On the relevance of soliton theory to tsunami modelling

Abstract We discuss the relevance of soliton theory to the modelling of tsunamis in the context of the two largest tsunamis for which records are available—the December 2004 and the May 1960 tsunami. Our contention is that in both cases the scales involved do not permit a balancing effect of dispersion and nonlinearity, and therefore soliton theory is not applicable.

[1]  L. Molinet,et al.  Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa–Holm equation , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  J. F. Toland,et al.  On solitary water-waves of finite amplitude , 1981 .

[3]  H. Segur.  Waves in shallow water, with emphasis on the tsunami of 2004 , 2007 .

[4]  Adrian Constantin,et al.  Exact steady periodic water waves with vorticity , 2004 .

[5]  R. S. Johnson,et al.  Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis , 2008 .

[6]  Anjan Kundu.  Tsunami and nonlinear waves , 2007 .

[7]  Hermann Fritz,et al.  Observations by the International Tsunami Survey Team in Sri Lanka , 2005, Science.

[8]  A. Bressan,et al.  GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION , 2007 .

[9]  J. Grue,et al.  Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami , 2008 .

[10]  Y. Martel,et al.  Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation , 2008, 0803.4020.

[11]  P. Sternberg,et al.  Symmetry of solitary waves , 1988 .

[12]  M. Lakshmanan.  Integrable Nonlinear Wave Equations and Possible Connections to Tsunami Dynamics , 2007 .

[13]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[14]  D. Henry,et al.  Solitons and Tsunamis , 2009 .

[15]  Matsuno.  Nonlinear evolutions of surface gravity waves on fluid of finite depth. , 1992, Physical review letters.

[16]  Vasily Titov,et al.  The Global Reach of the 26 December 2004 Sumatra Tsunami , 2005, Science.

[17]  W. Strauss,et al.  Rotational steady water waves near stagnation , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  E. Pelinovsky.  Hydrodynamics of Tsunami Waves , 2006 .

[19]  Thomas Kappeler,et al.  On geodesic exponential maps of the Virasoro group , 2007 .

[20]  Walter Craig,et al.  Surface Water Waves and Tsunamis , 2006 .

[21]  R. Grimshaw.  Solitary waves propagating over variable topography , 2007 .

[22]  R. S. Johnson,et al.  A Modern Introduction to the Mathematical Theory of Water Waves: Bibliography , 1997 .

[23]  Efim Pelinovsky,et al.  Asteroid impact tsunamis , 2005 .

[24]  A. Bressan,et al.  Global Conservative Solutions of the Camassa–Holm Equation , 2007 .

[25]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

[26]  Tsz Leung Yip,et al.  Predicting tsunami arrivals: Estimates and policy implications , 2009 .

[27]  F. Serre,et al.  CONTRIBUTION À L'ÉTUDE DES ÉCOULEMENTS PERMANENTS ET VARIABLES DANS LES CANAUX , 1953 .

[28]  Edward Bryant,et al.  Tsunami: The Underrated Hazard , 2001 .

[29]  T. Barnett,et al.  Recent advances in the study of wind waves , 1975 .

[30]  Harvey Segur,et al.  The Korteweg-de Vries equation and water waves. Part 3. Oscillatory waves , 1978 .

[31]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[32]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[33]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[34]  H. Segur.  Integrable models of waves in shallow water , 2008 .

[35]  T. Benjamin.  The stability of solitary waves , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[36]  Adrian Constantin,et al.  The trajectories of particles in Stokes waves , 2006 .

[37]  M. Lighthill,et al.  Waves In Fluids , 2002 .

[38]  J. Maddocks,et al.  On the stability of KdV multi‐solitons , 1993 .

[39]  Joachim Escher,et al.  Particle trajectories in solitary water waves , 2007 .

[40]  A. Constantin,et al.  Geodesic flow on the diffeomorphism group of the circle , 2003 .

[41]  R. Johnson,et al.  On the Non-Dimensionalisation, Scaling and Resulting Interpretation of the Classical Governing Equations for Water Waves , 2008 .

[42]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[43]  S. Barrientos.  Earthquakes: Giant returns in time , 2005, Nature.

[44]  Per A. Madsen,et al.  On the solitary wave paradigm for tsunamis , 2008 .

[45]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[46]  Adrian Constantin,et al.  Stability of the Camassa-Holm solitons , 2002, J. Nonlinear Sci..

[47]  A. Constantin,et al.  The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations , 2007, 0709.0905.

[48]  Yuki Sawai,et al.  Predecessors of the giant 1960 Chile earthquake , 2005, Nature.

[49]  Adrian Constantin,et al.  A shallow water equation on the circle , 1999 .

[50]  Walter Craig,et al.  Non–existence of solitary water waves in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[51]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.

[52]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[53]  J. Hammack.  A note on tsunamis: their generation and propagation in an ocean of uniform depth , 1973, Journal of Fluid Mechanics.

[54]  W. Strauss,et al.  Stability properties of steady water waves with vorticity , 2007 .

[55]  Peter J. Olver,et al.  Euler operators and conservation laws of the BBM equation , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.