

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue…

We study a spatially inhomogeneous coagulation model that contains a transport term in the spatial variable. The transport term models the vertical motion of particles due to gravity, thereby incorporating their fall into the dynamics.…

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the…

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces…

We give a streamlined account of $2$-spinors, up to and including the Dirac equation, using little more than the resources of linear algebra. We prove that the Dirac bundle is isomorphic to the associated bundles $\mathrm{SL}_2(\mathbb{C})…

We prove that the initial value problem for the Dirac equation $ \left ( -i\gamma^\mu \partial_\mu + m \right) \psi = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)\right) \psi \quad \text{in } \ \R^{1+3} $ is globally well-posed…

In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter…

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution $\partial_{t}^2u-\Delta u=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial solution $u$ with…

We consider the energy-supercritical nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori}…

In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schr\"odinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this…

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative…