# Questions tagged [nonlinear-eigenvalue]

The nonlinear-eigenvalue tag has no usage guidance.

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### Norm of “resolvent” for nonlinear eigenvalue problems

I have been looking at extensions of the usual Linear eigenvalue problems for (possibly unbounded but closed) operators. These are defined as follows. Let $T$ be an analytic function of $\lambda \in \...

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### Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...

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25 views

### Eigenfunction of Distorted Laplacian on Smooth compact domain

Setup
Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...

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22 views

### soliton solution of nonlinear convection-diffusion-reaction equation system?

I am interested in a flame propagation in 1D which is governed by a nonlinear convection-diffusion-reaction equation system:
$$
u_t+C(u)u_x+(D(u)\cdot u_x)_x + R(u) = 0,x\in (-\infty,+\infty) \\
u(-\...

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135 views

### The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...

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156 views

### Is this “stretched eigenvector” studied? (If so, what are its properties?)

An eigenvector is defined by
$$
\lambda \mathbf{v} = A\mathbf{v}.\tag{1}
$$
But suppose I change this to
$$
\lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2}
$$
for real $\alpha\ne 1$, where $\mathbf{v}^...

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61 views

### Polynomial Eigenvalue Problem with few non-zero coefficients

Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...

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659 views

### First eigenfunction of $p=3$-Laplacian of a square domain in $\Bbb R^2$ : reference for any work on this?

In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the ...

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165 views

### Simultaneous triangularization of two diagonal matrices and a symmetric matrix

I have the following quadratic eigenvalue problem:
$\det(\lambda^2M + \lambda D + J)=0$
where, M and D are both $n \times n$ real diagonal matrix with positive diagonal entries; $J$ is
1) a ...

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108 views

### How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...

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175 views

### significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...

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726 views

### Non-linear state-space model system stability using Lyapunov?

I have a non-linear system modelled in state-space as follow:
$$
\mathbf {\dot x} = \mathbf A(x) \mathbf x
$$
I need to find out if this system is stable, so I was thinking in using the Lyapunov ...

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231 views

### Distinct eigenvalues of the quadratic eigenvalue problem

Suppose we have a quadratic eigenvalue problem $(A_{0}+\lambda A_{1}+ \lambda^{2} A_{2})x=0$. I'd to know if there are conditions under which the problem is known to have a small number of distinct ...

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201 views

### Nonlinear eigenvalue problem - sorta

Suppose you have an equation of the form $Ax=f(x)$, where $A$ is a $n \times n$ matrix, $x$ is a vector of length $n$ and $f(\cdot)$ is some function. Is there a name for this sort of problem?