**73**

votes

**0**answers

5k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**19**

votes

**0**answers

307 views

### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each ...

**17**

votes

**0**answers

883 views

### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...

**14**

votes

**0**answers

1k views

### The radius of convergence of the p-adic exponential function.

As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is
$$\rho = p^{-1/(p-1)}.$$
This is typically proven by computing ...

**7**

votes

**0**answers

284 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**7**

votes

**0**answers

138 views

### What subspaces of n-tuples of rational functions can be the solution space to a system of differential equations?

Let $K=\mathbb{C}(x)$ denote the field of rational functions (in 1 variable), and let $K^n$ denote an $n$-dimensional $k$-vector space (with basis). For some integer $m$, let
$$\delta_{11}, ...

**6**

votes

**0**answers

226 views

### Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B.
Wonderfulness of roller comes from this property ...

**6**

votes

**0**answers

145 views

### Toda Flow Embeddings

What are strategies for generating the following types of pictures:
Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are:
...

**6**

votes

**0**answers

153 views

### The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...

**6**

votes

**0**answers

283 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**5**

votes

**0**answers

109 views

### Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...

**5**

votes

**0**answers

581 views

### A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
...

**5**

votes

**0**answers

95 views

### Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...

**5**

votes

**0**answers

202 views

### Linear ODEs in a locally convex vector space

Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...

**5**

votes

**0**answers

286 views

### Rigorous results on the method of multiple scales

The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation ...

**5**

votes

**0**answers

317 views

### When is a vector field on a surface a Lie bracket

On a Riemannian 2 manifold, when is a vector field (possibly defined only on the manifold minus a finite number of points) the Lie bracket of two mutually orthogonal vector fields?

**5**

votes

**0**answers

344 views

### Differential Equations Satisfied by Modular Forms

In Verrill's paper preprint here, she has the following theorem which is from a paper of Stiller. It states that
Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{R})$ commensurable with ...

**5**

votes

**0**answers

436 views

### When is the monodromy group of a linear differential equation dense in the Galois group?

Given a system $Y'=A(t)Y$ with only regular singular points, then a theorem of Schlesinger says that the Zariski closure of the monodromy group is equal to the Galois group of the corresponding ...

**4**

votes

**0**answers

428 views

### A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE):
$\frac{d}{dt}y= A(t) y$, where $A(t)$ is a ...

**4**

votes

**0**answers

169 views

### Constructing solutions to matrix equations

Let $k,n$ be integers, $u_1,\dots,u_n \in U(k)$, $d_1,\dots,d_n \in \mathbb Z$ with $\sum_{i=1}^{n} d_i =: d \neq 0$.
Consider the map $w:= U(k) \to U(k)$ with
$$w(v):= u_1 v^{d_1} \cdots u_n v^{d_n} ...

**3**

votes

**0**answers

64 views

### Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...

**3**

votes

**0**answers

137 views

### System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x ...

**3**

votes

**0**answers

139 views

### A special non vanishing vector field on $S^{3}$

Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for ...

**3**

votes

**0**answers

132 views

### Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an ...

**3**

votes

**0**answers

93 views

### Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.
It seems to me that something like the ...

**3**

votes

**0**answers

75 views

### Is there any weighted sobolev embedding with non-decaying weight

Is there any weighted sobolev embedding like
$$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$?
Here $s>0$, and for some appropriate $p, q$.

**3**

votes

**0**answers

128 views

### Nonexistence of Limit Cycle

Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where ...

**3**

votes

**0**answers

110 views

### Approximating solutions of non-linear differential equations

I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
...

**3**

votes

**0**answers

222 views

### May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa)

By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ...

**3**

votes

**0**answers

88 views

### Classical parabolic theory (PDEs)

I am reading an articule(http://link.springer.com/content/pdf/10.1007%2Fs00028-010-0085-8.pdf) so I almost done but I dont understand the next argument is in the page 8 "It is known from the ...

**3**

votes

**0**answers

231 views

### Periodic orbits of Hamiltonian systems

Consider a second order Hamiltonian system of the type
$$
\ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N.
$$
Under very `natural assumptions' it is possible to prove the existence of a non constant ...

**3**

votes

**0**answers

373 views

### PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let ...

**3**

votes

**0**answers

177 views

### closure properties of q-differential equations

I am interested in q-differential equations of the form
$p(f(z), f(qz),\dots,f(q^kz))=0$
where $p$ is a polynomial and $k$ an nonnegative integer. I wonder about the closure properties of the class ...

**3**

votes

**0**answers

877 views

### (Approximate) analytic solutions to the Mathieu equation

I'm trying to solve the driven Mathieu equation
$x''+\beta x'+(a-2q\cos{\Omega t})\frac{\Omega^2}{4}x=f(t)$
for both zero and non-zero $\beta$.
I can write down an analytic solution using the ...

**3**

votes

**0**answers

338 views

### Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs

$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...

**2**

votes

**0**answers

47 views

### About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper:
A concave–convex elliptic problem involving the fractional Laplacian -
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez.
At the Pgs 41, 42, the ...

**2**

votes

**0**answers

54 views

### Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...

**2**

votes

**0**answers

45 views

### Non-autonomous O.D.E with discontinuous and not integrable R.H.S

Consider the non-autonomous O.D.E
$\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$
where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$
and $h:\Bbb R^d \times S \to \Bbb R^d$
where ...

**2**

votes

**0**answers

95 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

**2**

votes

**0**answers

102 views

### Differential equation related to the Schwarzschild metric

How can one find solutions of the following second-order diﬀerential equation
$$\frac{d^2W}{dr^2}-\frac{1}{r}\frac{dW}{dr}=\frac{C}{W^2}\frac{dW}{dr}$$
with the boundary condition $W(r)\to r^2$ at ...

**2**

votes

**0**answers

112 views

### A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...

**2**

votes

**0**answers

69 views

### How to find a Lax Pair for the modified KdV equation

Question
I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as:
$
X = ...

**2**

votes

**0**answers

92 views

### Variant form of the gronwall inequality

I know the following statement for gronwall inequality:
Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have,
$f' \leq \phi f$ and $f(0)=0$ then $f=0$
Now is ...

**2**

votes

**0**answers

423 views

### The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**2**

votes

**0**answers

89 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**2**

votes

**0**answers

212 views

### J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle ...

**2**

votes

**0**answers

174 views

### Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...

**2**

votes

**0**answers

151 views

### Solving ODE with negative expansion power series

I'm moving this here, as suggested from physics.stackexchange. The original is here.
So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...

**2**

votes

**0**answers

139 views

### A natural tower of bundles over Grassmannian manifolds

There is a well-known exact sequence of vector bundles
$$
0\to U\to T\to N\to 0
$$
over a Grassmannian manifold $Gr(V,n)$: the fibers over $L\in Gr(V,n)$ are, respectively, $L$ itself, the whole $V$ ...

**2**

votes

**0**answers

107 views

### How to find solutions of non-linear ODE with particular BCs

What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a ...