The economical and technological benefits of controlling friction are enormous. For example, the effects of kinetic friction on energy dissipation and abrasive degradation of materials is ubiquitous in 1) mechanical and material engineering and 2) machine performance and lifetime. In addition friction is a major mechanism controlling the nucleation and development of earthquakes. The latter impacts the lives of millions of people around the globe. A better understanding of conditions responsible for the appearance and dynamics of regular earthquakes and transient fault slip will be one more step toward the long-term goal of prediction of the place and time of large catastrophic earthquakes.
The theory of friction is a fundamental problem in physics and geophysics. Macroscopic friction is a highly nonlinear process for which there are no governing equations describing it at either laboratory or crustal-fault scales. However, various models have been used to describe friction in a specific context. Our preliminary studies suggest that the Frenkel-Kontorova (FK) model, which was originally introduced to describe plasticity in crystals, is also an appropriate tool to model friction. In the continuum limit the FK model is described by the sine-Gordon (SG) equation, one of the fully integrable nonlinear equations of mathematical physics. This equation has been intensely investigated due to its exceptional importance and universality in physics. The results obtained and mathematical apparatus developed in various prior applications of both the FK model and the SG equation will be applied in this project to describe friction and compared with the results from laboratory experiments. In addition two specific applications of the model will be considered: (1) Episodic Tremor and Slip (ETS), and (2) triggering of non-volcanic tremor. The recently discovered phenomenon of ETS is a promising tool for monitoring previously unavailable regions of the Earth's crust. An ETS event consists of two parts: non-volcanic tremor and slow slip. These are coupled seismic and geodetic phenomena that are broadly correlated in space and time and strikingly periodic in the northern Cascadia and southwest Japan subduction zones. The latest observations reveal a complicated migration pattern of tremor during an ETS event. How a slip pulse generates tremor remains an open question; however, this question will be considered in this project in the framework of the FK model and the SG equation. Analytical solutions of the latter (allowing a clear analysis of parameter dependencies) will connect slip pulse and tremor parameters. Based on this, the migration pattern of tremor will be analyzed in detail. Also, there is growing evidence of teleseismic and tidal triggering of non-volcanic tremor, but the mechanisms of these phenomena are not clear. We suggest that tides or distant earthquakes may affect tremor production through interaction with slip pulses and this hypothesis will be examined in our model.