# Solitons in Hierarchical Systems (an example)

Monday, October 8, 2001 - 2:00pm - 3:30pm

Keller 3-180

Sergey Cherkis (Princeton University)

We apply techniques of conformal field theory and integrable systems to explore the following problem arising in seismology: prediction of a strong earthquake by the emergence of particular patters of seismic activity in a lower energy range. Seismic activity is known to exhibit, on average, scale invariance of the form dN(e)~E-c dE, where N is the annual number of earthquakes with energy E and c is a critical exponent. We model seismicity by a hierarchical model proposed by Belov. The model is integrable and displays scale invariance.

Using Lax formalism we find infinitely many conserved quantities and find solitonic solutions. A soliton solution is interpreted as free transfer of the abundance of defects on small scales to large scales. In other words, the original rise of seismic activity is a perturbation of N(E) which has a very special form. Such a perturbation propagates without dispersion from small to large energies.

In such a system, monitoring its behavior at small scales for solitonic excitations can provide criteria for predicting a large scale event. We search for complete Gelfand-Levitan-Marchenko transformation to scattering data, which would provide such quantitative criteria.

Even though the integrability of the considered system is a fine feature that is lost with a generic perturbation, by the universality principle we expect his model to provide a good description near the conformal point for all other systems in the same universality class.

Using Lax formalism we find infinitely many conserved quantities and find solitonic solutions. A soliton solution is interpreted as free transfer of the abundance of defects on small scales to large scales. In other words, the original rise of seismic activity is a perturbation of N(E) which has a very special form. Such a perturbation propagates without dispersion from small to large energies.

In such a system, monitoring its behavior at small scales for solitonic excitations can provide criteria for predicting a large scale event. We search for complete Gelfand-Levitan-Marchenko transformation to scattering data, which would provide such quantitative criteria.

Even though the integrability of the considered system is a fine feature that is lost with a generic perturbation, by the universality principle we expect his model to provide a good description near the conformal point for all other systems in the same universality class.