Related papers: Special solutions of nonlinear von Neumann equatio…
We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend…
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and…
We present a new complex non-stationary particle-like solution of the non-linear Klein-Gordon equation with several spatial variables. The construction is based on reduction to an ordinary differential equation.
We study elliptic and parabolic problems governed by singular elliptic operators \begin{equation*} \mathcal L =\sum_{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N,…
For a five-parameter discrete $\phi^4$ model, we derive various exact static solutions, including the staggered ones, in the form of the basic Jacobi elliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of their hyperbolic…
In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov…
The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of…
In this work we prove the uniqueness of solutions to the nonlocal linear equation $L \varphi - c(x)\varphi = 0$ in $\mathbb{R}$, where $L$ is an elliptic integro-differential operator, in the presence of a positive solution or of an odd…
This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented $n$ dimensional Riemannian manifold with a…
We consider a Cahn-Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is characterized by a Helmholtz free energy density, which can be of logarithmic type. Moreover, the…
We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary…
We study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schr\"odinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely…
In this paper, the trial function method is employed to find the exact solutions for high-order nonlinear Schr\"odinger equations with time-dependent coefficients. This system describes the propagation of ultrashort light pulses in…
Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~\Omega\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~\Omega\\ u,v>0~~\text{in}~~\Omega\\ u=v=0~\quad~\text{on}~~\partial\Omega, \end{cases} \] where $\Omega$ is an open…
The Relationship between the Neumann system and the Jacobi system in arbitrary dimensions is elucidated from the point of view of constrained Hamiltonian systems. Dirac brackets for canonical variables of both systems are derived from the…
This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new…
Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative…
The nonlinear Schr\"{o}dinger-Newton system \begin{equation*} \begin{cases} \Delta u- V(|x|)u + \Psi u=0, &~x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by…
We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators…
We prove the existence and uniqueness of the complexified Nonlinear Poisson-Boltzmann Equation (nPBE) in a bounded domain in $\mathbb{R}^3$. The nPBE is a model equation in nonlinear electrostatics. The standard convex optimization argument…