Soliton generation and nonlinear wave propagation

1985 ◽  
Vol 315 (1533) ◽  
pp. 367-377 ◽  
Keyword(s):  
Wave Propagation ◽  
Nonlinear Wave ◽  
Numerical Solutions ◽  
Simple Wave ◽  
Initial Boundary ◽  
Pulse Generation ◽  
Experimental Results ◽  
Boussinesq System ◽  
Nonlinear Wave Propagation ◽  
Soliton Generation

Soliton generation by various means is described. First, experimental results of J . V. Wehausen and coworkers on solitons generated by a ship model in a towing tank are presented. Then T. Y. Wu’s related Boussinesq system of equations for shallow water motion with a moving pressure disturbance on the free surface is introduced. Numerical solutions of this system by D. M . W u and T. Y. Wu are shown to compare well with the experimental results. Similar numerical results on an initial-boundary value problem for the K .d.V . equation by C. K. Chu and coworkers are presented, which also yield soliton generation. Then J . P. Keener and J . Rinzel’s analysis of pulse generation in the Fitzhugh—Naqumo model of nerve conduction is described. Next, G. B. W hitham ’s modulation theory of nonlinear wave propagation is explained and the problem of relating its results to initial and boundary data is mentioned. Asym ptotic methods for solving this problem for the K .d.V . equation are described. They include the Lax—Levermore theory for the case of small dispersion, its extension by S. Venakides, and the centered simple wave solution of the modulation equations by A. V. G urevitch and L. P. Pitaevskii. Finally, the theory of weakly nonlinear waves of Choquet—Bruhat and of J . K. Hunter and the present author is described.

A lattice method for seismic wave propagation in nonlinear viscoelastic media

2020 ◽  
Vol 224 (3) ◽  
pp. 1572-1587
Author(s):  
Gareth S O'Brien
Keyword(s):  
Wave Propagation ◽  
Nonlinear Wave ◽  
Numerical Solutions ◽  
Seismic Wave Propagation ◽  
Complex Model ◽  
Numerical Results ◽  
Static Deformation ◽  
Nonlinear Wave Propagation ◽  
Lattice Method ◽  
Nonlinear Stress

SUMMARY Nonlinear wave propagation is an important consideration in several geophysical problems as heterogeneities within the subsurface give rise to nonlinear stress–strain relationships in rocks. To address these problems, where the inclusion of subsurface complexity is accounted for, numerical solutions are required. Here, we outline a discrete particle or lattice numerical method that can simulate dynamic and static deformation in an isotropic viscoelastic nonlinear medium where the nonlinearity includes the fourth-order elastic tensor coefficients. The relationship between the local interaction constants in the lattice method and the macroscopic nonlinear coefficients is derived and the relationships presented. The theoretical dispersion properties of the system is determined from a nonlinear perturbation method and are validated against some numerical results. There is an excellent agreement between the theory and numerical results and the method reproduces the common features of nonlinear wave propagation, for example, harmonics, waveform distortion and spectra shifts. As expected the system is shown to be dependent on the wave amplitudes and an example is presented which shows the complexity that can arise in both dynamic and static deformation in a complex model where nonlinearity is included.

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