Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

488 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
23 votes
0 answers
688 views

What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...
user avatar
  • 7,057
22 votes
0 answers
481 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 ...
user avatar
16 votes
0 answers
2k 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 ...
user avatar
  • 12.5k
13 votes
1 answer
1k views

The perturbation of non-Hamiltonian algebraic vector fields

In this question, we are interested in the number of limit cycles which appears in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
user avatar
12 votes
0 answers
240 views

Galois groups of classical differential equations

I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following: Is there a book or article devoted (either partially or completely) to ...
user avatar
  • 4,420
11 votes
0 answers
415 views

Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
user avatar
  • 2,902
10 votes
0 answers
661 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
user avatar
10 votes
0 answers
257 views

Is there a classification of differential equations over the field of fractions of formal power series? (characteristic 0)

Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ...
user avatar
  • 7,057
10 votes
0 answers
679 views

Algebraic geometry and PDEs (reference-request)

Context: Let's say we have an affine algebraic variety corresponding to the zero set of an irreducible polynomial (over $\mathbb{C}$) in $n$ variables, denoted by $p(x_1, \dots, x_n)$. $$p(x_1, \dots, ...
user avatar
  • 2,477
8 votes
0 answers
315 views

On a remark of Langlands

I'm been wondering about this for a while and hope someone can enlighten me. In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I ...
user avatar
8 votes
0 answers
233 views

Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary. Let $M^2$ be a compact connected 2-manifold and $\...
user avatar
8 votes
0 answers
263 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 ...
user avatar
8 votes
0 answers
660 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 Picard-...
user avatar
7 votes
0 answers
322 views

On the solvability of a nonlinear differential system

A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
user avatar
  • 59
7 votes
0 answers
182 views

No intermediate denominators growth for holonomic functions?

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form $$ a(n+k) = \sum_{i=0}^{k-1} p_i(...
user avatar
7 votes
0 answers
357 views

What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?

I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations. Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the ...
user avatar
  • 4,281
7 votes
0 answers
243 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: $$\frac{d}{dt}a_k=2(b_k^2-b_{k-1}^2),$$ ...
user avatar
  • 4,822
7 votes
0 answers
1k 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 ...
user avatar
7 votes
0 answers
436 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 ...
user avatar
  • 1,163
7 votes
0 answers
203 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}, \delta_{...
user avatar
  • 12.2k
6 votes
0 answers
128 views

Do you recognize these numbers related to the higher Airy equations?

I'm studying the higher Airy equations $$\left[\big({-}\tfrac{\partial}{\partial y}\big)^{n-1} - y\right] \psi = 0$$ under a coordinate transformation. The interesting coefficients $c_n^{(1)}, \ldots, ...
user avatar
  • 2,159
6 votes
0 answers
110 views

Analysis of solutions to a system of nonlinear ODEs arising from differential geometry

Consider the system of ODEs: \begin{equation} \varphi''\varphi'^{q-1}\psi'^{p-2}=\varphi^{p-1}\psi^{q-1}, \end{equation} \begin{equation} \varphi'^2+\psi'^2=1, \end{equation} where $\varphi>0$, $\...
user avatar
6 votes
0 answers
158 views

Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

I have a question for you. BACKGROUND Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical ...
user avatar
  • 95
6 votes
0 answers
328 views

Had this theorem in Tresser's article been proven somewhere?

The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
user avatar
  • 165
6 votes
0 answers
293 views

Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?
user avatar
  • 61
6 votes
0 answers
729 views

Grothendieck problem

Could you suggest me a book or a link where I can find some information about the Grothendieck problem about differential equations? The Grothendieck problem that I am reffering to is the following: ...
user avatar
  • 299
6 votes
0 answers
2k 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 ...
user avatar
6 votes
0 answers
372 views

An algebraic 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 ...
user avatar
6 votes
0 answers
192 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 $\...
user avatar
6 votes
0 answers
136 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
user avatar
  • 13.2k
6 votes
0 answers
530 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?
user avatar
  • 429
5 votes
0 answers
36 views

An equality satisfied by the solutions to Kolmogorov forward and backward PDEs

Let $b: \mathbb R_+\times\mathbb R\to \mathbb R$ and $\sigma: \mathbb R_+\times\mathbb R\to (0,\infty)$ be functions as nice as possible (e.g. bounded and of bounded partial derivatives, and $\inf_{(t,...
user avatar
  • 1,386
5 votes
0 answers
77 views

Lie groupoids as symmetries of mechanical systems?

Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...
user avatar
  • 181
5 votes
0 answers
190 views

Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$

Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to $$ \Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
user avatar
  • 241
5 votes
0 answers
196 views

Nonlinear variation of constants formula

Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
user avatar
  • 51
5 votes
0 answers
142 views

Solve nonlinear, forced and damped Duffing oscillator

I am trying to solve a Duffing type equation by using Van Der Paul's method: \begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align} with $$x(t) = Re[A(t) \...
user avatar
5 votes
0 answers
171 views

Overtwisted contact forms on open manifolds

I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here. It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...
user avatar
5 votes
0 answers
161 views

Uniqueness of solutions of Young differential equations

Consider the following one dimensional Young differential equation: $$ Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1]. $$ Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\...
user avatar
  • 821
5 votes
0 answers
231 views

Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
user avatar
  • 325
5 votes
0 answers
135 views

Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
user avatar
5 votes
0 answers
171 views

Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
user avatar
5 votes
0 answers
307 views

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let $f$ be ...
user avatar
  • 1,494
5 votes
0 answers
112 views

A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$ Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...
user avatar
5 votes
0 answers
216 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 finite-...
user avatar
5 votes
0 answers
169 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 ...
user avatar
5 votes
0 answers
233 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 ...
user avatar
5 votes
0 answers
192 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} ...
user avatar
  • 24.5k
4 votes
0 answers
222 views

Generalising Bäcklund transform to solve $\omega''(t)=t\sin\omega(t)$

Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation $$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*}\label{...
user avatar
4 votes
0 answers
105 views

The semiclassical limit of Virasoro reps $\varphi_{n,1}$ produces certain $\mathfrak{sl}_2$ reps — what is the connection to KdV?

The semiclassical ("light") limit $c\to \infty$ of the irreducible Virasoro representation $\varphi_{n,1}$ with highest weight $h_{n,1}\to -\frac{n-1}{2}$ is $\mathbb{C}[L_{-1},L_{-2},\dotsc]...
user avatar
4 votes
0 answers
58 views

A continuity argument for a dispersive $gKdV$ estimate

I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The ...
user avatar

1
2 3 4 5
10