Dynamical Systems Commons^™

An Integrated Computational Pipeline To Construct Patient-Specific Cancer Models, Daniel Plaugher

Theses and Dissertations--Mathematics

Precision oncology largely involves tumor genomics to guide therapy protocols. Yet, it is well known that many commonly mutated genes cannot be easily targeted. Even when genes can be targeted, resistance to therapy is quite common. A rising field with promising results is that of mathematical biology, where in silico models are often used for the discovery of general principles and novel hypotheses that can guide the development of new treatments. A major goal is that eventually in silico models will accurately predict clinically relevant endpoints and find optimal control interventions to stop (or reverse) disease progression. Thus, it is …

The Direct Scattering Map For The Intermediate Long Wave Equation, Joel Klipfel

Theses and Dissertations--Mathematics

In the early 1980's, Kodama, Ablowitz and Satsuma, together with Santini, Ablowitz and Fokas, developed the formal inverse scattering theory of the Intermediate Long Wave (ILW) equation and explored its connections with the Benjamin-Ono (BO) and KdV equations. The ILW equation\begin{align*} u_t + \frac{1}{\delta} u_x + 2 u u_x + Tu_{xx} = 0, \end{align*} models the behavior of long internal gravitational waves in stratified fluids of depth $0< \delta < \infty$, where $T$ is a singular operator which depends on the depth $\delta$. In the limit $\delta \to 0$, the ILW reduces to the Korteweg de Vries (KdV) equation, and in the limit $\delta \to \infty$, the ILW (at least formally) reduces to the Benjamin-Ono (BO) equation.

While the KdV equation is very well understood, a rigorous analysis of inverse scattering for the ILW equation remains to be accomplished. There is currently no rigorous proof that the Inverse Scattering …

Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren

Theses and Dissertations--Mathematics

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.

For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.