Stability Criteria for Multi-Dimensional Waves and Patterns

May 16 to May 20, 2005

at the

American Institute of Mathematics, Palo Alto, California

organized by

Christopher K.R.T. Jones, Yuri Latushkin, Robert Pego, Arnd Scheel, and Bjorn Sandstede

This workshop, sponsored by AIM and the NSF, will be devoted to the study of stability of nonlinear waves in partial differential equations, with a particular focus on multidimensional questions. Prominent examples of nonlinear waves in higher-dimensional media include vortices and spiral waves. Multidimensional stability is an important issue also for many nonlinear waves with one-dimensional structure, such as solitary waves in the shallow-water equation, viscous shock waves, solitons in integrable dispersive systems, and fronts and pulses in reaction-diffusion systems.

Substantial progress in the understanding of the spectral stability of nonlinear waves in one-dimensional media has recently been achieved by systematically extending and refining the Evans function, an analytic function whose roots are in one-to-one correspondence with isolated eigenvalues of the linearization about a wave. Using robustness properties of topological indices and the computational advantages associated with the fact that a single analytic function captures all eigenvalues, stability and instability criteria for shock waves in viscous conservation laws, for solitons in coupled nonlinear Schrodinger equations, and for fronts and pulses in singularly perturbed reaction-diffusion systems have been derived.

The main emphasis of this workshop is to extend the construction of the Evans function to several space dimensions and to develop easily computable stability criteria for nonlinear waves such as vortices in the family of nonlinear Schrodinger equations, spiral waves in reaction-diffusion systems, and lump solutions in dispersive systems.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and working sessions.

Participants include J. Albert, S. Benzoni-Gavage, D. Bindel, J. Deng, G. Derks, A. Doelman, R. Gardner, F. Gesztesy, A. Ghazaryan, M. Haragus, P. Howard, J. Humpherys, C. Jones, T. Kapitula, R. Kollar, Y. Latushkin, Y. Li, S. Malham, V. Manukian, Y. Nishiura, M. Oh, R. Pego, K. Promislow, B. Sandstede, A. Scheel, P. Szmolyan, H. Warchall, M. Wechselberger, M. Weinstein, M. Williams, and K. Zumbrun.

The deadline to apply for support to participate in this workshop has passed.

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