Global attraction to solitary waves in nonlinear dispersive Hamiltonian systems
 
BIRS, Banff, Alberta, Canada. May 20-30, 2007

On the web: http://www.math.tamu.edu/~comech/proposals/birs-2007/

1. Title of proposal:

Global attraction to solitary waves in nonlinear dispersive Hamiltonian systems

2. Type of meeting: Focussed research group

3. Organizers:

1. Vladimir Buslaev, St. Petersburg University, buslaev@mph.phys.spbu.ru
2. Andrew Comech, Texas A&M University, comech@math.tamu.edu
3. Alexander Komech, University of Vienna, alexander.komech@univie.ac.at
4. Boris Vainberg, UNC - Charlotte, brvainbe@mosaic.uncc.edu

4. Primary subject area from the 2-digit AMS Classification: 35

5. A short overview of the subject area:

We are interested in the stability properties of solitary waves and long-time asymptotics of finite energy solutions. According to ``Derrick's theorem'', time-independent soliton-like solutions to Hamiltonian systems, under rather general assumptions, are unstable. On the other hand, if the system possesses symmetries, then quasi-stationary solutions may be stable. This is caused by the additional conservation laws, which may prevent a slightly perturbed solitary wave from tumbling in the direction of lower energy states. This stimulated the study of the existence and stability properties of solitary waves in the Hamiltonian systems with symmetries. This field remains very active for the last thirty years, and yet many questions are not understood. Let us illustrate the state of things on the example of the Nonlinear Klein-Gordon Equation:

NLKG: $$\partial\sb t^2 u(x,t)=\partial\sb x^2 u+F(u), \qquad x\in{\mathbf R},\quad t\ge 0, \quad u(x,t)\in{\mathbf C}.$$

This equation describes oscillations of a string with certain elasticity properties (represented by a smooth function $F(u)$, ``the nonlinearity''). The string is infinite and stretched along the $x$-axis. Real and imaginary parts of $u(x,t)$ are the $y$- and $z$-coordinates of the piece of the string above the point $x$ at the moment $t$. We assume that $F$ is smooth and satisfies $F(e^{is}u)=e^{is}F(u)$, $s\in{\mathbf R}$, so that the equation is ${\mathbf U}(1)$-invariant. For a particular $F(u)$, one would like to know:

1. Are there solutions of the form $\phi\sb\omega(x)e^{-i\omega t}$ with $\phi\sb\omega(x)$ localized (``solitary waves'')?

2. Which solitary waves are orbitally stable (so that small perturbations do not grow)?

3. Which solitary waves are asymptotically stable (so that small perturbations disperse)?

4. As $t\to\infty$, does any finite energy solution look like outgoing solitary waves?

These most natural questions in the context of nonlinear dispersive $U(1)$-invariant Hamiltonian systems are the central questions of the PDE Theory. It is mainly due to the research of W. Strauss and his school (since the seventies) that Questions 1, 2 are understood rather well. In particular, it is known that for the same nonlinearity some solitary waves could be stable while others could be unstable.

The asymptotic stability (Question 3) could be viewed as the local attraction to solitary waves. At present, not much is known about asymptotic stability, especially in the translation-invariant case; only the nonlinear Schrödinger equation has been studied in the works by Buslaev and Perelman; this was developed by Cuccagna and others. (No results for NLKG as of yet.) The asymptotic stability is proved under very strong assumptions on the spectrum and the order of vanishing of the nonlinearity. At the same time, one expects asymptotic stability for any orbitally stable solitary wave.

The last question (Question 4) is about the global attraction. Namely, we would like to know whether the system has a finite-dimensional attracting set, and whether it is formed by solitary waves. Properties of global attractors are well-understood for the dissipative systems (such as Navier-Stokes equation). Yet, essentially nothing is known even for NLKG in 1D.

Both the asymptotic stability and the global attraction are based on the ``disersive damping'': the nonlinearity increases the spectrum of the perturbation, creating the dispersive waves that carry the excess energy and charge away. This mechanism is still not described rigorously.

6. A statement of the objectives of the workshop and an indication of its relevance, importance, and timeliness:

Objectives. We are interested in properties of solitary waves. In particular systems, we will study the long-time asymptotics of finite energy solutions and properties of global attractors. Our ultimate goal is to prove that the attractor is finite-dimensional and consists of the set of all solitary waves; that is, that each finite energy solution approaches the set of ``nonlinear eigenfunctions''. More precisely, in the context of the NLKG described above, we would like to prove that any finite energy solution asymptotically looks like a superposition of scattering solitary waves plus dispersion:

$$\psi(x,t)\sim\sum\sb{n=1}\sp{N}\Psi\sb n(x,t)+\Psi\sb{disp}(x,t)+r(x,t),$$

where $\Psi\sb n(x)=\phi\sb n((x-v\sb n t)/\sqrt{1-v^2}\!\!\!\sb n) e^{-i\omega_n {t}/{\sqrt{1-v^2}\!\!\!\sb n}}$, $1\le n\le N$, are solitary waves traveling with the speeds $v\sb n$, $\Psi\sb{disp}$ is a dispersive wave (solution to a linear equation), and $r(x,t)$ is a small remainder that goes to zero as $t\to\infty$ in the global energy norm. Let us point out that this asymptotic behavior is a purely nonlinear effect.

We are going to consider such long-time asymptotics and the global attraction in several models based on the nonlinear Klein-Gordon equation in one and also in higher dimensions. The proposed research is on the borderline of Mathematics and Physics and is directly related to Optics, Waveguide Theory, Field Theories, Solid Matter Physics. Its mathematical tools are rooted in Harmonic and Functional Analysis, Spectral Theory, Partial Differential Equations. This interconnection of several disciplines allows to engage top specialists from adjacent fields, who represent the main body of the group. Their collaboration (for example, the focused research group of A. Komech in Wolfgang Pauli Institute in Vienna University in 2004-2005 that included V. Buslaev and B. Vainberg) already proved extremely fruitful.

Relevance and importance. Long-time asymptotics and stability of solitary waves in Hamiltonian systems with dispersion is a fundamental question of the PDE theory, and is of tremendous value for natural sciences because of the ubiquity of the dispersive equations which describe all sorts of oscillations. The ultimate goal is to prove that the global attractor for a dispersive system is formed by the solitary waves (that is, any finite energy solution asymptotically looks like a superposition of leaving solitary waves and dispersive waves). The global attractors were extensively studied for dissipative systems (such as the Ginzburg-Landau equation from Solid State Physics, the Kuramoto-Sivashinsky equation introduced in the study of phase turbulance and thermal diffusive instabilities, and the two-dimensional forced Navier-Stokes equation). Yet, the results are absent for dispersive systems, except for completely integrable models. The research in this direction stimulates the development of a variety of mathematical tools aimed at nonlinear problems. In particular, the application of the Titchmarsh Convolution Theorem for the analysis of the global attractors of models based on the Klein-Gordon equation gives yet another link of the Harmonic and Complex Analysis to the PDE Theory.

Timeliness. Absence of adequate description of nonlinear effects was seriously hindering the science for the last hundred years. While the traveling waves of the Korteweg - de Vries equation date back to 1895, it is only in the last thirty years that we start understanding their stability properties. At the same time, numerical simulations do not allow adequate long-time description of nonlinear equations in the infinite space; long-time asymptotics and stability issues are to be tackled analytically.

Today, in a great part due to pioneering works of W. Strauss and his school, these important problems of nonlinear Science seem to be within the mathematical reach. In the last several years, the group participants considered different aspects of attraction to solitary waves in a number of dispersive Hamiltonian systems. In particular, the global attraction to solitary waves was rigorously proved for several dispersive Hamiltonian systems; this is the first result of this kind. The models are based on nonlinear Schrödinger and Klein-Gordon equations, with different nonlinear parts. We expect that the recent progress we made will allow us to extend our methods for more general systems, such as nonlinear translation-invariant Schrödinger and Klein-Gordon equations.

7. We have a policy of sending out a 1-2 paragraph press release for each BIRS 5-Day Workshop. The press seems to be interested and we think it's a good idea to advertise our scientists. Please submit a 1-2 paragraph piece that is understandable to the general populace.

When the amplitude of the waves increases (be it the gravitational waves or the waves in plasmas), their interaction with the medium (or self-interaction) becomes important. This interaction may seriously change the behavior of the waves and lead to the appearance of nonlinear solitary waves, or solitons. Solitary waves describe numerous natural phenomena of purely nonlinear origin. They appear in the Ocean Dynamics (surface waves, including rogue waves; tidal bores; undersea internal waves), in the Atmosphere Studies (such as Morning Glory cloud), and in the Quantum Field Theories. Solitons exist in plasmas and crystal lattices. Recently they came under most scrupulous attention due to application of fiber optics for data transmissions.

We are interested in the stability properties and large time asymptotics of solitary waves. These properties are related to the following most natural questions: How often do solitary waves form? How easily to they dissolve under small perturbations? Detailed knowledge of stability properties of solitary waves will help predict sudden weather changes, develop optical waveguides, and describe quantum effects on the scales being inexorably approached by today's electronics and chip manufacturers, let alone the Experimental Physics. The nonlinear Analysis is a blend of ideas from many different branches of today's Science. This blend provides a fruitful interdisciplinary environment for a research, and we intend to continue the involvement of the top specialists in several adjacent fields.

8. Total number of participants including organizers: 8

9. A list of possible participants and their affiliation:

1. Vladimir Buslaev, St.Petersburg University, buslaev@mph.phys.spbu.ru
2. Andrew Comech, Texas A&M University, comech@math.tamu.edu
3. Vladimir Georgiev, University of Pisa, georgiev@dm.unipi.it
4. Valera Imaykin, WPI at University of Vienna, ivm@infoline.su
5. Alexander Komech, University of Vienna, alexander.komech@univie.ac.at
6. Elena Kopylova, Vladimir State University, ek@vpti.vladimir.ru
7. Galina Perelman, Ecole Polytechnique, perelman@math.polytechnique.fr
8. David Stuart, University of Cambridge, D.M.A.Stuart@damtp.cam.ac.uk
9. Boris Vainberg, UNC at Charlotte, brvainbe@mosaic.uncc.edu

10. Year, preferred dates:

2007

May 13 - May 27

February 25 - March 11

March 4 - March 18

May 27 - June 10

June 10 - June 24

Additional comments:

We indicated the number of participants to be 8 since two or three of the proposed participants will not be able to attend the whole length of the workshop, while some people from the list still have not confirmed whether they intend to participate. We expect that the average number of participants is going to be 8.