Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
587
questions with no upvoted or accepted answers
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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 ...
22
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491
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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 ...
16
votes
0
answers
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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 ...
14
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1
answer
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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) }
\...
12
votes
0
answers
268
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
12
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0
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257
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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 ...
11
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452
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 $Y$ ...
10
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783
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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 ...
10
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262
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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 ...
10
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865
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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, ...
9
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When is the solution to a linear system of ODEs an algebraic variety?
Question: Are the following observations well known, and in what general context?
Let $A$ be a diagonalizable $n\times n$ matrix over $\mathbb{C}$ and consider the following system of differential ...
9
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474
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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 ...
8
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397
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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 ...
8
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251
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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 $\...
8
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0
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265
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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 ...
8
votes
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712
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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-...
7
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353
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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 ...
7
votes
0
answers
199
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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(...
7
votes
0
answers
451
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 ...
7
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265
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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:
$$\frac{d}{dt}a_k=2(b_k^2-b_{k-1}^2),$$
...
7
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0
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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
$$\delta_{11}, \delta_{...
6
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166
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Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
6
votes
0
answers
137
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Do you recognize these numbers related to the higher Airy equations?
I'm studying the higher Airy equations
$$\left[\big({-}\tfrac{\partial}{\partial y}\big)^{n-1} - y\right] \psi = 0$$
under a coordinate transformation. The interesting coefficients $c_n^{(1)}, \ldots, ...
6
votes
0
answers
96
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A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
6
votes
0
answers
126
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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$, $\...
6
votes
0
answers
210
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
6
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180
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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 ...
6
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335
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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 ...
6
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297
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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?
6
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774
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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: ...
6
votes
0
answers
3k
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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 ...
6
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454
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 ...
6
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answers
200
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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 $\...
6
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0
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140
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.
$$
...
6
votes
0
answers
557
views
When is a vector field on a surface a Lie bracket
On a Riemannian 2 manifold, when is a vector field (possibly defined only on the manifold minus a finite number of points) the Lie bracket of two mutually orthogonal vector fields?
5
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206
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
5
votes
0
answers
211
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
5
votes
0
answers
110
views
Lie groupoids as symmetries of mechanical systems?
Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...
5
votes
0
answers
196
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
5
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352
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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, ...
5
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167
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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 $$x(t) = Re[A(t) \...
5
votes
0
answers
231
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Overtwisted contact forms on open manifolds
I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here.
It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...
5
votes
0
answers
1k
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Are those distributional solutions that are functions, the same as weak solutions?
There are two closely related concepts and I am not sure exactly how close. Consider a partial differential equation. (The coefficients need not be constant but assume they are functions, and not ...
5
votes
0
answers
254
views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
5
votes
0
answers
138
views
Algebraic independence of limit cycles of Lienard equation
It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related ...
5
votes
0
answers
210
views
Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$
I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
5
votes
0
answers
127
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A criterion for a differential equation to be realized as an Euler-Lagrange equation on the infinite dimensional space
I study PDEs that arise in fluid dynamics in an infinite dimensional Riemannian geometric perspective. For example, Ebin-Marsden(1970) showed that the group of volume preserving diffeomorphisms has an ...
5
votes
0
answers
305
views
Is the closed orbit of the Van der Pol equation a stable periodic orbit?
We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$
It is well known that this equation has a unique limit ...
5
votes
0
answers
353
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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 ...
5
votes
0
answers
114
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 ...