This repository contains the MATLAB codes associated with the paper:
"Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schroedinger equations".
by Jonathan Jaquette
SUMMARY
This code computes the space-time Fourier coefficients of monochromatic
initial data to the 1d quadratic NLS:
i u_t = d_xx u + u^2
where x is taken in the 1-torus R / 2 \pi Z.
In particular, this means that the variable omega in the paper is taken to equal 1, and as such does not appear as a variable in the code. The initial data considered in this software is u_0(x) := A e^{i x} for a complex scalar A.
With the Fourier coefficients computed, this code is set up to (1) Produce a computer-assisted-proof of the existence of a periodic orbit (2) Generate figure displaying the periodic orbits (3) Estimate the critical value of A* between where the Fourier coefficients grow/decay exponentially.
These code for part (1) requires the INTLAB interval arithmetic library: http://www.ti3.tu-harburg.de/rump/intlab/
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Data Files
Coeff_110_intval.mat -- The Fourier coefficients c_{n,j} computed
USING interval arithmetic,
for A=3 & 1<=n<=110
Coeff_300.mat -- The Fourier coefficients c_{n,j} computed
NOT USING interval arithmetic,
for A=1 & 1<=n<=300
(Note: By the rescaling in Theorem 3.1, the coefficients for one value of A can be easily used to calculate the coefficients for another value of A. This is utilized in the Script_ProduceFigures. However the computation for the rescaling will introduce some rounding error.)
Script files
ComputeCoeff - Used to create the data file "Coeff_110_intval.mat"
ComputeCoeff_intval - Used to create the data file "Coeff_300.mat"
script_CAP - Produce a computer-assisted-proof of the existence of a periodic orbit, with the data Coeff_110_intval
script_ProduceFigures - Generates figures displaying the periodic orbits
script_Rate_of_Decay - Estimate the critical value of A* between where the Fourier coefficients grow/decay exponentially.
Auxiliary Functions
quadratic_cauchy_product ... -- Computes the Cauchy product of two sequences (vectors) with possibly different lengths
Y_bound -- Computes the Y_0 constant from Lemma 3.6
Z_bound -- Computes the Z_1 constant and Z_2(r) polynomial from Lemma 3.6
===========================================================================
Copyright (C) 2021 Jonathan Jaquette.
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.