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Interactions of supersonic uniform streams with cylindrical bodies, placed in open channels, are studied. Channels of rotation with the interval of cross-sectional area decreasing are considered. Two-dimensional Euler equations are solved…
In this paper, traveling wave solutions to the nonlinearly dispersive Schr\"odinger equation are given in the case of one-dimensional non-relativistic electron confined to a cylindrical quantum well. Investigations gave evidence to the…
This paper investigates inverse potential problems of wave equations with cubic nonlinearity. We develop a methodology for establishing stability estimates for inversion of lower order coefficients. The new ingredients of our approach…
From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, new theories state that transitions between extreme waves are allowed. However, these have never been experimentally observed…
An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting…
Extracorporeal Shock Wave Therapy (ESWT) is a noninvasive treatment for a variety of musculoskeletal ailments. A shock wave is generated in water and then focused using an acoustic lens or reflector so the energy of the wave is concentrated…
Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time…
The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two…
We study the stability of an explicitly known, non-trivial self-similar blowup solution of the quadratic wave equation in the lowest energy supercritical dimension $d = 7$. This solution blows up at a single point and extends naturally away…
We formulate a smoothed-particle hydrodynamics numerical method, traditionally used for the Euler equations for fluid dynamics in the context of astrophysical simulations, to solve the non-linear Schrodinger equation in the Madelung…
We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data,…
A technique is given to generate coupled scalar field solutions in colliding Einstein - Maxwell (EM) waves. By employing the Bell - Szekeres solution as seed and depending on the chosen scalar field it is possible to construct nonsingular…
We introduce a new class of numerical methods for solving McKean-Vlasov stochastic differential equations, which are relevant in the context of distribution-dependent or mean-field models, under super-linear growth conditions for both the…
The Harten-Lax-van Leer with contact (HLLC) scheme is known to be plagued by various forms of numerical shock instabilities. In this paper, we propose a new framework for developing shock stable, contact and shear preserving approximate…
This paper presents a new method allowing for a numerical study of the interaction between a submerged solid structure and a current with surface waves. The method involves a Navier-Stokes equation solver for a two-fluid formulation with…
We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the nonlinear wave system. This shock diffraction problem can be formulated as a…
In this article we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg-de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a…
One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a…
In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well known cone solution, which is locally…
High Mach number shocks are ubiquitous in interstellar turbulence. The Pencil Code is particularly well suited to the study of magnetohydrodynamics in weakly compressible turbulence and the numerical investigation of dynamos because of its…