Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn self-similar solutions of the vortex filament equation
67
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study self-similar solutions of the binormal curvature flow which governs the evolution of vortex filaments and is equivalent to the Landau-Lifshitz equation. The corresponding dynamics is described by the real solutions of $sigma$-Painlev{e} IV equation with two real parameters. Connection formulae for Painlev{e} IV transcendents allow for a complete characterization of the asymptotic properties of the curvature and torsion of the filament. We also provide compact hypergeometric expressions for self-similar solutions corresponding to corner initial conditions.
rate research
Read More
We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0<s ll 1$; they are exact corners of slope $s$ at $t=0$ and become smooth in $x$ for $t>0$.
begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrodinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.
We consider the nonlinear heat equation $u_t = Delta u + |u|^alpha u$ with $alpha >0$, either on ${mathbb R}^N $, $Nge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) alpha <4$, for every $mu in {mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = mu |x-x_0|^{-frac {2} {alpha }}$ in a neighborhood of some $x_0in Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $mu |x-x_0|^{-frac {2} {alpha }}$ on ${mathbb R}^N $. Moreover, if $mu ge mu _0$ for a certain $ mu _0( N, alpha )ge 0$, and $u_0 Ige 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
