اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites
129
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We have derived exact solutions of the isothermal Lane--Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are most relevant to the dynamics of the protosolar nebula before and during the stages of planet and satellite formation. The nonrotating solution for the mass density is analytic, nonsingular, monotonically decreasing with radius, and it satisfies easily the usual physical boundary conditions at the center. When differential rotation is added to the Lane--Emden equation, an entire class of exact solutions for the mass density appears. We have determined all of these solutions analytically as well. Within this class, solutions that are power laws or combinations of power laws are not capable of satisfying the associated boundary--value problem, but they are nonetheless of profound importance because they constitute baselines to which the actual solutions approach when the central boundary conditions are imposed. Numerical integrations that enforce such physical boundary conditions show that the actual radial equilibrium density profiles emerge from the center close to the nonrotating solution, but once they cross below the corresponding baselines, they cease to be monotonic. The actual solutions are forced to oscillate permanently about the baseline solutions without ever settling onto them because the central boundary conditions strictly prohibit such settling, even in the asymptotic regime of large radii. Based on our results, we expect that quasistatically--evolving protoplanetary disks should develop oscillatory density profiles in their midplanes during their isothermal phase. The peaks in these profiles correspond to local potential minima and their locations are ideal sites for the formation of protoplanets ...
قيم البحث
اقرأ أيضاً
We classify solutions of finite Morse index of the fractional Lane- Emden equation
We study positive solutions to the fractional Lane-Emden system begin{equation*} tag{S}label{S} left{ begin{aligned} (-Delta)^s u &= v^p+mu quad &&text{in } Omega (-Delta)^s v &= u^q+ u quad &&text{in } Omega u = v &= 0 quad &&text{in } Omega^c={mat
hbb R}^N setminus Omega, end{aligned} right. end{equation*} where $Omega$ is a $C^2$ bounded domains in ${mathbb R}^N$, $sin(0,1)$, $N>2s$, $p>0$, $q>0$ and $mu,, u$ are positive measures in $Omega$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $mu$ and $ u$. Furthermore, if $p,q in (1,frac{N+s}{N-s})$ and $0 leq mu,, uin L^r(Omega)$ for some $r>frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on fini
te strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
We study stable positive radially symmetric solutions for the Lane-Emden system $-Delta u=v^p$ in $R^N$, $-Delta v=u^q$ in $R^N$, where $p,qgeq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
The singularly perturbed Riccati equation is the first-order nonlinear ODE $hbar partial_x f = af^2 + bf + c$ in the complex domain where $hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with presc
ribed asymptotics as $hbar to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $hbar$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrodinger equation with a rational potential.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
