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Questions tagged [nonlinear-analysis]

For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.

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Is the nonlinear Schrödinger equation solved?

Consider the following initial value problem: $i\psi_t = -\psi_{xx} - 2|\psi|^2\psi$ with $x\in[0,2\pi)$ and $t\geq 0$, and $\psi(x,0) = \frac{3}{2}\left(1 - \frac{1}{10}\cos(x-\pi)\right)$. The ...
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Local invertibility of general nonlinear maps and the existence of a nonlinear map.

Consider the following nonlinear maps: $y = f(x)$, where $f:R^n\mapsto R^q$, $q\leq n$, and $z = g(x)$, where $g:R^n\mapsto R^r$, $r\leq q$. The Implicit Function Theorem provides a sufficient ...
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1answer
87 views

Discretizing a mathematical equation

This is a 3D map that maps every $(x,y,z)\to (x',y',z')$ uniquely. If i want to implement it's discrete counterpart on matlab platform, i do the following $$\text{if} (i<=\dfrac{n}{2} \wedge ...
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34 views

Why the dimension of the objective space of Multi-objective optimization problems is usually lower than the design space?

I was reading a book on non-linear multiobjective optimization by Kaisa M. Miettinen and in a paragraph the author says: "In single objective optimization problems, the main focus is on the decision ...
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64 views

Using homotopy continuation on nonlinear underdetermined systems.

I am asking this question to determine whether or not I should learn the basics of nonlinear algebra for a problem I am facing. I need to solve a set of polynomials with M equations and M+N unknowns. ...
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21 views

Track an object's position - nonlinear system of difference equations

Suppose I have to track the (2D) position of an object, that moves along a steady but curved path. The object can record its own position coordinates, however, the measurement sensor is located at a ...
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1answer
53 views

de-coupling non-linear odes using change of variables

I have a pair of odes in $x$, $y$ and $t$ along with an extra variable $z$. It looks as follows: $\frac{dx}{dt} = \frac{1}{z}f(x,y)$, $\frac{dy}{dt} = \frac{1}{z}g(x,y)$ where $f$ and $g$ are real ...
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25 views

Doubt on momentum conservation in nonlinear schrodinger

I want to prove the momentum conservation of the nonlinear Schrodinger eq. $u_t=i\Delta u + i|u|^{p-1}u$. The momentum is gives by $$ Pu=2Im \int_{R^n} \overline{u}\nabla u\ dx$$ I have read that in ...
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19 views

Linear parameterization of nonlinear functions

Suppose $f(t,x(t)) \in \mathbb{R}^m$ and $x(t) \in \mathbb{R}^n$ $(n \geq m)$ are continuous in their arguments and bounded, and that $f(t,0) = 0$. Further, assume that the partial derivatives of $f$ ...
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Regularity of Solution for the Kdv equation

Let $u_{t}+u_{xxx}=f,\,\, u(x,0)=0,\,x\in(0,1), \, t\in[0,T]$ $u(0,t)=0,u(1,t)=0, u_{x}(1,t)=0$. Prove that \begin{equation} \boxed{\lVert u \rVert_{L^{2}(0,1;H^{2}(0,1))}\leq C\lVert f\rVert_{ L^{...
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Maintaining probability distribution under transform

The context of this question is correct dithering of color $c$ with gamma correction. We can only ever output integers $\lfloor c \rfloor - 1,\lfloor c \rfloor,\lfloor c \rfloor + 1, \dots$, but wish ...
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40 views

Volume of a solid from rotating a function around a non-linear axis of rotation

In Calculus class, we are all taught how to find the volume of a solid of rotation around any line of the form $y=a$ or $x=b$. Then in linear algebra, we are taught how to find the volume of a solid ...
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1answer
16 views

Test if a function given as a non-integrable ode set is Bijective

Given that state space trajectories of an autonomous system do not cross, can I deduce that a mapping function f:(x,y)→(x',y') given by a solution of an ODE of an autonomous system is bijective? ...
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24 views

Imitation Dynamics: Rest Points and Jacobian

Questions: Show that for a given matrix $A$ the imitation dynamics in the following equations: $$\dot{x}_i=x_i\sum_j x_j\psi((A\mathbf{x})_i-(A\mathbf{x})_j) $$ $$\dot{x}=x_i((A\mathbf{x})_i-\mathbf{x}...
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1answer
27 views

Does Lyapunov Stability imply Attractivity for intervals on the real line?

For intervals on a real line, I have found a result which states that for a continuous map, attracting fixed points are Lyapunov stable. However, I found no result about the converse. So, is the ...
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42 views

Solving underdetermined nonlinear system of 2 equation with 3 unknowns.

I've gotten into a problem I haven't really worked with before in my numerics classes. I have an underdetermined nonlinear system of equations with 3 real-valued parameters, precisely those ...
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1answer
82 views

Let $A: X \rightarrow Y$ a bijective continous map between two Banach spaces X and Y. Then, $A^{-1}$ is also continous?

We know if A is a map continuous, bijective and linear then the answer is yes, $A^{-1}$ is continuous. But, if $A$ is no linear then $A^{-1}$ is also continuous ?
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1answer
55 views

Solution to Nonlinear System of Differential Equations

I am working on a optimal control problem, specifically the design of a hypersonic aircraft nose. When minimizing the drag coefficient, I am led to the following system of differential equations: $$ ...
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12 views

Linear extension of a vector field

I would like to know if there is a document written in a friendly tutorial form (so that humble engineers like me can understand it) to obtain linear extensions of vector fields of whom I know a ...
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2answers
43 views

Nonlinear transformation of region from $\mathbb R^2\to\mathbb R^2$

If I have a given continuous nonlinear map $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, and a region $D \subset \mathbb{R}^2$, is it necessarily true that $T(\partial D)=\partial T(D)$? That is, do ...
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1answer
57 views

Squared-derivative PDEs

Is there a general theory for equations of the type $ f_y^2 = A(x,y) f_x$? where one first derivative is expressed as a multiple of the other one. Concretely, I'm interested in the equation $$ ( x+...
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How to do a sensitivity analysis on a non-linear equation?

In the company, it is very difficult to actually do quotations for our customers properly because we do not have perfect information regarding the factors that affect the cost and profit. So I created ...
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99 views

Nonexistance result of elliptic equation

I want to prove that there is no $L^2(\mathbb{R}^N)$-solution of the equation $−\Delta \phi =\lambda \phi$ for every $\lambda \in \mathbb{R}$. I know that the Pohozaev identity asserts for $N\geq 3$, $...
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25 views

About the Frechet derivative of a functional

How can I compute the Frechet derivative of the functional $$ I(u)=\frac{1}{2}\int_{\Omega} \vert \nabla u\vert ^2\ dx \ + \ \int_{\Omega}\left[1-|u|^2\right]^2\ dx$$ for $u$ in a functional ...
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1answer
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How to show uniqueness of the non-degenerate solution of $g(x)=0$?

Let $g(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$. We call a root $x_*$ of $g(x)=0$ non-degenerate if $Jg(x_*)$ is invertible, where $Jg(x_*)$ is the Jacobian at $x_*$. How can we show if $x_*$ is ...
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25 views

A continuity result for functions on a Sobolev space

Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let $$F:[0,T] \...
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42 views

Is there such branch as Nonlinear Matrix Algebra?

Is there such branch as nonlinear matrix algebra, that researches nonlinear functions with matrix (tensor) valued arguments and outputs where nonlinearities are applied componentwise. Such functions ...
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1answer
42 views

How can a function including a sin operation be linearly transformable for any offset?

In the paper "Attention is all you need" the authors have chosen a function to encode the position of a word in a sequence (section 3.5). The following encoding is chosen: $ PE(pos, 2dim) = sin(pos / ...
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1answer
18 views

Inner direct sum and nonlinear operators

Suppose we have a not necessarily linear operator $A:X\to X'$, where X is a real Hilbert space that can be written as the inner direct sum $X=V \oplus V^{\perp}$. Consider $u\in X$, so $u$ can be ...
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46 views

Optimization Methods in Banach Spaces

does anyone know if there's a theory for the following problem: Optimize the task $\begin{align*} T_\phi(\tilde{u})&=\inf\limits_u T_\phi(u)\\ Au&=b\\ u&\in L^p(\Omega),\,\Omega\subset \...
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Solving non-linear Leonard Jones potential using approximations

Can you help me solving this problem, I want to find the values of $K$ and $r_{\max}$ in which $f(K)=0$, to simplify the calculations let $a=b=1$ $$f(K) = \int_{0.73}^{r_{\max}}\sqrt{K-\Big(\dfrac{a}{...
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1answer
29 views

Solution of a system of linear equations when matrix is a function of a vector variable

My problem is actually a nonlinear equation: $A(x)x=0$, where x is a vector and matrix A is a function of vector variable x. Are there any works related to this kind of problem??
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27 views

The application of Gronwall Inequality in $y(t)<\lambda(t)+\rho(t)\int\mu(\tau)y(\tau)d\tau$

Assume $y(t)$ is nonnegative scalar function and: $y(t)\leq k_1 \exp(-\alpha(t-t_0))+\int_{t_0}^t \exp(-\alpha(t-\tau))[k_2y(\tau)+k_3]d\tau$ where $k_1,k_2,k_3$ are all nonnegative and $\alpha>...
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1answer
80 views

Solve differential equation $xyy'=x^4+y^4$

How to find general solution to this differential equation (if it exists): $$ xyy'=x^4+y^4 ?$$ I do not know how to even approach it since I never dealt with nonlinear equations. Only thing that I ...
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30 views

How to find explicit solution for recursive sequences with nonlinear terms?

Are there any general known methodologies or algorithms to derive an explicit equation for a nonlinear sequence? As an example, I am struggling to find an explicit expression for the sequence below: $...
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1answer
42 views

Riccati and Linear 1st Order ODE Parallel

I have noticed a certain similarity between Riccati 1st Order ODEs and linear 1st Order ODEs. Specifically, the general solution for each is given by any particular solution plus some function of the ...
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12 views

Lookup tables inversion

I'm having troubles solving the following problem. I have 2 2D lookup tables of the form $\lambda_1 = f_1(x, y) \\ \lambda_2 = f_2(x, y)$ where $x\in \{x_1, x_2,...,x_N\}$ and $y\in \{y_1, y_2,...,...
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1answer
35 views

Continuity for a nonlinear functional

I'd like to check the continuity for the nonlinear functional $T: (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) \rightarrow (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) $ , with $T(f)(x)=\arctan(f(x))$. I ...
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1answer
24 views

Determining components of Halley's method

I'm working on a homework problem about Halley's method and I'm not quite sure where to start and how I prove a cubic convergence. Consider the iterative method $$x_{n+1} = x_{n} − \frac{h_n}{Ah_n + ...
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1answer
52 views

What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?

I have two biosignals recording the same phenomenon with different methods. Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
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19 views

Uniqueness and existence of this system, verifying my answer

I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve. 1)Getting the Jacobian, I obtain $$ J= \begin{bmatrix} 0 & 1\\ -1-2xy & 1-...
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1answer
42 views

Nonlinear equation analysis withe epsilon value [closed]

Consider the nonlinear equation $$\frac{d^2x}{dt^2}+\epsilon\sin(x)=0,~~\epsilon \ll 1\\ x(0)=0,~~\dot{x}(0)=1$$ and find... A. The value of $x_0$ as $\epsilon$ goes to $0$ B. The first order term $...
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1answer
43 views

Solve an equation involving the error function

Let $0<a<1$ be given. The equation: $$a = 1 - \frac{2\sqrt{x/\pi}}{\mathrm e^x \mathrm{erf}(\sqrt x)}$$ has a unique root $x$, because the right-hand side is increasing in $x$, and goes to $0,...
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27 views

boundary problem: use main theorem of monotone operators

I am trying to investigate for which $\alpha \in \mathbb R$ the boundary problem $$-u''(x)+\alpha sin(u(x))u'(x)=f(x)$$ $$u(a) = u(b) = 0$$ is weakly solvable using the main theorem of monotone ...
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35 views

How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
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27 views

Fourier Transform of B-splines linear combination, any application to spectral analysis of non stationary series?

In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the ...
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1answer
68 views

Transform a non-linear differential equation into a linear equation

Following the work of Yaoji Lu - 1967 (here's a link to the full paper) I got stuck at the step when the author transform a non-linear differential equation into a linear equation (eq. 3.9 pag. 19). ...
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55 views

How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
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0answers
12 views

Positive and negative eigenvalues via Brouwer degree

The problem is: if $B=B(0,1)\subset\mathbb{R}^n$ is the open unit ball and $f$ is a continuous function defined on the closed unit ball $\overline{B}$ with $0\notin f(\overline{B})$, then there exists ...
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0answers
31 views

Property of continuous functions defined on a sphere of R^n where n is odd

If $B=B(0,1)$ is the unit open ball in $\mathbb{R}^n$ with $n$ odd and $f:\partial B \rightarrow \partial B$ is continuous, then exists $x\in \partial B $ such that $f(x)=x$ or $f(x)=-x$. Any idea of ...