Related papers: Exponentially Localized Solutions of Mel'nikov Equ…
In this thesis we consider so-called linear evolutionary problems, a class of linear partial differential equations covering classical elliptic, parabolic and hyperbolic equations from mathematical physics as well as classes of…
We consider the helical reduction of the wave equation with an arbitrary source on $(n+1)$-dimensional Minkowski space, $n\geq2$. The reduced equation is of mixed elliptic-hyperbolic type on ${\bf R}^n$. We obtain a uniqueness theorem for…
In this paper we show the existence of strictly monotone heteroclinic type solutions of semilinear elliptic equations in cylinders. The motivation of this construction is twofold: first, it implies the existence of an entire bounded…
In this paper we study the numerical method for approximating the random periodic solution of semiliear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating…
This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$,…
Ishimori equation is a $(2+1)$ dimensional generalization of the $(1+1)$ dimensional integrable classical continuous Heisenberg ferromagnetic spin equation. The richness of the coherent structures admitted by Ishimori equation I such as…
This paper develops a new class of exponential-type integrators where all the matrix exponentiations are performed in a single Krylov space of low dimension. The new family, called Lightly Implicit Krylov-Exponential (LIKE), is well suited…
We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of…
A system of linear equations in $\mathbb{F}_p^n$ is \textit{Sidorenko} if any subset of $\mathbb{F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty$. A system of…
We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of…
We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. We do not need to know the explicit formulas for the homoclinic orbits prior…
We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…
In this work, we proved the existence of a unique global mild solution of the d-dimensional incompressible Navier-Stokes equations, for small initial data in Besov type spaces based on mixed-Lebesgue spaces; namely, mixed-norm…
We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be…
We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are…
In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlev\'e VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux…
The Goldberg-Ostrovskii problem asks whether finite-order solutions of a linear differential equation inherit the property of completely regular growth (c.r.g.) from its coefficients. While Bergweiler's counterexample demonstrated that the…
We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space…