The Inverse Scattering Transform for the Nonlinear Schrödinger Equation

Abstract: The Nonlinear Schrödinger (NLS) Equation

$$i \psi_t+\frac{1}{2}\psi_{xx}\pm \lvert \psi \rvert^2 \psi = 0$$

is a nonlinear partial differential equation which is used to model physical phenomena such as nonlinear effects inside of optical fibers and the formation of rogue waves in shallow water. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, we study a particular class of solutions called solitons and multi-solitons. Solitons, also called solitary travelling waves, are localized traveling waves which arise as solutions to several nonlinear dispersive partial differential equations. We use the Dressing Method to numerically compute 100-soliton solutions to the NLS equation with 500 digits of precision. We then analyze the statistical properties of these multi-soliton solutions and compare them to some established results computed using other methods.

The minimum degree of $(K_s, K_t)$-co-critical graphs

Abstract: Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every {red, blue}-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if
$G \nrightarrow ({H}_1, H_2),$ but $G+e\rightarrow ({H}_1, H_2)$ for every edge $e$ in the complement of $G$. Galluccio, Simonovits and Simonyi in 1992 proved that every $(K_3, K_3)$-co-critical graph on $n\ge6$ vertices has minimum degree at least four, and this bound is sharp for all $n\ge 6$. In this paper, we first extend this result to all $(K_s, K_t)$-co-critical graphs by showing that every $(K_s, K_t)$-co-critical graph $n\ge9$ has minimum degree at least $2t+s-5$, where $t\ge s\ge 3$. We then prove that every $(K_3, K_4)$-co-critical graph has minimum degree at least seven, and this bound is sharp for all $n\ge 9$. This answers a question of the third author in the positive for the case $s=3$ and $t=4$.