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

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2
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0answers
87 views

What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time). Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...
2
votes
1answer
129 views

A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...
5
votes
0answers
488 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: ...
2
votes
0answers
84 views

Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...
1
vote
0answers
72 views

Criterion for existence of solution to nonlinear second order ODE

Good day. I am asking this question as an issue of concern for determining whether an ordinary differential equation can be solved. Existence, and not uniqueness of solution, is what is required. ...
4
votes
1answer
63 views

Problem on differential inclusion

For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the ...
2
votes
0answers
60 views

The motivation of Weyl-Titchmarsh function

Given a second linear differential operator, $(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$, where $V$ is a bounded and real valued function, $f$ lies in $L^2(\mathbb{R})$. For an $z$ with $Im(z)\neq ...
1
vote
0answers
133 views

Green's function on sphere

Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian ...
1
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0answers
31 views

Question about a specific differential equation [closed]

I have the following differential equation (1+e^x+xe^xy)dx + (xe^x+2)dy=0 I need to check whether it is full and to find its general solution. Can you help me ?
1
vote
1answer
113 views

Solving Schroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [closed]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
2
votes
0answers
85 views

Homogeneous regular manifolds

In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way: ...
2
votes
1answer
207 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
2
votes
1answer
179 views

Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
1
vote
0answers
91 views

Closed form answer to a naive integral [closed]

Let a and b be positive real numbers. How to find a closed form answer to the integral $$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$ If it is not possible to find a ...
15
votes
1answer
136 views

Commuting ODE's implies existence of nonzero vanishing two variable polynomial?

Write $\partial := d/dt$, fix $m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ...
1
vote
0answers
91 views

Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
0
votes
0answers
82 views

Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular: $$ \int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M}, $$ in which ...
0
votes
1answer
158 views

How to prove that a non-linear differential equation has a solution

I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$, $$ \frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1], $$ where $w:[0,1] \to [0,1]$ and $v:[0,1] \to ...
0
votes
0answers
95 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 ...
3
votes
2answers
120 views

Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
0
votes
1answer
87 views

Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
0
votes
0answers
34 views

Explicit solution for a variational inequality?

Assume $\eta_t \in \mathbb{R}$ is a given continuous function depending on time. It is known that if we look for a continuous solution $\zeta_t \in \mathbb{R}$ depending on time of the following ...
3
votes
1answer
71 views

Examples of systems with stable equilibria at the boundary of the phase space

Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...
2
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0answers
70 views

Point Spectrum of a Second Order System of Differential Equations

Consider the following operator acting on $H^1(\mathbb{R})$ $$ \mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) ...
3
votes
4answers
471 views

Solution of second order differential equation with singularities at 0,1, and ∞

I am trying to solve the following equation; $$ U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0 $$ where U is a function of t and C is constant. The above ...
1
vote
0answers
60 views

asymptotic behavior of the solution of an ordinary differential equation

I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem. Consider the following autonomous system of two non-linear ordinary differential ...
2
votes
1answer
79 views

Coupled differential equations

I'm looking at the following coupled set of differential equations. Because of the symmetry, I'm hoping to be able to write down the solution for $x_n(t)$ and $y_n(t)$ in terms of $f(t)$ and $g(t)$, ...
1
vote
1answer
78 views

Runge-Kutta convergence [closed]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
1
vote
1answer
59 views

Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed. In particular it is said that the poles form a deformed lattice that ...
6
votes
1answer
147 views

Writing the quotient of solutions of linear ODEs as the solution of a nonlinear ODE

Let $A(x)$ and $B(x)$ be solutions to homogeneous linear ODEs with polynomial coefficients, i.e., $A(x)$ satisfies $$p_KA^{(K)} + p_{K-1}A^{(K-1)} + \cdots + p_1A' + p_0A=0$$ and $B(x)$ satisfies ...
2
votes
1answer
186 views

Partial differential equation from Kirchhoff system

I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$ $$\partial_t \vec{u} = \partial_s \vec{w} - [\vec{w} \times \vec{u}].$$ Using ...
2
votes
1answer
314 views

Hilbert 16th problem, distribution of Limit cycles

Edit: Can one help for translation of the link in Russian(comment by Dimitry Todorov)(Or at least a summary of it)? It seems that the second part of the Hilbert 16th problem is solved or is going to ...
5
votes
3answers
663 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0. $$ Then, to find ''bound states'', you solve on the right and find the ...
0
votes
2answers
108 views

Class of analytically-integrable divergence-free vector fields?

Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$? That is, I'm looking for a large class of vector fields given by ...
0
votes
1answer
92 views

Method of characteristics [closed]

I have difficulties understanding how to solve a PDE in $\mathbb{R}^{4}$ using the method of characteristics. I have a limited background in solving PDEs. I have seen only 2-dim examples and none for ...
0
votes
0answers
114 views

The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...
1
vote
1answer
234 views

Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post I apologize in advance, if this question is obvious: 1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...
1
vote
0answers
85 views

An algebraics 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 ...
3
votes
2answers
122 views

Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$. Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths ...
1
vote
0answers
61 views

Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems. Are there any papers or books that ...
0
votes
0answers
41 views

Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...
4
votes
1answer
141 views

A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation $$ \frac{d x (t)}{dt} = f(x(t)) $$ with some initial condition $x(0)=x_0$ has no solution?
0
votes
0answers
75 views

Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation $$ -\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x) $$ with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...
5
votes
1answer
374 views

Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$ M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0. $$ My question is to ...
5
votes
1answer
173 views

Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
1
vote
1answer
202 views

First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation $$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$ where $A_0, A_1, A_2, B$ are some constants, ...
5
votes
1answer
149 views

[This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question: Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as ...
15
votes
1answer
860 views

Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
0
votes
0answers
76 views

Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
11
votes
1answer
233 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...