Related papers: Landis-type results for discrete equations
We consider the semilinear heat equation $u_t=\Delta u+u^p$ on ${\mathbb R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions defined for all…
A linearized Vlasov-Poisson system of equations is transformed into a Schr\"{o}dinger equation, which is used to demonstrate that the fluctuation theorem holds for the relative stochastic entropy, defined in terms of the probability density…
We revisit aspects of dynamics and stability of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. By a combination of analytic and numerical techniques, we show that localized initial conditions…
Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…
We consider the focusing nonlinear Schr\"odinger equations $i\partial_t u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$.…
We consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schr\"odinger equations \[ i\partial_t u - \Delta^2 u + \mu \Delta u = - |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq…
We study the convergence rates of the semi-discrete (SD) method originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics,…
We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has…
We consider the one-dimensional discrete linear Schrodinger (DLS) equation perturbed by a conservative stochastic dynamics, that changes the phase of each particles, conserving the total norm (or number of particles). The resulting total…
In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schr\"odinger equations $iu_t + \Delta u = |u|^{p-1}u$, given initial data which is analytic with fixed radius. It is shown that the uniform radius of…
Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately…
We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional…
We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schr\"odinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling…
We propose and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and…
We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to the cubic…
We study solutions of the 2D Ginzburg-Landau equation -\Delta u+\frac{1}{\ve^2}u(|u|^2-1)=0 subject to "semi-stiff" boundary conditions: the Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann condition for the phase.…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N},…
We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schr\"odinger equations of the…
We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the classical continuous setting benefits from Gaussian…