Related papers: Slightly Two or Three Dimensional Self-Similar Sol…
In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right.…
In this article we find explicit formulae for spherically symmetric solutions of the multidimensional zero-pressure gas dynamics system and its adhesion approximation. The asymptotic behaviour of the explicit solutions of the adhesion…
Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a…
Approximations of the Sedov self-similar solution for a strong point explosion in a medium with the power-law density distribution \rho^o\propto r^{-m} are reviewed and their accuracy are analyzed. Taylor approximation is extended to cases…
In this paper we propose an accurate, and computationally efficient method for incorporating adaptive spatial resolution into weakly-compressible Smoothed Particle Hydrodynamics (SPH) schemes. Particles are adaptively split and merged in an…
We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…
We present n-dimensional vortex-ring-like and potential-like solutions with unusual properties related to some elliptical differential equations with compact sources. Solutions have almost 3- or 2-dimensional behaviour in the spaces with…
We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} d\theta_t +…
We compare the statistical properties of galaxies found in two different models of hierarchical galaxy formation: the semi-analytic model of Cole et al. and the smoothed particle hydrodynamics (SPH) simulations of Pearce et al. Using a…
We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a…
It is shown how a complete set of hydrodynamic equations describing an unsteady three-dimensional viscous flow nearby a solid body, can be reduced to a closed system of surface equations using the method of dimension reduction of…
In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra…
We highlight some of the interesting properties of a new and finite, exact family of solutions of 1 + 1 dimensional perfect fluid relativistic hydrodynamics. After reviewing the main properties of this family of solutions, we present the…
Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the…
This paper is devoted to find out cylindrically symmetric kinematic self-similar perfect fluid and dust solutions. We study the cylindrically symmetric solutions which admit kinematic self-similar vectors of second, zeroth and infinite…
Self-diffusion in a two-dimensional simple fluid is investigated by both analytical and numerical means. We investigate the anomalous aspects of self-diffusion in two-dimensional fluids with regards to the mean square displacement, the…
This paper discusses some of the physical properties of plane symmetric self-similar solutions of the first kind (i.e., homothetic solutions). We are interested in calculating the expansion, the acceleration, the rotation, the shear tensor,…
Three (2+1)-dimensional equations, they are KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by different transformation of variables respectively. Since the single solitary wave…
Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the…
The objective of this paper is to study the plane symmetric kinematic self-similar heat conducting fluid and charge dust solutions of the Einstein field equations. These solutions are classified according to self-similarity of the first,…