## Strongly Nonlinear Phenomena and Singularities in Optical, Hydrodynamic and Biological Systems

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/25774

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Title

Strongly Nonlinear Phenomena and Singularities in Optical, Hydrodynamic and Biological Systems

Author(s)

Dyachenko, Sergey

Advisor(s)

Lushnikov, Pavel

Korotkevich, Alexander

Korotkevich, Alexander

Committee Member(s)

Coutsias, Evangelos

Diels, Jean-Claude

Diels, Jean-Claude

Department

University of New Mexico. Dept. of Mathematics and Statistics

Degree Level

Doctoral

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 $t\to t_c$.
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 $t\to t_c$. 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 $t\to t_c$.
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.

Date

December 2014

Subject(s)

nonlinear waves

nonlinear Schrodinger Equation

wave collapse

water waves

singularities

self focusing

nonlinear Schrodinger Equation

wave collapse

water waves

singularities

self focusing

##### Collections

- Mathematics [34]