**1**

vote

**0**answers

88 views

### Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...

**0**

votes

**0**answers

81 views

### Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular:
$$
\int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M},
$$
in which ...

**0**

votes

**1**answer

145 views

### How to prove that a non-linear differential equation has a solution

I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$,
$$
\frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1],
$$
where $w:[0,1] \to [0,1]$ and $v:[0,1] \to ...

**0**

votes

**0**answers

93 views

### What is the Beltrami differential?

Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$.
Local ...

**3**

votes

**2**answers

117 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

**0**

votes

**1**answer

87 views

### Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...

**0**

votes

**0**answers

33 views

### Explicit solution for a variational inequality?

Assume $\eta_t \in \mathbb{R}$ is a given continuous function depending on time.
It is known that if we look for a continuous solution $\zeta_t \in \mathbb{R}$ depending on time of the following ...

**3**

votes

**1**answer

71 views

### Examples of systems with stable equilibria at the boundary of the phase space

Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...

**2**

votes

**0**answers

69 views

### Point Spectrum of a Second Order System of Differential Equations

Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) ...

**3**

votes

**4**answers

467 views

### Solution of second order differential equation with singularities at 0,1, and ∞

I am trying to solve the following equation;
$$
U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0
$$
where U is a function of t and C is constant.
The above ...

**1**

vote

**0**answers

60 views

### asymptotic behavior of the solution of an ordinary differential equation

I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem.
Consider the following autonomous system of two non-linear ordinary differential ...

**2**

votes

**1**answer

79 views

### Coupled differential equations

I'm looking at the following coupled set of differential equations. Because of the symmetry, I'm hoping to be able to write down the solution for $x_n(t)$ and $y_n(t)$ in terms of $f(t)$ and $g(t)$, ...

**1**

vote

**1**answer

78 views

### Runge-Kutta convergence [closed]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method:
The expression in order to solve is :
\begin{equation}
Ay^{''}+By^{'}+Cy= Cu
\end{equation}
\begin{equation}
y =OUTPUT
...

**1**

vote

**1**answer

59 views

### Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed.
In particular it is said that the poles form a deformed lattice that ...

**6**

votes

**1**answer

146 views

### Writing the quotient of solutions of linear ODEs as the solution of a nonlinear ODE

Let $A(x)$ and $B(x)$ be solutions to homogeneous linear ODEs with polynomial coefficients, i.e., $A(x)$ satisfies
$$p_KA^{(K)} + p_{K-1}A^{(K-1)} + \cdots + p_1A' + p_0A=0$$
and $B(x)$ satisfies
...

**2**

votes

**1**answer

186 views

### Partial differential equation from Kirchhoff system

I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$
$$\partial_t \vec{u} = \partial_s \vec{w} - [\vec{w} \times \vec{u}].$$
Using ...

**2**

votes

**1**answer

310 views

### Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)?
It seems that the second part of the Hilbert 16th problem is solved or is going to ...

**5**

votes

**3**answers

662 views

### Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by
$$
y''+(\delta(x)-\lambda^2)y=0.
$$
Then, to find ''bound states'', you solve on the right and find the ...

**0**

votes

**2**answers

104 views

### Class of analytically-integrable divergence-free vector fields?

Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$?
That is, I'm looking for a large class of vector fields given by ...

**0**

votes

**1**answer

91 views

### Method of characteristics [closed]

I have difficulties understanding how to solve a PDE in $\mathbb{R}^{4}$ using the method of characteristics. I have a limited background in solving PDEs. I have seen only 2-dim examples and none for ...

**0**

votes

**0**answers

114 views

### The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...

**1**

vote

**1**answer

231 views

### Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

**1**

vote

**0**answers

84 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**3**

votes

**2**answers

119 views

### Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.
Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$.
However, can all these paths ...

**1**

vote

**0**answers

61 views

### Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems.
Are there any papers or books that ...

**0**

votes

**0**answers

41 views

### Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...

**4**

votes

**1**answer

141 views

### A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x_0$ has no solution?

**0**

votes

**0**answers

75 views

### Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation
$$
-\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x)
$$
with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...

**5**

votes

**1**answer

373 views

### Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.
$$
My question is to ...

**5**

votes

**1**answer

167 views

### Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...

**1**

vote

**1**answer

175 views

### First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation
$$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$
where
$A_0, A_1, A_2, B$ are some constants, ...

**5**

votes

**1**answer

148 views

### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question:
Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as
...

**15**

votes

**1**answer

854 views

### Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes:
This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...

**0**

votes

**0**answers

76 views

### Energy Oscillations in a One Dimensional Crystal

Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...

**11**

votes

**1**answer

232 views

### Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...

**1**

vote

**1**answer

178 views

### Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...

**3**

votes

**1**answer

86 views

### Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...

**4**

votes

**2**answers

146 views

### How to find an ODE with prescribed terminal values?

Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...

**1**

vote

**1**answer

117 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**0**

votes

**1**answer

126 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**1**

vote

**2**answers

204 views

### Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation:
$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$
which is essentiall in matrix notation:
$\dot{\mathbf{x}} = ...

**-1**

votes

**1**answer

293 views

### Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...

**8**

votes

**1**answer

373 views

### Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg:
Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...

**5**

votes

**0**answers

139 views

### Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...

**1**

vote

**1**answer

140 views

### How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE
$$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$
using Charpit's method. The ...

**2**

votes

**0**answers

93 views

### Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
...

**10**

votes

**3**answers

594 views

### Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...

**0**

votes

**0**answers

67 views

### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...

**2**

votes

**1**answer

148 views

### acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...

**32**

votes

**2**answers

3k views

### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...