Abstract
We develop the Riemann–Hilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann–Hilbert problem with a shift posed on a torus.
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Ablowitz M.J., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Aki K., Richards P.: Quantitative Seismology. Freeman, San Francisco (1980)
Fokas, A.S.: A unified approach to integrability: Fourier transform and beyond. In: Proceedings of the Conference in Honor P. Lax and L. Nirenberg, Venice, 1996
Fokas A.S.: A unified transform method for solving linear and certain nonlinear PDE’s. Proc. R. Soc. Ser. A 453, 1411–1443 (1997)
Fokas A.S.: Lax pairs and a new spectral method for linear and integrable nonlinear PDEs. Sel. Math. New Ser. 4, 31–68 (1998)
Fokas, A.S.: A Unified Approach to Boundary Value Problems. CBMS-SIAM (2008)
Fokas A.S.: Two Dimensional Linear PDEs in a Convex Polygon. Proc. R. Soc. A 457, 371–393 (2001)
Its, E.: Lax Pair and the Riemann–Hilbert method for solving diffraction and scattering problems in geophysics. In: Proceedings of the SAGEEP, March 2001, 9 pp
Its, E.: Riemann–Hilbert approach to the elastodynamic equation in a quarter-space, Part I. Pr07-05. http://www.math.iupui.edu
Knopoff L., Gangi A.F.: Transmission and reflection of Rayleigh waves by wedges. Geophysics 25, 1203–1214 (1960)
Lamb H.: On the propagation of tremors over a surface of an elastic solid. Phil. Trans. R. Soc. Lond. A203, 1–42 (1904)
Levshin, A.L., Yanovskaya, T., Lander, A., Bukchin, B., Ratnikova, L., Its E.: In: Keilis-Borok, V.I. (ed.) Seismic Surface Waves in Horizontally Inhomogeneous Earth. Elsevier Science Publishers, Netherlands (1989)
Momoi T.: Scattering of Rayleigh waves in an elastic quarter space. J. Phys. Earth 28, 385–413 (1980)
Novikov S.P., Zakharov V.E., Manakov S.V., Pitaevski L.V.: Soliton Theory: The Inverse Scattering Method. Plenum, NY (1984)
Sommerfeld A.: Partial Differential Equations in Physics. Academic Press, New York (1949)
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Its, A., Its, E. & Kaplunov, J. Riemann–Hilbert Approach to the Elastodynamic Equation: Part I. Lett Math Phys 96, 53–83 (2011). https://doi.org/10.1007/s11005-010-0448-7
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DOI: https://doi.org/10.1007/s11005-010-0448-7