# Questions tagged [differential-equations]

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

315 questions
439 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 ...
494 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 ...
609 views

117 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 ...
315 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 ...
266 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?
617 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: ...
181 views

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 $\... 0answers 1k views ### The Perturbation of Non Hamiltonian algebraic Vector fields In this question we are interested in the number of limit cycles which appear in the following perturbational system: \cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \... 0answers 117 views ### Lagrangean uniqueness versus Eulerian uniqueness (1) Lagrangean description. Let us consider a$N\times N$system of autonomous ODE: $$\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.$$ ... 0answers 465 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? 0answers 102 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 ... 0answers 107 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 ... 0answers 184 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-... 0answers 155 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 ... 0answers 230 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 ... 0answers 187 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 $$w(v):= u_1 v^{d_1} \cdots u_n v^{d_n} ... 0answers 112 views ### Weighted reverse Poincare inequality over a function class of neural networks We consider a probability measure supported on the whole space \mathbb{R}^n, whose density is p(x). We also consider a (one-layer) neural network function class \mathcal{C}, whose elements have ... 0answers 66 views ### Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel Let \Omega be an open subset of \mathbb{R}^d. Under regularity conditions, we know that the s-th order Sobolev space H^s(\Omega) with s \geq d/2 is a reproducing kernel Hilbert space. In ... 0answers 262 views ### Spectral Gap of Elliptic Operator Under what conditions on a(x) and domain D, the spectral gap of the elliptic operator \nabla \cdot(a(x)\cdot \nabla) defined on D, can be controlled? The boundary condition is that the ... 0answers 104 views ### An embedding question: Morrey spaces Question. If u\in L^1 and Du is in the dual of the Holder space C^\alpha, then is it possible to say u belongs to some Morrey space L^{1, \delta}? 0answers 286 views ### On modified Bessel solutions to complex ODE's using Kummer's series I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$I tried to solve it via the standard method, i.e.,... 0answers 189 views ### Does the divergent solution of this equation :f'=e^{f^{-1}} of Gevrey type and could be Borel summation applied for it? This question was asked here in MO by someone seeking for the solution of the functional -differential:f'=e^{f^{-1}} not exactly an O.D.E, and again here seeking for the growth rate of it solution ... 0answers 127 views ### Symmetry-finding with SAGE? On pp. 152-3 of Hydon's Symmetry Methods for Differential Equations (2000 ed.), he lists some computer packages for symmetry-finding. This related Mathematica StackExchange question mentions the SYM ... 0answers 77 views ### Dynamics of pairwise distances in the n-body problem Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the n-body problem, so I decided to cross post here as well. ... 0answers 124 views ### Connection between cardiac equations and untangling knots? I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul Sutcliffe. ... 0answers 155 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 ... 0answers 562 views ### Models used for the Zika virus? I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ... 0answers 122 views ### Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric? My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ... 0answers 183 views ### System of linear ODEs with hypergeometric coefficients For quite some time I have been trying to solve the following system of differential equations for the two functions G and H defined on the interval [0,1]:$$ \begin{align}x G''(x)=&\mathscr{... 0answers 407 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 \left\{ \begin{array}{rl} ... 0answers 301 views ### May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa) By a "globally boundedG$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on$\mathbb{P}^1$(then necessarily of the Fuchsian type and with ... 0answers 120 views ### Solutions to$w_x=CA_x$,$w_y=CA_y$other than$w=f(A)$and$C=f'(A)$? Let$R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let$A=A(x,y) \in R[x,y]$with$\deg(A) \geq 1$(total degree). I wish to prove or find a counterexample to the following claim: If ... 0answers 59 views ### Limits of a simple damped system Definition: Let$F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$be the Laplace transform of$f_n(t)$. Required Result: To show$\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$. Ideas: Let$G_n(s)=\frac{1}{s^{n+...
Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, \$\Phi^...