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Questions tagged [differential-equations]

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

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700 views

Has this Peculiar Property of Unit Circles Already been Noticed?

Yesterday I needed to do some calculations with circles and "ventured" to calculate the arc length via the $\int{\sqrt{1+\left(f'(x)\right)^2}}$ formula and was baffled to see that in the case of unit ...
3
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1answer
222 views

Analytic solutions to algebraic differential equation

Dear Colleagues and Friends, Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive. I believe that it's known that if a ...
1
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0answers
51 views

Existence of a solution to a boundary value problem

Consider an ODE of the form $$y''=f(x,y,y') $$ with $x \geq 0$, and initial conditions of the form $$y(0)=y_0>0, \\y'(0)=m_0. $$ I want to claim that under reasonable conditions (which I can't ...
2
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1answer
223 views

A cubic system with two nested limit cycles with opposite orientations

What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
5
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2answers
242 views

Flow of a nowhere vanishing complete vector field

Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, ...
3
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1answer
189 views

what is about the corresponding power series?

According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes: periodicity ~ potential well or lattice (order) lack of absolutely continued spectrum ~ Anderson ...
3
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2answers
285 views

Symbol of the Laplace-Beltrami on $\mathbb{S}^2$

This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e. A differential operator $P=\sum_{|\...
3
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0answers
79 views

Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE $$\frac{\mathrm{d}^2 y}{\...
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2answers
238 views

Simple-looking nonlinear ODE with fractional power

I am trying to solve the following nonlinear ODE for a function $P(x)$: $$\left(1-x^2\right)P''(x)+k(k+1)P(x)=cP(x)^\frac{k-1}{k+1}.$$ Here, $k$ and $c$ are arbitrary parameters. By rescaling $P(x)$,...
2
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1answer
37 views

Purpose of using a saturable logistic like term

I would like to know what is the purpose of using the term $P\over (k+P)$ in the following. I found it when reading the article found here but it was commonly used in few other related articles . Is ...
2
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2answers
493 views

Sobolev trace theorem on Lipschitz domains

Supposing that D is a bounded Lipschitz domain (and not smooth) in $\mathbb{R}^d$. From what I know, it is known that the trace operator is well-defined and continuous from $H^s(D)$ to $H^l(\partial D)...
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1answer
77 views

Does there exist a comparison principle for vector differential equations?

As is well known, there is a comparison principle for scalar differential equations $\dot x(t)=a(x(t))$ and $\dot y(t)=b(y(t))$ with $x(t_0)=y(t_0)$ and $a(\cdot)\leq b(\cdot)$, with the result being ...
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0answers
75 views

Diffraction across an absorbing wall

I need help finding the procedure for the solution of the following differential equation. This is equation is: Find $u:\mathbb{R}^2 \to \mathbb{C}$ such that for $C>0$ $ \begin{cases} u_{xx}+ ...
3
votes
1answer
158 views

“Canonical” form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$

Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions. Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ are ...
13
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3answers
406 views

Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
3
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1answer
81 views

Higher order Lyapunov equation

Let $A$ be a (finite) Hurwitz matrix. In this related question of mine, (see also https://en.wikipedia.org/wiki/Lyapunov_equation) it is shown that $$ \int_0^\infty \sum_{j,k} (e^{At})_{ij} Q_{jk} (...
5
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1answer
182 views

Regular holonomic D-modules as generalisation of regular singular points

I'm trying to understand why the definition of a regular holonomic D-module is a good generalisation of the usual definition of a regular singular point for a differential equation. More precisely, ...
3
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0answers
105 views

Wolff's article: Note on counterexamples in strong unique continuation problems

I am reading Wolff's Note on counterexamples in strong unique continuation problems: http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf On Page 3, ...
3
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1answer
215 views

Continuation (Extension) of Harmonic Functions

Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
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0answers
84 views

How can I show the principal symbol is not elliptic?

Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is $$\sigma‎_{‎E‎}(p):‎\Gamma (‎T^*M‎)\times \Gamma (T^*‎M‎)\to \Gamma (T^*‎M‎)‎$$‎ which acts as follows: ‎‎‎...
2
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1answer
176 views

Ground state for a double well potential (Schrödinger)

Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e. $V(x)=V(-x)$ for all $x\in\mathbb{R}$ $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$ $V(x)\in \mathbb{...
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0answers
199 views

Solutions for an ODE

I would like to know if someone could provide me the solution(s) for the following equation: \begin{align*} \frac{d^{2}w}{dx^{2}}\cdot\frac{dw}{dx} = e^{-x}w \end{align*} Where $w > 0$, $w''>0$ ...
3
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0answers
62 views

Well-posedness of a differential equation: Cauchy vs Dirichlet [closed]

I am struggling to understand what makes some differential equations well-posed Cauchy (initial value) problem rather than a Dirichlet (boundary value) problem. Consider, for example, a simple ...
5
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1answer
164 views

Riemann surface from Riccati equation

I have quite a practical question motivated by physics. Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum: $$ (p(x))^2 + \dfrac{\hbar}{i}...
5
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2answers
275 views

On Wilson's claim that Lyapunov function level sets are not exotic spheres

In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (...
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0answers
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 ...
11
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3answers
640 views

Vector field with holomorphic flow

Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms. I know one reference ...
5
votes
1answer
342 views

Fredholm index vs. Limit cycle theory

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$. Let $B $ be ...
7
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2answers
445 views

Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?

I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$, $$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$ I ...
3
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1answer
160 views

Reference request: Original source of Yosida approximation

Numerous papers/books(citation needed) refer to the operator $$A_\lambda := \lambda AR_\lambda (A) = \lambda^2 R_\lambda(A) - \lambda I$$ where $R_\lambda(A)=(I+\lambda A)^{-1}$ is the resolvent, as ...
5
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2answers
509 views

An ordinary differential equation

While I was working on a variational problem, I met this equation as its Euler-Lagrange equation, but I cannot solve it: $ x= \frac{af'(x)}{\sqrt{1+af'(x)^{2}}} + \frac{bf'(x)}{\sqrt{1+bf'(x)^{2}}} \ (...
6
votes
1answer
261 views

Gauge equivalence between operators

I have tried to figure out the following problem for some time now, but with little success: Let $ \mathcal{L} $ be a third order linear differential operator with coefficients in $ \mathbb{C}(X) $. ...
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0answers
215 views

Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?

Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...
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0answers
35 views

A uniqueness result for a BVP over a semi-infinite interval

Let $q:[0,\infty) \to \mathbb{R}$, and consider the ODE $$u''(x)=q(x)u(x) $$ with the boundary conditions $u(0)=\lim_{x \to \infty} u(x)=0$. Under what conditions on $q$ is $u \equiv 0$ the only ...
4
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0answers
285 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.,...
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0answers
34 views

Uniqueness of positive solutions to the n-vortex type equation

The $n$-vortex equation, in the context of optical vortex solitons, is of the form \begin{equation} -(ru_{r})_r+\dfrac{n^2}{r}u+\beta ru= f(u^2)ru,\quad r\in (0,\infty),\\ u(0)=0=u(\infty), \end{...
1
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1answer
91 views

Does this equation have an explicit solution? [closed]

For a positive constant $C$: \begin{align} y(x)+C\ln y(x)=f(x). \end{align} At least from specific $f(x)$, such as piece-wise linear function, is there an explicit solution for $y(x)$?
3
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2answers
206 views

Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator $$ -\frac{d^2}{dx^2} + V(x) $$ Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $. Now of course,the ...
3
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0answers
64 views

Convergence of Bessel (Sturm-Liouville) Expansions at the End Points

I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
4
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1answer
136 views

A non vanishing vector field on $S^3$ with a periodic attractor

Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case?
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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 ...
1
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1answer
150 views

Solutions to linear SDE with many noise sources

It is well known how to solve the linear stochastic ODEs with one source of noise $$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$ See, for instance, https://math.stackexchange.com/questions/1788853/...
3
votes
1answer
216 views

Hörmander's hypoellipticity theorem for complex coefficients

Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
2
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0answers
87 views

A global geometric formulation of the fundamental theorem of Picard Vessiot theory?

Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can ...
6
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0answers
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 ...
4
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1answer
198 views

Spectral growth of One dimensional Schrodinger Operator

Conside the One dimentional Schrodinger Operator $$ -\frac{d^2}{dx^2} + ( V(x) + E ) $$ Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $. What is known ...
3
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0answers
147 views

Is the closed orbit of the Vander pol equation a stable periodic orbit?

We consider the Vander 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 ...
1
vote
0answers
49 views

Control of the Hessian of the distance function at infinity

Let $(M,g)$ be a complete noncompact manifold and $r$ is the distance function to a fixed point. We assume that the sectional curvature $|Rm|(x) \le K(r(x))$ where $K(r)=\frac{1}{1+r^k}$ for some $k&...
3
votes
1answer
120 views

An indefinite integral containing functions that are solutions to a 2nd order linear ODE

I am trying to evaluate an indefinite integral of the form $\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$ where $u_1$ and $u_2$ are two independent solutions to the ODE $u'' + F(z)u = 0$ This ...
4
votes
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 ...