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We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a…
We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space…
In this article, we study the following problem $$-{\rm div} (\omega(x)|\nabla u|^{N-2} \nabla u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R^{N}}$,…
In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using…
We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the…
We investigate the existence of local holomorphic solutions $Y$ of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety $\Theta$ in $\mathbb{C}^{2}$. The coefficients are…
A class of nonlocal nonlinear wave equation arises from the modeling of a one dimensional motion in a nonlinearly, nonlocally elastic medium. The equation involves a kernel function with nonnegative Fourier transform. We discretize the…
We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…
We study semilinear elliptic equations with the fractional Laplacian in $\mathbb{R}$. The equations with single power nonlinearities have been observed by Weinstein(1987), Frank--Lenzmann(2013) and so on. We focus on the equations with…
Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions…
The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces.
New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…
We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
The nonlinear signal processing has achieved a rapid process in the recent years. A family of nonlinear Fourier bases, as a typical family of mono-component signals, has been constructed and applied to signal processing. In this paper, the…
Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series…
This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = \alpha^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $\alpha$ is a…