Related papers: Single-valley-extended solutions with platforms of…
Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave…
The Gas-Kinetic Scheme (GKS), widely used in computational fluid dynamics for simulating hypersonic and other complicated flow phenomena, is extended in this work to electromagnetic problems by solving Maxwell's equations. In contrast to…
A periodic two-phase algebro-geometric solution of the focusing nonlinear Schr\"odinger equation is constructed in terms of elliptic Jacobi theta-functions. A dependence of this solution on the parameters of a spectral curve is…
In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme…
Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter $N$ or continuous parameter $x$. The analysis of these equations reveals to which order they…
We consider the nonlinear Schr\"{o}dinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2\pi$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system…
In this note we use the normal forms of the completely resonant non--linear Schr\"odinger equation on a torus (NLS) derived in previous work in order to produce, under a KAM algorithm, large families of stable and unstable quasi periodic…
The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a $d$--dimensional torus $\T^d$. The main point is to prove that prove that the normal form is…
Spatial chaos as a phenomenon of ultimate complexity requires the efficient numerical algorithms. For this purpose iterative low-dimensional maps have demonstrated high efficiency. Natural generalization of Feigenbaum and Ikeda maps may…
Using the Riemann-Liouville and Caputo Fractional Standard Maps (FSM) and the Fractional Dissipative Standard Map (FDSM) as examples, we investigate types of solutions of non-linear fractional differential equations. They include periodic…
Rarefied gas effects are of critical importance for the aerodynamic performance of hypersonic vehicles operating at high altitudes. In these scenarios, conventional computational fluid dynamics (CFD) solvers break down as the linear…
Some multidimensional generalizations of the Fokker-Planck equation used by R. Friedrich and J. Peinke for the description of a turbulent cascade as a stochastic process of Markovian type, are considered. The exact solutions of the Cauchy…
We study the emergence of pattern formation and chaotic dynamics in the one-dimensional (1D) generalized Kuramoto-Sivashinsky (gKS) equation by means of a time-series analysis, in particular a nonlinear forecasting method which is based on…
We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order.…
We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…
Quasi-periodic solutions with Liouvillean frequency of forced nonlinear Schr\"odinger equation are constructed. This is based on an infinite dimensional KAM theory for Liouvillean frequency.
In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove…
All complex fluid motions, such as transition and turbulence, obeying the Navier-Stokes equations are non-linear phenomena. Some aspects of the non-linear terms of these equations are not well understood and are, in fact, misunderstood. The…
We found a new kind of soliton solutions for the 5-parameter family of the potential-free Stenflo-Sabatier-Doebner-Goldin nonlinear modifications of the Schr\"odinger equation. In contradistinction to the "usual'' solitons like…