**2**

votes

**1**answer

125 views

### Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...

**0**

votes

**0**answers

54 views

### Reference request: Heat kernel regularity near the boundary

Let $D$ be a domain in $\mathbb{R^d}$ and $p(t,x,y)$ be the heat kernel of $D$ (for the Dirichlet problem). I was told that if the boundary of $D$ is real-analytic, then the function $y\mapsto ...

**0**

votes

**1**answer

70 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 ...

**1**

vote

**0**answers

48 views

### About the integral of the plurisubharmonic functions on the whole manifold

Let $(M,\omega)$ be a Kahler manifold with positive first Chern class. $\omega\in c_1(M)$ is a Kahler current. $D$ is a smooth simple divisor in $-K_M$. $s$ is a determining section of $D$. The ...

**0**

votes

**0**answers

46 views

### Existence of BVP solutions for non-linear Bessel-like equation

The basic problem: we have the differential equation
$$
g''(r) + \frac{2}{r} g'(r) - \frac{2}{r^2} g(r) = F(g(r))
$$
in which $F(g)$ satisfies $F(0)=F(1)= 0$ and $F(x) > 0$ for $0<x<1$. ...

**-1**

votes

**0**answers

26 views

### Is there a program for convenient working with equations and coefficients? [migrated]

I perform some calculations with one differential equation. Then I got a huge expression depending on $x$ and its degrees/powers. E.g.
$$\alpha x+(4-x+\sqrt[3]{x})^2-(\beta\sqrt{x}+\frac12(x^3+1))^3 + ...

**1**

vote

**1**answer

55 views

### Solving Bessel-like equation by using Bessel Kernel [closed]

Consider the drifted-Bessel equation as follows.
\begin{equation}
x^2\ddot y + x\dot y + (x^2-n^2)y=f,
\end{equation}
where $n$ is an integer and $f$ is a known function. If $f\equiv 0$, the solution ...

**7**

votes

**1**answer

465 views

### Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$:
$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...

**5**

votes

**0**answers

74 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

35 views

### Is there an explicit ODE solution for this system?

y'(t)=A(t)*y(t)+g(t),
A(t)=[-t^2, t;
sin(t) cos(t)]
g(t)=(1+t/2;
-1)
y(0)=(3;
1)
I actually do not know how to solve an ODE with variable constants, but when I use the ...

**2**

votes

**1**answer

80 views

### Global Solutions of Ordinary Differential Equations

Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...

**2**

votes

**0**answers

150 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

**3**answers

229 views

### Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...

**1**

vote

**0**answers

67 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 ...

**1**

vote

**1**answer

139 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 ...

**0**

votes

**1**answer

320 views

### Teaching profession:Differential Equations and Mean Value Theorems

Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...

**0**

votes

**0**answers

58 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.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**18**

votes

**0**answers

825 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 ...

**4**

votes

**2**answers

177 views

### Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...

**3**

votes

**1**answer

615 views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**2**

votes

**1**answer

172 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^+ = ...

**1**

vote

**0**answers

44 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 ...

**9**

votes

**1**answer

267 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

**3**

votes

**0**answers

90 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 ...

**7**

votes

**1**answer

351 views

### Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...

**0**

votes

**1**answer

298 views

### phase portrait of system of differential equations

Is there a full classification of the phase portraits of the following systems of differential equations
\begin{equation}
\cases{
\dot x=a_{11}x+a_{12}y+a_{13}z \\
\dot y=a_{2 1}x+a_{22}y+a_{23}z\\
...

**4**

votes

**2**answers

195 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) ...

**2**

votes

**2**answers

352 views

### Does a (smooth, constant-rank, integrable) distribution have a (local) basis of divergence-free vector fields?

In coordinate-free language, my question is as follows. Let $M$ be an $n$-dimensional manifold with volume form, and let $\mathcal D$ be a smooth (integrable, if necessary) distribution with constant ...

**0**

votes

**0**answers

47 views

### ODE system with a very large number of equations

Are there any theory about a ode system (linear or nonlinear) with a very large number of equations, e.g. more than $10^5$ equations ?

**2**

votes

**1**answer

120 views

### how to solve this system of nonlinear differential equations

I want to solve this system
\begin{align*}\tag{*}
x'(s)=x^2(s)+y(s),
y'(s)=x(s)y(s)
\end{align*}
with initial conditions $$x(0)=t, y(0)=t,$$
where $t\not=0.$
With the help of Maple, the solution is
...

**15**

votes

**6**answers

1k views

### Catenary curve under non-uniform gravitational field

The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ...

**2**

votes

**0**answers

128 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}
...

**0**

votes

**0**answers

43 views

### Solvability and uniqueness of Fokker-Planck BVP

I have been searching for solvability for the following BVP, I believe that this is a Fokker Planck's equation but I can't find any comprehensible text on existence and uniqueness of the solutions of ...

**0**

votes

**0**answers

72 views

### Wave operator for focusing NLS

Consider the NLS equation
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u+u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where ...

**0**

votes

**0**answers

29 views

### Green function and translational symmetry

I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ...

**5**

votes

**1**answer

190 views

### Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...

**2**

votes

**0**answers

224 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) }
...

**0**

votes

**1**answer

108 views

### Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation :
$\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...

**5**

votes

**3**answers

143 views

### Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...

**0**

votes

**0**answers

41 views

### Mathieu equation

I have come across different versions of Mathieu equation. I am trying to solve for the stability of an inverted pendulum with oscillating base. If there any correspondence between different version ...

**1**

vote

**3**answers

146 views

### General systems of linear differential equations with variable coefficients

I am wondering what can be said in general about the fundamental matrix of a system of linear differential equations. For simplicity, let $A(t)\in\mathbb{C}^{n\times n}$ be a time dependent matrix, ...

**1**

vote

**1**answer

182 views

### Simple example of why Differential Equations can be NP Hard [closed]

Just looking for a simple example of why Differential Equations can be NP hard
Edit:
It appears that the answer below may be what I was looking for, but I am clarifying just in case:
Slides ...

**0**

votes

**1**answer

381 views

### Continuous variation from solution of easy problem to solution of hard problem

I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...

**3**

votes

**1**answer

194 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...

**3**

votes

**2**answers

231 views

### existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a ...

**3**

votes

**1**answer

59 views

### Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...

**1**

vote

**0**answers

37 views

### Id monodromy in hamiltonian dynamics

In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...

**5**

votes

**1**answer

515 views

### Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?

**0**

votes

**0**answers

46 views

### How to apply initial conditions to the Mathieu differential equation?

The Mathieu equation was a result of the separation of variables method used on a wave equation for a string with sinusoidally changing cross-section with the coordinate, $S(x)=S_1-S_2 \cos{x}$. The ...

**0**

votes

**0**answers

381 views

### Inconsistency in definition of characteristics for a linear PDE? Folland versus Fritz John.

There seems to be a major inconsistency (perhaps due to my lack of understanding) between what Folland calls a "characteristic" and what I had previously thought was a characteristic.
For example, ...