Publication Date

1-28-2015

Abstract

Singularity formation is an inherent feature of equations in nonlinear physics, in many situations such as in self-focusing of light nonlinearity is essential part of the model and physical events cannot be captured by linearized equations. There are nonlinear systems, such as $1$D NLSE where singularities in analytic continuation would follow a soliton solution at fixed distances and while it is true that soliton determines the position of all the singularities, it is also true that evolution of singularities determines the solution on the real axis. Before we go further to discuss $2$D problems, we want to be more specific about analytic continuation in a $2$D problem: it is well-known that collapse in $2$D NLSE is radially symmetric and introducing radial variable $r = \\sqrt{x^2 + y^2}$ the problem becomes effectively one-dimensional. If we expand the interval spanned by $r$ from $[0,+\\infty)$ to $(-\\infty,+\\infty)$ and continue all the functions evenly across the origin, it starts to make sense to further expand $r$ to complex plane $\\mathbb{C}$ and talk about analytic continuation of functions in $r\\in\\mathbb{C}$. In $2$D nonlinear Schr\\"odinger equation (NLSE) it is common to think that a singularity appears in finite time, but one can also say that a singularity already exists in the analytic continuation of initial data and at critical time $t_c$, the singularity touches the real axis and solution reaches its maximal interval of existence. The latter point of view captures evolution in more detail, in particular it allows to ask many questions that would seem quite meaningless if you follow the "philosophy'' of a singularity just appearing at a finite time. In particular, one can ask question what is the trajectory of singularity in complex plane and how does the type of singularity change as $t\ o t_c$. For $2$D focusing NLSE and Keller-Segel model (KSE) of chemotactic bacteria, the singularities evolve towards the real axis if sufficient conditions are met by initial distribution of laser intensity (NLSE) and bacteria density (KSE) respectively. Well-established conditions are included in the text and are cited upon in corresponding sections. The central subject of this work is the study of onset of singularity towards the real axis in radially symmetric $2$D NLSE and $2$D reduced KSE models (RKSE) combining two approaches: direct numerical simulations of collapse and asymptotic analysis in the limit $t\ o t_c$. The benefit of this two-sided approach is evident when comparing results of classic theory of critical collapse in $2$D NLSE to numerical simulations: the collapse exhibits dependence on initial data even when intensity reaches enormous magnitudes and as a result is inconsistent with classical theory (e.g loglog law). An intervention of numeric approach allowed us to perform sanity checks of many assumptions and estimate regions of applicability of approximations that were used in asymptotic approach and resulted in a new corrected theory that is able to consistently describe the onset of singularity even for moderate-amplitude, developed collapse while still recovering classic theory in the limit $t\ o t_c$. The problem of $2$D potential flow of ideal fluid in free surface hydrodynamics is another example of nonlinear system containing solutions with singularities. The focus of our investigation lies in fully nonlinear travelling waves on the surface of fluid also known as Stokes waves and in particular we are interested in singularities that are present in the analytic continuation of Stokes waves. These waves computed as a part of this dissertation range from linear waves to the limit of extremely nonlinear waves that were never observed before, in addition a predicted phenomenon of parameter oscillation was confirmed for strongly nonlinear waves.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Advisor

Lushnikov, Pavel

Second Advisor

Korotkevich, Alexander

First Committee Member (Chair)

Coutsias, Evangelos

Second Committee Member

Diels, Jean-Claude

Language

English

Keywords

nonlinear waves, nonlinear Schrodinger Equation, wave collapse, water waves, singularities, self focusing

Document Type

Dissertation

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