Sine-Gordon equation and its application to tectonic stress transfer

Abstract

An overview is given on remarkable progress that has been made in theoretical studies of solitons and other nonlinear wave patterns, excited during the deformation of fault block (fragmented) geological media. The models that are compliant with the classical and perturbed sine-Gordon equations have only been chosen. In these mathematical models, the rotation angle of blocks (fragments) and their translatory displacement of the medium are used as dynamic variables. A brief description of the known models and their geophysical and geodynamic applications is given. These models reproduce the kinematic and dynamic features of the traveling deformation front (kink, soliton) generated in the fragmented media. It is demonstrated that the sine-Gordon equation is applicable to the description of series of the observed seismic data, modeling of strain waves, as well as the features related to fault dynamics and the subduction slab, including slow earthquakes, periodicity of episodic tremor and slow slip (ETS) events, and migration pattern of tremors. The study shows that simple heuristic models and analytical and numerical computations can explain triggering of seismicity by transient processes, such as stress changes associated with solitary strain waves in crustal faults. The need to develop the above-mentioned new (nonlinear) mathematical models of the deformed fault and fragmented media was caused by the reason that it is impossible to explain a lot of the observed effects, particularly, slow redistribution and migration of stresses in the lithosphere, within the framework of the linear elasticity theory.

Appendix

Here, only those analytical solutions of the sine-Gordon equation are given that have been used by different researchers for developing models of seismic activation of faults, demonstrate the main features of the deformation process in fault zones and explore a relative role of different factors in wave dynamics of the earthquake source.

The classical sine-Gordon equation has the following form:

$$ \frac{\partial^2U}{\partial {\xi}^2}-\frac{\partial^2U}{\partial {\eta}^2}= \sin U, $$

(25)

where ξ and η are the spatial and temporal coordinates; U is the dynamic variable (the rotation angle or displacement of the block or fragment of the medium). If to search for the solution in the shape of a traveling wave (β is the wave velocity),

$$ U=U\left(\tau \right)=U\left(\xi -\beta \eta \right). $$

Equation (25) turns into

$$ \frac{d^2U}{d{\tau}^2}=\frac{ \sin U}{1-{\beta}^2}. $$

(26)

Equation (26) has the following well-known solutions:

• 1. Periodic fast cnoidal waves (0 < k < 1; β ² > 1):

$$ U=2 \arcsin \left( ksn\left[-\frac{1}{k}\left(\frac{\xi -\beta \eta}{{\left(1-{\beta}^2\right)}^{1/2}}\right);k\right]\right), $$

(27)

$$ V=\frac{\partial U}{\partial \eta }=-\frac{\beta k}{\pi {\left({\beta}^2-1\right)}^{1/2}}\mathrm{cn}\left[\left(\frac{\xi -\beta \eta}{{\left({\beta}^2-1\right)}^{1/2}}\right);k\right]. $$

(28)

Solution (27) appears as a traveling wave oscillating close to the value U = 0. Solution (28) corresponds to a periodic wave with an average zero value. V is the velocity of dynamic variable U (the rotation angle or displacement of the block of the geological medium).

• 2. Periodic slow cnoidal waves (0 < k < 1; β ² < 1):

$$ U= \arcsin \left\{\pm \mathrm{cn}\left[-\frac{1}{k}\left(\frac{\xi -\beta \eta}{{\left(1-{\beta}^2\right)}^{1/2}}\right);k\right]\right\}, $$

(29)

$$ V=\frac{\partial U}{\partial \eta }=\pm \frac{\beta }{\pi k{\left(1-{\beta}^2\right)}^{1/2}}\mathrm{dn}\left[\frac{1}{k}\left(\frac{\xi -\beta \eta}{{\left(1-{\beta}^2\right)}^{1/2}}\right);k\right]. $$

(30)

Solution (30) represents a periodic sequence of pulses with a spatial period 2k(1 − β ²)^1/2 K(k), where K(k) is the complete elliptical integral of the first kind. In expressions (27)–(30), the notations sn(ξ, k), cn(ξ, k), and dn(ξ, k) are the Jacobian elliptic functions; k is the modulus of the elliptic function.

• 3. Solitary waves (solitons) ($\begin{array}{cc}\hfill k\to 1;\hfill & \hfill {\beta}^2<1\hfill \end{array}$):

$$ U=4\mathrm{arctg}\left[ \exp \left(\pm \frac{\xi -\beta \eta}{{\left(1-{\beta}^2\right)}^{1/2}}\right)\right], $$

(31)

$$ V=\frac{\partial U}{\partial \eta }=\pm \frac{\beta }{\pi {\left(1-{\beta}^2\right)}^{1/2}}\mathrm{sech}\left(\frac{\xi -\beta \eta}{{\left(1-{\beta}^2\right)}^{1/2}}\right). $$

(32)

Solutions (31) and (32) are most frequently encountered in the present overview and have the proper names: the first one is a kink, a wave with invariant profile in the shape of a twist by variable U; the second one is a soliton, a solitary wave, transmitting at velocity β. The above solutions are schematically shown in Fig. 3. Figure 3b, c, corresponding to the solutions (31) and (32) of the sine-Gordon equation, coincides in their shapes with the displacements and velocities of stick-slip at the contact of blocks of rocks, observed in the laboratory experiments (Bykov 2008).

Fig. 3

Profiles of velocities V and displacements U of particles in waves. a “Fast” (1) and “slow” (2) cnoidal waves. b Solitary waves. c Displacements U of particles (kink)