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In this paper we consider a multi-dimensional damped semiliear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined…
The paper deals with the problem of existence of a convergent "strong" normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear term. The…
In 1+1-dimensions, an extension of the canonical solitonic Dym equation has previously been derived both in a geometric torsion evolution context and in the analysis of peakon solitonic phenomena in hydrodynamics. Here, a novel…
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear…
Semilinear stochastic evolution equations with multiplicative L\'evy noise and monotone nonlinear drift are considered. Unlike other similar work we do not impose coercivity conditions on coefficients. Existence and uniqueness of the mild…
Quasi-periodic solutions of a nonlinear polyharmonic equation for the case $4l>n+1$ in $\R^n$, $n>1$, are studied. This includes Gross-Pitaevskii equation in dimension two ($l=1,n=2$). It is proven that there is an extensive "non-resonant"…
For solutions of (inviscid, forceless, one dimensional) Burgers equation with random initial condition, it is heuristically shown that a stationary Feller-Markov property (with respect to the space variable) at some time is conserved at…
We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…
A (2+1)-dimensional linear ultra-parabolic Fokker--Planck--Kolmogorov equation is investigated from the group-theoretical point of view. By using the Berest--Aksenov approach, an algebra of invariance of fundamental solutions of the…
In this article, we prove existence and uniqueness of the Barenblatt solution of the evolution equation on the whole Euclidean space where the principle part is the nonlocal fractional p-Laplacian composed with a power function. Our proof…
We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev…
We study a homogenisation problem for problems of mixed type in the framework of evolutionary equations. The change of type is highly oscillatory. The numerical treatment is done by a discontinuous Galerkin method in time and a continuous…
In this short note we discuss a new formula for solving the nonlocal $\overline{\partial}$-problem, and discuss application to the Manakov--Zakharov dressing method. We then explicitly apply this formula to solving the complex (2+1)D…
In this paper, we consider the existence of localized sign-changing solutions for the $p-$Laplacian nonlinear Schr\"odinger equation $$ -\epsilon^p\Delta_pu+V(x)|u|^{p-2}u=|u|^{p^*-2}u+\mu|u|^{q-2}u,~~u\in W^{1,p}(\mathbb{R}^N), $$ where…
Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature…
We investigate the Balitsky-Kovchegov (BK) equation for D=3 space-time dimensions, corresponding to one transverse coordinate, and we show that it can be solved analytically. The explicit solutions are found in the linear approximation and…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated…
We present an adaptive delaminating Levin method for evaluating bivariate oscillatory integrals over rectangular domains. Whereas previous analyses of Levin methods impose non-resonance conditions that exclude stationary and resonance…