**2**

votes

**0**answers

117 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 ...

**1**

vote

**2**answers

194 views

### Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...

**2**

votes

**0**answers

123 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 ...

**4**

votes

**1**answer

195 views

### Reference request: Invariant sets of dynamical systems

(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...

**2**

votes

**1**answer

110 views

### hypergeometric at nearest singularity

Reference request. A prototype case:
In
$$
{}_2F_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the ...

**0**

votes

**1**answer

81 views

### A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable

First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
...

**2**

votes

**0**answers

68 views

### Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...

**2**

votes

**2**answers

304 views

### Monge-Ampere type PDE

NB: I have edited this question to clarify what the OP is asking – Robert Bryant
Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = ...

**1**

vote

**1**answer

38 views

### Can we implicitly fit a system of linear ODEs by reduced information?

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...

**-3**

votes

**1**answer

169 views

### The logarith map as a contraction [closed]

Two Questions:
(1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping?
Or more ...

**4**

votes

**1**answer

207 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 ...

**1**

vote

**0**answers

44 views

### singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation
$$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$
where $\gamma\in (1, 2]$ is some ...

**0**

votes

**0**answers

221 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**20**

votes

**8**answers

7k views

### Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such.
Yet, they could have been just ...

**0**

votes

**0**answers

45 views

### A hyperbolic partial differential equation

How solve this equation (numeral or analytical)?
$u(t,x)=\int_{t-x}^{t}{a \cdot e^{b \cdot s} \cdot \int_{0}^{s-(t-x)}{u(s,z)dz}ds}+\int_{t-x}^{t}{c \cdot e^{d \cdot s} \cdot ...

**6**

votes

**5**answers

399 views

### Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...

**13**

votes

**2**answers

889 views

### What justification can you give for the fact that “most ODEs do not have an explicit solution”?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?

**9**

votes

**2**answers

237 views

### curvature flow for loops in S^2

Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...

**1**

vote

**0**answers

65 views

### Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...

**2**

votes

**0**answers

89 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 = ...

**0**

votes

**1**answer

155 views

### Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...

**3**

votes

**0**answers

159 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 ...

**11**

votes

**2**answers

1k views

### How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...

**43**

votes

**2**answers

2k views

### What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE.
In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...

**0**

votes

**0**answers

34 views

### Appropriately choosing the parameter so that one function maximizes while other minimizes

Suppose we have two functions $f$ and $g$ of some variables $x,y,z\in R$. Our goal is to appropriately choose the values of $x,y,z$ such that $f$ is maximized while $g$ is minimized.
Here, I am not ...

**-2**

votes

**1**answer

121 views

### reducing an n-order differential equation to a first order system of equations using either sagemath or sympy [closed]

I want to reduce a n-order ordinary differential equation into a first order system of equations. This is in preparation for numerical analysis. I use both Sympy and Sagemath for Computer Algebra, but ...

**9**

votes

**1**answer

712 views

### Examples of continuous differential equations with no solution

Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem.
...

**2**

votes

**2**answers

113 views

### Making a system of second-order ODEs chaotic

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x_1, x_2, ..., x_N)$, we have
...

**4**

votes

**1**answer

79 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

**3**

votes

**1**answer

458 views

### Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is
\begin{align}
\nabla \times B + i\omega E &= 0\\
\nabla \times E - i\omega B &= 0 \\
\nabla \cdot B &= 0 \\
\nabla ...

**0**

votes

**2**answers

151 views

### Blow up solution for a Riccati's equation

I apologize for the problem too simple, but I'm not able at solving it.
Consider the Cauchy problem
$$
\left\{
\begin{array}{l}
\dot x=x(t)^2+t\\
x(0)=0
\end{array}
\right.
$$
Show that its solution ...

**3**

votes

**0**answers

96 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 ...

**-1**

votes

**1**answer

133 views

### Solution to simple first-order partial differential equations [closed]

Is there a general solution for first-order partial differential equations of the form
$$m(x) \partial_x f(x,y) = n(y) \partial_y f(x,y)$$
for given $m(x),n(y)$ and reasonable boundary conditions ...

**5**

votes

**2**answers

253 views

### Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& ...

**0**

votes

**1**answer

94 views

### equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...

**6**

votes

**0**answers

147 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:
...

**1**

vote

**1**answer

55 views

### Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...

**3**

votes

**2**answers

85 views

### Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation
$$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$
If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate ...

**0**

votes

**0**answers

37 views

### Uniqueness of Newton (modulo a constant) series on a compact set

Good morning everybody. My question is as follows:
Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$.
Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It ...

**0**

votes

**0**answers

35 views

### Solution of ODE related to normal density

This question below is from mathse. I reqeustioned here because very limited respond there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...

**1**

vote

**0**answers

126 views

### A differential equation on toric Kahler manifolds

Let $M$ be a toric Kahler manifold with $\text{dim}_{\mathbb{R}} = 4$. Let $V$ be a Killing vector field associated to the action of the two-torus $\mathbb{T}^2$ on $M$. We also assume the existence ...

**2**

votes

**2**answers

401 views

### How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations:
$$
\left[
\begin{array}{ccccccc}
\text{No.}& t & y_1(t)&y_2(t) & ...

**2**

votes

**1**answer

169 views

### The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...

**1**

vote

**0**answers

84 views

### Smooth normal forms of vector fields (the path method)

I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of ...

**0**

votes

**0**answers

136 views

### Existence of the Dirichlet heat kernel for arbitrary open subsets?

consider first of all an open and bounded subset $\Omega\subset\mathbb{R}^n$, s.t. the boundary $\partial \Omega$ is a manifold of class $C^2$. Then I know that there exists a Dirichlet heat kernel, ...

**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

215 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 ...