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Distinguished Lecture Series

     Vladimir E. Zakharov
       Regent's Professor
      University of Arizona

November 13, 2009
Talk - 1:45 p.m in PSA 118

Workshop on Nonlinear Science - Event Flyer

 

Title:
Free Surface Hydrodynamics in Conformal Variables

 

 
Abstract:

We study the potential flow of deep 2-D fluid with free surface in the presence of gravity. The area $ -\infty<y>&lt;\eta(x, t) $ filled with fluid is conformally mapped onto the lower half-plane $ v $<$ 0 $ on "mathematical" plane $ w=u+i v $. If $ \Phi $ is the hydrodynamic potential, in Dyachenko variables $ R(w, t)=\frac{1}{z'},\,\,V(w, t)=i\frac{\partial\Phi}{\partial z} $, the Euler equations take an unusual but very elegant form. These equations are suitable both for analytical and numerical study. Analytical in the lower half-plane functions $ R(w,t) $ and $ V(w,t) $ have moving singularities in the upper half-plane. Zeros of $ R(w,t) $ in the upper half-plane are also important. We are interested in the "robust" singularities only, which preserve their type with time, and can formulate the following rigorous analytical results:

$ \quad $ 1. Poles of $ R,\,V $ are not robust; they immediately turn to cuts.

$ \quad $ 2. Cuts are robust if they end with simple root-square type branch points

 $ R=a(t)\,\left(w -\lambda (t)\right)^{1/2} $, $ V=V_0(t)+b(t)\left(w -\lambda(t)\right)^{1/2} $.

$ \quad $ 3. Zeros of $ R(w, t) $ are robust; they generate motion constants. In the vicinity of the moving zero $ w=a_n(t) $,

$$ R=R_n(w-a_n(t)),\quad V=V_n(t), $$

where

$$ \frac{d R_n}{dt}=0,\quad \frac{dV_n}{dt}=-g,\quad V=-gt+V_n^{(0)}. $$

Each moving root $ a_n(t) $ generates two complex motion constants $ R_n ,\,V_n^{(0)} $.

$ \quad $ 4. If the cuts are narrow, initial integro-differential equations turn to a system of PDE's. In absence of gravity this system is integrable.

In spite of the progress, central questions of the theory are still unanswered:

$ \quad $ A. Can singularity of the surface occur in a framework of exact equations in a finite time?

$ \quad $ B. Are Euler equations for potential flow of deep fluid with free surface an integrable system?

The most plausible answer on both questions is positive.

As part of Workshop on Nonlinear Sciences and to recognize the impact of Zakharov research onto Applied Math and Physics, the DLS will be followed by 2 lectures 3:30-5pm Justin Holmer(Brown Univ.) and Pavel Lushnikov (Univ. of New Mexico/LANL). Click here for more information.

*This event is sponsored by School of Mathematical and Statistical Sciences and Mathematical, Computational and Modeling Sciences Center

Pavel Lushnikov

Title:
Finite time singularities: from individual collapses to collapse turbulence

Abstract:

Many nonlinear systems have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which has been called collapse since the pioneering work of Vladimir Zakharov in 1972. Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability and other mechanisms become important such as inelastic collisions in the Bose-Einstein condensate; optical breakdown and dissipation in nonlinear optical media and plasma, wave breaking in hydrodynamics. Collapses occur in many physical and biological systems including a nonlinear Schroedinger equation (NLS), a Kadomtsev-Petviashvili equation with higher order nonlinearity, a Keller-Segel equation and many others. We will focus on NLS with dissipation and forcing in critical dimension. Without both linear and nonlinear dissipation NLS results in a finite-time singularity (collapse) for any initial conditions. Dissipation ensures collapse regularization. If dissipation is small then multiple near-singular collapses are randomly distributed in space and time forming collapse turbulence. Collapses are responsible for non-Gaussian tails in the probability distribution function of amplitude fluctuations which makes turbulence strong. Power law of non-Gaussian tails is obtained for strong NLS turbulence.

Justin Holmer

Title:
Dynamics of blow-up for the nonlinear Schroedinger equation

Abstract:

We begin by surveying basic structural properties of the nonlinear Schroedinger equation in dimensions one, two, and three. Specifically, we mention ground state solitons, complete integrability and multisolitons in one dimension (discovered by Zakharov-Shabat (1972)), and sufficient conditions for finite-time blow-up in two and three dimensions. We describe the Hamiltonian formulation and discuss the relation between symmetries and conservation laws. In the last part of the talk, we apply these concepts to give an interpretation of the derivation of blow-up dynamics in two dimensions obtained recently by Merle-Raphael (2001-2005) following the heuristic and numerical studies by Landman-Papanicolaou-Sulem-Sulem (1988) and Dyachenko-Newell-Pushkarev-Zakharov (1992). We also discuss our construction of a solution to the three dimensional equation blowing up on a circle. This is joint work with Svetlana Roudenko and is based on a result of Raphael (2006).