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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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| December 23, 2024 |
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AbstractsInvited Speakers:Robin Ming Chen (University of Minnesota) We consider the continuity of the solution map to the Camassa-Holm
equation, which was shown to be nonuniformly continuous in the solution
space. We prove that in a weaker topology the solution map is Holder
continous. _______________________________ _______________________________ We propose a complete set of Zakharov's equations type describing
the Raman amplification. We point out that the Raman instability
gives rise to three components. The first one is collinear to the
incident laser pulse and counter propagates. In 2-D, the two other
ones make a non-zero angle with the initial pulse and propagate
forward. We discuss the local Cauchy problem and present some numerical
simulations. Furthermore, we address the question of existence and
stability for different type of solitary waves for an approximate
system made with Schr\"odinger equations. _______________________________ Nick Constanzino (Pennsylvania State University) The most fundamental equations describing the propagation of light in a dielectric medium are the nonlinear Maxwell equations, which are a system of coupled nonlinear hyperbolic equations. I will discuss some recent progress in deriving reduced models for the propagation of pulses that are relativley localized in space (and hence relatively broad in frequency). This leads to several different competing models, each incorporating different phenomena and different scales. I will then discuss some results on solitary wave propagation of a particular dispersive regularization of the Short Pulse equation, _______________________________ Walter Craig (McMaster University) _______________________________ Alex Himonas (University of Notre Dame) We shall discuss well-posedness of the initial value problem for a class of weakly dispersive nonlinear evolution equations, including the Camassa-Holm, the Degasperis-Procesi, and the Novikov equation. The focus will be continuity properties of the data-to-solution map in Sobolev spaces. This talk is based on work in collaboration with Carlos Kenig, Gerard Misiolek and Curtis Holliman. _______________________________ John Hunter (University of California) Waves with constant frequency form an interesting class of nondispersive
waves with qualitatively different properties from nondispersive
hyperbolic waves. In a first approximation, a constant-frequency
wave motion consists of spatially decoupled oscillations with the
same frequency, and these oscillations are weakly coupled by spatial
nonlinearities. We will describe some model equations for constant-frequency
waves, including a Burgers-Hilbert equation that consists of the
inviscid Burgers equation with a lower-order oscillatory Hilbert
transform term, and give a number of physical applications to surface
waves on vorticity discontinuities and inertial oscillations in
rotation-dominated shallow water waves. _______________________________ Slim Ibragim (University of Victoria) _______________________________ Nathan Kutz (University of Washington) _______________________________ Didier Pilod (Federal University of Rio de Janeiro) _______________________________ Zhijun Qiao (University of Texas - Pan American) In this talk, we will report an interesting integrable equation that has both classical solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions. _______________________________ Tobias Schafer (City University of New York) After a brief review of the derivation of the cubic nonlinear Schroedinger
equation (NLSE) and the short pulse equation (SPE) we discuss higher
order deterministic terms for both models. In order to cope with
stochastic perturbations, we develop a method to coarse-grain noise
in the presence of multiple scales and discuss the application of
this method to both the NLSE and the SPE. _______________________________ Guido Schneider (University of Stuttgart) The short pulse equation takes the role of the NLS equationas amplitude
equation if the number of oscillations $ N $ under the envelope
becomes to small. From a mathematical point of view in the NLS-limit,
$ N $ must be rather large. However, numerical experiments show
that the NLS equation makes correct predictions even for small $
N $. After giving an overview about existing approximation results
we explain a number of reasonsfor this fact and explain why the
NLS equation makes good predictions far beyond its formal validity.
Finally we present an alternative approach for the description of
short pulses. _______________________________ Atanas Stefanov (University of Kansas) We consider the generalized Ostrovsky (gO) equation $u_{t x}=u+(u^p)_{xx}$
for $p\geq 2$. In the case $p\geq 4$, we show that for small and
sufficiently smooth and decaying data, the solutions exist globally
and decay at the rate of the free solutions. The proof is based
on a fairly general energy estimate scheme, which is supplemented
by newly established decay and Strichartz estimates for (gO). We
will also elucidate the reason for the restriction $p\geq 4$ and
what one may be able to do about the important cases $p=2,3$. _______________________________ Feride Tiglay (Fields Institute) In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study the equations of motion of a rigid body in $\mathbb{R}^3$ and the equations of ideal hydrodynamics. I will describe how to extend his formalism and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bihamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and some results on blow-up as well as global existence of solutions. Time permitting, I will describe the peakon solutions for these equations. _______________________________ Eugene Wayne (Boston University) I will describe a new method for studying the stability of traveling
wave solutions of nonlinear, dispersive partial differential equations
which possess a B\"acklund transformation. We will consider
the case of the $m$-soliton solution of the Toda lattice and use
the linearization of the B\"acklund transformation to construct
a conjugation of the Toda flow linearized about an $m$-soliton with
the Toda flow linearized about an $m-1$-soliton. Applying this procedure
inductively we can relate the linearization of the Toda flow about
an $m$-soliton to the linearization about the zero-solution, whose
stability properties can be determined by explicit calculation.
This is joint work with N. Benes and A. Hoffman.
Graduate Student Speakers_______________________________ Edwin Ding (University of Washington) A generalized master mode-locking model is presented to characterize the pulse evolution in a ring cavity laser passively mode-locked by a series of waveplates and a polarizer, and the equation is referred to as the sinusoidal Ginzburg-Landau equation (SGLE). The SGLE gives a better description of the cavity dynamics by accounting explicitly for the full periodic transmission generated by the waveplates and polarizer. The SGLE model supports intense, short pulses with large amount of energy that are not predicted by the conventional master mode-locking theory, thus providing a platform for optimizing the laser performance. _______________________________ Curtis Holliman (University of Notre Dame) We consider the initial value problem for the periodic Hunter-Saxton
(HS) equation. For $s>3/2$, we demonstrate that the data-to-solution
map for HS from $H^s$ into $C([0,T]; H^s)$ is continuous but no
better. In addition to this fact, we will also show that when the
topology of the codomain of this map is weakened to $C([0,T];H^r)$,
$r < s$, this map becomes in fact becomes H\"{o}lder continuous. ____________________________ Levant Kurt (City University of New York) We study a stochastic version of the short pulse equation (SPE), which models ultra-short pulse propagation in cubic nonlinear media. The derivation of the stochastic short pulse equation and the sources of stochasticity in Maxwell's equation are discussed. The effects of random variations on the evolution of a SPE soliton in both the stochastic SPE and stochastic Maxwell's equation are characterized. Our numerical work shows that the SPE solitons propagate stably in both nonlinear Maxwell's equations and the stochastic SPE. _______________________________ Seungly Oh (University of Kansas) We study local well-posedness (l.w.p.) for the quadratic Schroedinger equations in 1+1 dimension. Earlier result in the subject (by Kenig-Ponce-Vega, late 90's and Bejenaru-Tao, '05) showed l.w.p. for data in the Sobolev space $H^{-1+}$, which is sharp. The method of proof is a fixed point argument in spaces inspired by the standard Bourgain spaces $X^{s,b}$, modified to accommodate the low regularity. We consider more general models (with up to half of extra derivative added) and prove l.w.p. for a range of indices, in particular recovering Bejenaru-Tao's result. Our method of proof is totally different, since we work in the standard Bourgain spaces, but we precondition the equation by a special change of the variables - normal forms. We also get some Lipschitzness statements of the solution map in smoother spaces, which do not follow from the Bejenaru-Tao's proof. _______________________________ Anton Sakovich (McMaster University) _______________________________ Alessandro Selvitella (McMaster University) _______________________________ Yannan Shen (University of Massachusetts at Amherst) We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. We derive two short-pulse equations (SPEs) for the high- and low-frequency band for 1 dimensional case. Then we generalize this into 2 dimensional case and give a 2D version of the short pulse equation. For 1D soutions, we will discuss the connection with the soliton solution of the nonlinear Schr¨odinger equation, also we will discuss the robustness of various solutions emanating from the sine-Gordon equation and their periodic generalizations. For the 2D short pulse equation, we will discuss several possible forms and their solutions. _______________________________ Matt Williams (University of Washington) A reduced order model (ROM) based on the proper orthogonal decomposition (POD) is used to describe the multi-pulse transition in a waveguide array mode-locked laser. The reduced order model qualitatively reproduces the dynamics observed in the full system. Furthermore, it reveals that the multi-pulse transition is instigated by a Hopf bifurcation, followed by period-doubling and torus bifurcations, and then chaos.
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