# Tagged Questions

**3**

votes

**1**answer

83 views

### Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...

**4**

votes

**2**answers

143 views

### How to find an ODE with prescribed terminal values?

Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...

**1**

vote

**1**answer

117 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**0**

votes

**1**answer

114 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**1**

vote

**2**answers

170 views

### Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation:
$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$
which is essentiall in matrix notation:
$\dot{\mathbf{x}} = ...

**-1**

votes

**1**answer

289 views

### Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...

**8**

votes

**1**answer

369 views

### Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg:
Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...

**5**

votes

**0**answers

129 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 ...

**1**

vote

**1**answer

136 views

### How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE
$$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$
using Charpit's method. The ...

**2**

votes

**0**answers

92 views

### Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
...

**10**

votes

**3**answers

587 views

### Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...

**0**

votes

**0**answers

63 views

### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...

**2**

votes

**1**answer

144 views

### acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...

**31**

votes

**2**answers

3k views

### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...

**2**

votes

**0**answers

59 views

### About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper:
A concave–convex elliptic problem involving the fractional Laplacian -
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez.
At the Pgs 41, 42, the ...

**7**

votes

**0**answers

245 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 ...

**1**

vote

**1**answer

94 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...

**5**

votes

**1**answer

203 views

### Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...

**1**

vote

**0**answers

21 views

### Solution to Helmholtz equation with non-circular boundary

Let $D$ be an homogeneous 2D domain with non-circular boundary $\partial D$.
I am trying to solve the Helmholtz equation
$$
\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)
$$
in which ...

**0**

votes

**0**answers

113 views

### Is the trivial solution the unique solution to the following initial value problem?

This question is a duplicate one asked by myself elsewhere. But there are no answers or comments so far. The initial value problem that I am considering is:
$$ y'' (3y+2x)^2=3(3y'-1)(9yy' + 4xy' -y), ...

**1**

vote

**1**answer

107 views

### Heat kernel for non bounded domains

consider a domain $U\subset \mathbb{R}^n$ which is not bounded and denote its boundary by $\partial U$. (The boundary I'm confronted with is actually not very irregular. Think of a smooth manifold ...

**0**

votes

**0**answers

79 views

### Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...

**1**

vote

**0**answers

79 views

### Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...

**1**

vote

**1**answer

96 views

### Fundamental solution to the heat equation with zero boundary values

let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ ...

**0**

votes

**0**answers

49 views

### Causal (Volterra type) differential equation with local Lipschitz condition

Consider the equation
$$
u'(t) = (Fu)(t)
$$
where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type)
nonlinear operator. It means that the value of $(Fu)(t_0)$ ...

**2**

votes

**1**answer

161 views

### What are some good references on the Galois theory, factorization, or minimality of differential equations?

Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u ...

**3**

votes

**0**answers

99 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 ...

**-2**

votes

**1**answer

551 views

### BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2?
Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better?
The BDF2 method requires the ...

**9**

votes

**2**answers

307 views

### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...

**0**

votes

**1**answer

189 views

### Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...

**4**

votes

**2**answers

192 views

### Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote:
...

**0**

votes

**2**answers

518 views

### General description of surface with zero gaussian curvature

Let suppose function $g(s,t)$ satisfies partial differential equations:
$g_{ss} g_{tt} - g_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution ...

**0**

votes

**0**answers

52 views

### Shooting method for non-oscillatory solutions

I am having some trouble with using the shooting method :
I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I ...

**2**

votes

**1**answer

202 views

### Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation
$$\frac{\partial f ...

**1**

vote

**0**answers

66 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**0**

votes

**1**answer

185 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**1**

vote

**0**answers

138 views

### Floquet-Bloch solutions of the quasiperiodic Schrödinger equation

I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional ...

**1**

vote

**1**answer

191 views

### Vector fields whose divergence are proper maps

Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of ...

**-1**

votes

**1**answer

73 views

### Integrating factors and integrability of an ODE system

The following argument is from a paper about the Bendixson-Dulac Theorem.
Consider a smooth differential equation on the plane
$$
x'=g(x,y),\quad y'=h(x,y).
$$
Suppose there exists a function ...

**1**

vote

**0**answers

65 views

### Perturbation method of a boundary value problem

Let $u(x, \epsilon, \theta)$ be the solution of $$u''+(\epsilon \cos(x)+\theta-u)u=0$$ with boundary conditions $u'(0)=0$ and $u'(\pi)=0$. Here $\theta\in [0, 1]$.
I tried to put the solution in ...

**1**

vote

**0**answers

71 views

### Regularity of Schrödinger Resolvent

The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...

**1**

vote

**2**answers

92 views

### Finding conditions to guarantee existence of solutions to IVP [closed]

Consider the following IVP:
$x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.
Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.
In order for the ...

**0**

votes

**1**answer

53 views

### Solution of a second order nonlinear ode [closed]

I encountered the following ode in the attempt to solve the cauchy problem of Liouville equation. I have tried for a long time to give it a solution but failed.
$(K e^f h + ...

**11**

votes

**1**answer

331 views

### applications of C$^*$-algebras in the field of PDEs

I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...

**3**

votes

**0**answers

84 views

### Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...

**1**

vote

**1**answer

64 views

### SIRS Stability Analysis

I have set up the following ODE's for a SIRS model:
$$\frac{dS}{dt} =-\alpha SI + \zeta R$$
$$\frac{dI}{dt} = \alpha SI - \beta I - \rho I$$
$$\frac{dR}{dt} = \beta I - \zeta R$$
...

**2**

votes

**0**answers

104 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

**3**

votes

**0**answers

156 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 ...

**7**

votes

**1**answer

367 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**0**

votes

**1**answer

268 views

### About an integral equation

I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...