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On spatio-temporal dynamics of sine-Gordon soliton in nonlinear non-homogeneous media using fully implicit spectral element scheme

One- and two-dimensional sine-Gordon equation in non-homogeneous media is considered. Sine-Gordon equation exhibits soliton-like solution whose existence and behaviour in non-homogeneous media is studied. Various non-homogeneous media are discussed which include nonlinear smooth as well as discontinuous density variations. Time-dependent continuous and discontinuous density variations which can model geodynamical processes like lithospheric deformation, fault dynamics, etc. are also discussed. In one dimension, dynamics of kink which are topological soliton is studied under such density variations, whereas in two dimensions, circular-ring soliton dynamics has been scrutinized. The governing sine-Gordon equation is solved numerically using higher order Legendre polynomial-based spectral element method. Spectral stability analysis of the numerical scheme shows the strong dependence of a critical time step not only on density value but also on the nature of the distribution. It is shown that for smooth density variation the critical time step reduces drastically which makes explicit scheme very expensive. To reduce the computational cost, unconditionally stable fully implicit spectral element scheme is proposed which is also energy conservative.