# User:Ninetiger

My name is Xiao Liu.

I'm a PhD. student in Math of University of Toronto.

## Orbital stability

Definition: Let $U=U(t,x)$ be a solitary wave solution of the PDE. We say that $U(t)$ is orbitally stable for the PDE if for any $\epsilon>0$ there exists $\delta>0$ such that if $u_0\in H^1(\mathbb{R})$ satisfies $||u_0-U(0)||_{H^1}<\delta$, then the solution $u(t)$ of the PDE with initial data $u(0)=u_0$ exists globally in time and satisfies $$\sup_{t\geq 0} \inf_{(\theta,y)\in\mathbb{R}^2}||u(t)-e^{i\theta} U(t,.-y)||_{H^1}<\epsilon.$$ Otherwise, $U(t)$ is said to be orbitally unstable.

This definition is from the paper Stability of solitary waves for derivative nonlinear Schrodinger equation (2006) by Colin and Ohta.

Here is the definition of orbital stability on wikipedia:orbital stability

## Regularity and Fourier transform

Last week (Jan 27), Jim showed some relation between regularity and Fourier transform. I try to explain his idea by Matlab.

Consider three functions:

$f_1(x)=8-|x|,x\in(-8,8)$ and $f_1(x)=0$ otherwise.

$f_2(x)=8((\frac{x}{8})^4-2(\frac{x}{8})^2+1),x\in[-8,8]$ and $f_2(x)=0$ otherwise.

$f_3(x)=e^{-x^2}.$

Then we have $f_1(x)\in \mathbf{C}(\mathbb{R})$, $f_2(x)\in \mathbf{C}^1(\mathbb{R})$ and $f_2(x)\in \mathbf{C}^{\infty}(\mathbb{R})$.

Here is the loglog graph of $f_1(x),f_2(x)$ and $f_3(x)$ after Fourier Transform:

From the graph, we find that the $f_2(x)$ decays fast than $f_1(x)$, and $f_3(x)$, which is smooth, decays fastest, as we expect.

Thanks Xiao! I am happy to see these examples which help to develop our intuition about how to understand regularity properties of functions when viewing their Fourier transform. The log-frequency versus log-Fourier-transform-modulus plots I described in class on 27 January are called Bode plots in engineering, especially appearing in discussions of filter designs. Colliand 10:07, 1 February 2011 (EST)