# SCIENTIFIC PROGRAMS AND ACTIVITIES

December 20, 2014

## May 2-5, 2011 Workshop on Wave Breaking and Global Solutions in the Short-Pulse Dispersive Equations at the Fields Institute , 222 College Street, Toronto

Organizer: Dmitry Pelinovsky, McMaster University (Canada)

### Invited Speakers:

Robin Ming Chen (University of Minnesota)
The Holder continuity of Camassa-Holm solution map

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.

_______________________________

Yeo-Jin Chung (Southern Methodist University)
Strong Collapse Turbulence in Quintic Nonlinear Schroedinger Equation.

We consider the quintic one dimensional nonlinear Schr\"odinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. We obtain the tail of PDF corresponding to intermittency in the collapse-dominated NLS turbulence.

_______________________________

Mathieu Colin (University of Bordeaux)
Laser Plasma interactions : Zakharov's System and solitary waves.

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)
Analysis of solitary waves of the regularized short pulse equation

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)
Birkhoff normal forms for the problem of water waves

_______________________________

Alex Himonas (University of Notre Dame)
The Cauchy problem of weakly dispersive equations

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

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)
Global small solutions to the Navier-Stokes-Maxwell equations

We consider a full system of magneto-hydro-dynamic equations. The system formally satisfies an energy estimate. Nevertheless, the existence of global weak solution seems to remain an interesting open problem in both two and three space dimension. In 3D, we show the existence of global small solutions (Kato-type). In 2D, we prove the same result in a space "close" to the energy space. This is joint work with S. Keraani (University of Lille 1, France).

_______________________________

Nathan Kutz (University of Washington)
Modeling Mode-Locked Lasers in the Few Femtosecond Regime

We propose a model that is valid for ultrafast pulse propagation in a mode-locked laser cavity in the fewfemtosecond pulse regime, thus deriving the equivalent of the master mode-locking equation (complex Ginzburg-Landau equation with L$^2$ norm control) for ultrashort pulses that has dominated mode-locking theory for two decades. The short-pulse equation with dissipative gain and loss terms allows for the generation of stable ultrashort optical pulses from initial white noise, thus providing the first theoretical framework for quantifying the pulse dynamics and stability as pulses widths approach the attosecond regime.

_______________________________

Didier Pilod (Federal University of Rio de Janeiro)
On Hirota-Satsuma's equation

_______________________________

Zhijun Qiao (University of Texas - Pan American)
Negative order KdV equation and its soliton and kink solutions

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)
The NLSE and SPE as approximations of a nonlinear wave equation

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)
Why makes the NLS equation correct predictions beyond its range of validity?

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)
Well-posedness and small data scattering for the generalized Ostrovsky equation

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)
Integrable evolution equations on spaces of tensor densities and their peakon solutions

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)
Asymptotic Stability of the Toda $m$-soliton

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.

_______________________________

Edwin Ding (University of Washington)
High-Energy Passive Mode-Locking with the Sinusoidal Ginzburg-Landau Model

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)
Continuity properties of the data-to-solution map for the Hunter-Saxton Equation

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)
Randomness and Stochastic Short Pulse Equation

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)
On quadratic Schroedinger equations in 1D: a normal form approach

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)
Wave breaking in the Ostrovsky--Hunter equation

The Ostrovsky--Hunter equation governs evolution of shallow water waves on a rotating fluid in the limit of small high-frequency dispersion. Sufficient conditions for the wave breaking in the Ostrovsky--Hunter equation are found both on an infinite line and in a periodic domain. Using the method of characteristics, we also specify the blow-up rate at which the waves break. Numerical illustrations of the finite-time wave breaking are given in a periodic domain.

_______________________________

Alessandro Selvitella (McMaster University)
Some problems concerning a Quasilinear Schroeinger Equation

I will talk about some problems concerning a Quasilinear Schroeinger Equation. In particular, I will discuss a joint work with Prof. Louis Jeanjean about uniqueness and nondegeneracy of the ground state. I will also summarize results on the Cauchy Problem for this equation.

_______________________________

Yannan Shen (University of Massachusetts at Amherst)
Short Pulse Equation in High/Low Frequency Band of Nonlinear Metamaterials

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 for the Multi-Pulse Transition in Mode-Locked Lasers

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.