GitHub

Week 10 Lab: Self-similar blowup

The goal of this lab is to start on the implementation of a computer-assisted proof for the self-similar blowup scenario for the CGL equation discussed in the last lecture. The paper itself, Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations, might be useful as a reference in some cases.

Setup

Recall from last lecture that proving the existence of self-similar blowup reduces to proving the existence of a solution to the second order non-linear ODE

\[(1 - i\epsilon)\left(Q'' + \frac{d - 1}{\xi}Q'\right) + i\kappa\xi Q' + i \frac{\kappa}{\sigma}Q - \omega Q + (1 + i\delta)|Q|^{2\sigma}Q = 0,\]

with appropriate boundary conditions at zero and infinity.

Using a shooting method approach, we introduced the function

\[G(\mu, \gamma, \kappa) = \left( Q_{0}(\mu, \kappa; \xi_{1}) - Q_{\infty}(\gamma, \kappa; \xi_{1}), Q_{0}'(\mu, \kappa; \xi_{1}) - Q_{\infty}'(\gamma, \kappa; \xi_{1}) \right),\]

with $Q_0$ and $Q_\infty$ corresponding to solutions satisfying the boundary conditions at zero and infinity, respectively. Proving the existence of a solution satisfying both boundary conditions then reduces to proving the existence of a zero of $G$.

To apply this approach, we need to implement an interval arithmetic version of $G$, as well as a multi-dimensional version of the interval Newton method to rigorously prove the existence of a zero. The aim of this lab is to implement a skeleton for this setup. We won't have the time to properly implement rigorous evaluations of $Q_0$ or $Q_\infty$; that will be the content of future discussions.

Overview of implementation

A very rough outline of the implementation is given in lab-10.jl in the notebooks directory. We will start to fill in parts of this outline.

For the purposes of this lab, we will use the parameters

\[\epsilon = 0,\ \delta = 0,\ \sigma = 1,\ d = 3,\ \omega = 1.\]

To simplify the implementation, we will hard-code these values in the code whenever they are used. When looking for a zero of $G$, we will use $\xi_1 = 60$ and the approximate zero $(\mu_0, \gamma_0, \kappa_0) = (1.88565, 1.71360 - 1.49179i, 0.91735)$ as an initial guess.

We split the implementation into four parts:

  1. Implementation of $Q_0$
  2. Implementation of $Q_\infty$
  3. Implementation of $G$
  4. Implementation of interval Newton

The first part will be to implement a skeleton for the full computation. For this first step, our implementations of $Q_0$ and $Q_\infty$ will simply return zero. The goal will be to get a proper implementation of everything else. We will roughly follow this order:

  1. Skeletons of $Q_0$ and $Q_\infty$
  2. Implementation of $G$
  3. Start implementing interval Newton
  4. Skeletons of Jacobian of $Q_0$ and $Q_\infty$
  5. Implementation of Jacobian of $G$
  6. Finish implementing interval Newton

An important part of the implementation at this point will be to figure out (and decide) what the input arguments and the type of the outputs of the functions should be.

The next step is to properly implement $Q_0$ and $Q_\infty$, as well as their Jacobians. If we have time, we will start to look at this. To begin with, we will implement non-rigorous versions of the functions, then discuss what is needed to make this fully rigorous.