**7**

votes

**3**answers

449 views

### Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**1**

vote

**1**answer

216 views

### Interior gradient estimate for uniformly elliptic equations

I am struggling with a problem like this: In dimension $n\geq 3$,
For the following uniformly elliptic equation, do we have interior gradient estimates?
$$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where ...

**-2**

votes

**2**answers

135 views

### Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?

**1**

vote

**0**answers

120 views

### Calogero-Moser eigenfunction

The folllowing function
\begin{equation}
J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...

**-6**

votes

**1**answer

86 views

### Non Linear PDE's [closed]

I want to solve two systems of questions being
$$\frac{C(r,y)''}{C(r,y)} + \frac{C(r,y)'^2}{C(r,y)^2}=0$$
and
$$A(r,y)'' + 2A(r,y)' \frac{C(r,y)'}{C(r,y)}=0$$
where ' is differentiating with respect ...

**1**

vote

**1**answer

450 views

### Global version of the Picard-Lindelöf theorem [closed]

Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) intervall and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ ...

**1**

vote

**0**answers

32 views

### Backlund transformation related to two NL differential equations

I'm looking for a Backlund transformation linking the following two nonlinear differential equations for real $t$:
$$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$
...

**6**

votes

**0**answers

159 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 ...

**1**

vote

**1**answer

274 views

### Quadratic PDE Systems

(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)
I have a problem that leads me to the following quadratic system of PDEs:-
$
c_1 ...

**3**

votes

**0**answers

76 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 $p, q$.

**2**

votes

**0**answers

97 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 ...

**1**

vote

**1**answer

198 views

### Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...

**4**

votes

**1**answer

209 views

### Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude).
Given $g \in (0,\infty), k \in C^1( [0, ...

**1**

vote

**0**answers

148 views

### What transformations preserve the von Mises distribution?

The von Mises distribution is entirely defined on the circle with a density given by
$$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$
where $x$ is in an arbitrary real interval of ...

**2**

votes

**0**answers

435 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

**1**answer

214 views

### How to find the general solution of Mathieu differential equation?

Two solutions of the equation are the Mathieu sine and cosine and they form an orthogonal basis. The equation has arisen from the wave equation for an infinite string with a sinusoidal cross-section. ...

**1**

vote

**1**answer

96 views

### A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...

**1**

vote

**3**answers

171 views

### Harmonic Function with special property

I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...

**6**

votes

**1**answer

173 views

### Are there any explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to add spin into account in this scheme, it ...

**2**

votes

**1**answer

116 views

### First order ODE and possible periodic solutions

I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions:
$\partial_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$.
This equation roughly ...

**0**

votes

**1**answer

176 views

### Legendre differential equation with additional term

In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( ...

**8**

votes

**1**answer

188 views

### Differential analogue of the newton polygon

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows:
Let us recall the ...

**3**

votes

**1**answer

92 views

### Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...

**0**

votes

**1**answer

106 views

### first order differential equation

Is there any solution formula for a differential equation like this
$$
y'(x)=f(x)y(x)+g(x)y(ax)\qquad \text{where}\qquad a\gt1.
$$
I have a differential equation a little more complicated than this. ...

**2**

votes

**1**answer

136 views

### General four-term recurrence relations

I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form
...

**3**

votes

**2**answers

256 views

### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

**2**

votes

**0**answers

80 views

### Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of ...

**0**

votes

**0**answers

51 views

### how can i solve this nonlinear problem?

let be the differential equation in dimension $ n > 3 $ or $ n =3 $
$$ -\Delta u =|grau|^{2}u $$ (1)
with the constraint that the vector 'u' $ |u|=1 $
then how could i prove that the unitary ...

**1**

vote

**0**answers

86 views

### G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...

**1**

vote

**0**answers

62 views

### Single parameter bifurcations caused by a simple additive term

Note: I asked this question on Math.SE over two months ago, and it has not received any answers.
Motivation: A practical dynamical system is often described by an ODE that has a parameter that ...

**1**

vote

**1**answer

241 views

### A property of groups of operators

Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...

**1**

vote

**0**answers

61 views

### Allen Cahn Equation with Dirichlet Data

consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions ...

**0**

votes

**1**answer

185 views

### Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ ...

**5**

votes

**1**answer

449 views

### Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange.
My question is where the following differential equation arises naturally and where it ...

**2**

votes

**0**answers

92 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 ...

**1**

vote

**0**answers

48 views

### Special case of Forced Harmonic Oscillator

$$ \frac{d^2l}{dt^2} + c \frac{dl}{dt} +k= F_0\sqrt{1-l^2}$$
Either a closed form or something like given initial conditions, the max value of $l$. For small $l$, this boils down to simple step ...

**6**

votes

**3**answers

232 views

### On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...

**-2**

votes

**1**answer

170 views

### This theorem is true or false in infinite dimensions?

Consider a vector space $E$ with finite dimension and linear map $A: E \rightarrow E$. The following statements are equivalent:
$x'(t)=A \circ x(t)$ defines an attractor.
All eigenvalues of $A$ have ...

**1**

vote

**0**answers

140 views

### Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the
Navier Stokes equations can be written as a semilinear symmetric first
order system
$$
u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u)
$$
for a 9 by 1 vector $u$ containing the ...

**15**

votes

**3**answers

986 views

### I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension

Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?
Here, Peano's theorem is:
Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re ...

**2**

votes

**1**answer

312 views

### Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...

**5**

votes

**0**answers

121 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 ...

**2**

votes

**0**answers

227 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 ...

**1**

vote

**0**answers

96 views

### Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...

**0**

votes

**0**answers

66 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**1**

vote

**0**answers

75 views

### Implementing boundary conditions to an ODE (involving elliptic integrals)

I am trying to solve the following differential equation:
$$
\frac{\mathrm{d} f}{\mathrm{d} x} = \frac{x^2-2 a}{\sqrt{4k^2-(x^2-2 a)^2}},
$$
where $a$ and $k$ are constants ($k$ is known and $a$ is ...

**2**

votes

**1**answer

250 views

### Solution to Seiberg-Witten monopole equation

To understand some physics problem, I want to know if there is (non-$L^2$ or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$
\begin{equation}
{D}_A \psi = 0\\
F_A^+ = ...

**3**

votes

**0**answers

138 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 ...

**4**

votes

**2**answers

242 views

### Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...

**1**

vote

**1**answer

187 views

### Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...