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Propagation of fractal tsunami solitary waves

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Abstract

In this paper, tsunami dynamics and solitary waves are constructed in fractal dimensions based on the concept of product-like fractal measure introduced recently by Li and Ostoja–Starzewski in their formulation of anisotropic continuum media. The tsunami waves are comparable to some extent to varying-speed wave equations which are used in several studies including the spreading of shallow-water wave in certain continuous depth variations, tsunami and Rayleigh waves. Nonlinear effects were studied and the associated solitary wave in fractal dimension has been derived. It was observed that the shape of the solitary wave is affected by the fractal dimension and is analogous to an asymmetric wave train. Besides, its period is long and differs from the famous Korteweg–De Vries soliton. Such a deformation in the wave profile results in a steep wave front configuration which is observed during large tsunamis such as the 2011 Tohoku tsunami and the 2004 Indian Ocean tsunami. Our analysis shows also that short tsunamis wave could drastically enhance the maximum flow velocities, a result which is in agreement with several analytical and experimental studies.

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References

  • Abdalazeez AA, Didenkulova I, Dutykh (2019) Nonlinear deformation and run-up of single tsunami waves of positive polarity: numerical simulations and analytical predictions. Nat. Hazards Earth Syst. Sci. 19: 2905-2913

  • Abramowitz M, Stegun IA (eds) (1972) Exponential integral and related functions, ch. 5 in handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing. Dover, New York, pp 227–233

    Google Scholar 

  • Adams WM, Jordaan JM (2005) Tsunamis and Tsunami Warning Systems. In Hydraulic Structures, Equipment and Water Data Acquisition Systems, Eds. Hydraulic Structures, Equipment and Water Data Acquisition Systems–Tsunamis and Tsunami Warning Systems, United States, Educational, Scientific and Cultural Organization, Encyclopedia of Life Support Systems ELOSS.

  • Ajiwibowo H (2003) Fractals and nonlinearity of ocean waves. J Tek Sipil 10:93–98

    Article  Google Scholar 

  • Alberti T, Donner RV, Vannitsem S (2021a) Multiscale fractal dimension analysis of a reduced order model of coupled ocean-atmosphere dynamics. Earth Syst Dyn 12:837–855

    Article  Google Scholar 

  • Alberti T, Donner RV, Vannistem S (2021b) Multiscale fractal dimension analysis of a reduced order model of coupled ocean-atmosphere dynamics. Earth Syst Dynam 12:837–855

    Article  Google Scholar 

  • Babich VM, Buldyrev VS (1991) Short-wavelength Diffraction Theory. Springer-Verlag, New York, Asymptotic Methods

    Book  Google Scholar 

  • Bains PG (1997) A fractal world of cloistered waves. Nat 388:518–519

    Article  Google Scholar 

  • Balankin AS, Elizarraraz BE (2012) Map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 85:056314

    Article  Google Scholar 

  • Balankin AS, Susarrey O, Mora Santos CA, Patiño J, Yoguez A, García EI (2011) Stress concentration and size effect in fracture of notched heterogeneous material. Phys Rev E 83:015101

    Article  Google Scholar 

  • Balankin AS, Nena B, Susarrey O, Samayoa D (2017) Steady laminar flow of fractal fluids. Phys Lett A 381:623–638

    Article  MathSciNet  MATH  Google Scholar 

  • Baldassarri A, Montuori M, Prieto-Ballesteros O, Manrubia SC (2008) Fractal properties of isolines at varying altitude reveals different dominant geological processes on Earth. J Geophys Res 113:E09002

    Google Scholar 

  • Barclay DW, Moodie TB, Rogers C (1987) Cylindrical impact waves in inhomogeneous viscoelastic media. Acta Mech 29:93–117

    Article  MATH  Google Scholar 

  • Benoit M, Raoult C, Yates ML (2017) Analysis of the linear version of a highly dispersive potential water wave model using a spectral approach in the vertical. Wave Motion 74:159–181

    Article  MathSciNet  MATH  Google Scholar 

  • Berizzi F, Dalle Mese E, Pinelli G (1999) One-dimensional fractal model of the sea surface. IEEE Proc.-Radar. Sonar and Navigation 146:55–64

    Article  Google Scholar 

  • Billingham J, King AC (2012) Wave Motion. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Boshenyatov BV, Zhiltsov KN (2019) Investigation of non-linear effects resulting from the interaction of tsunami like waves and underwater barriers. Vestn Tomsk Gos Univ Mat Mekh 59:37–52

    Google Scholar 

  • Boyd JP (1998) Weakly nonlocal solitary waves and beyond-all-orders asymptotics, mathematics and its applications 442. Kluwer

    Google Scholar 

  • Britt S, Tsynkov S, Turkel E (2018) Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials. J Comp Phys 354:26–42

    Article  MathSciNet  MATH  Google Scholar 

  • Cabrera-Brito L, Rodriguez G, Garcia-Weil L, Pacheco M, Perez E, Waniek JJ (2017) Fractal analysis of deep ocean current speed time series. J Atmosph Ocean Tech 34:817–827

    Article  Google Scholar 

  • Chakraborty S, Chattopadhyay S (2021) Exploring the Indian summer monsoon rainfall through multifractal detrended fluctuation analysis and the principle of entropy maximization. Earth Sci Inform 14:1571–1577

    Article  Google Scholar 

  • Chattopadhyay G, Chattopadhyay S, Midya SK (2021) Fuzzy binary relation based elucidation of air quality over a highly polluted urban region of India. Earth Sci, Inform 14:1625–1631

    Article  Google Scholar 

  • Clements DL, Rogers C (1975) Analytic solution of the linearized shallow-water wave equations for certain continuous depth variations. J Aust Math Soc B 19:81–94

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin A, Henry D (2009) Solitons and tsunamis. Z Naturforsch 64:65–68

    Article  Google Scholar 

  • Craig W (2006) Surface water waves and tsunamis. J Dyn Diff Equat 18:525–549

    Article  MathSciNet  MATH  Google Scholar 

  • Demmie PN, Ostoja-Starzewski M (2011) Waves in fractal media. J Elasticity 104:187

    Article  MathSciNet  MATH  Google Scholar 

  • Didenkulova I (2009) New trends in the analytical theory of long sea wave runup. In: Quak E, Soomere T (eds) Applied Wave Mathematics. Springer, Berlin, Heidelberg, Germany, pp 265–296. https://doi.org/10.1007/978-3-642-00585-5_14.

  • Didenkulova I, Pelinovsky E, Soomere T, Zahibo N (2007) Run-up of nonlinear asymmetric waves on a plane beach, in: Tsunami and nonlinear waves, edited by: Kundu A, Springer, Berlin, Heidelberg, Germany, 175–190, https://doi.org/10.1007/978-3-540-71256-5_8.

  • Didenkulova I, Denissenko P, Rodin A, Pelinovsky E (2013) Effect of asymmetry of incident wave on the maximum runup height. J Coastal Res 65:207–212

    Article  Google Scholar 

  • Dimri VP, Srivastava K (2015) Introduction for the special issue: the role of fractals in seismology. Nat Hazards 77:1–4

    Article  Google Scholar 

  • Dimri VP (2005) Fractals in Geophysics and Seismology: An Introduction. In: Dimri V.P. (eds) Fractal Behaviour of the Earth System. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26536-8_1.

  • Domany E, Alexander S, Bensimon D, Kadanoff LP (1983) Solutions to the Schrödinger equation on some fractal lattices. Phys Rev B 28:3110

    Article  MathSciNet  Google Scholar 

  • Donadio C, Paliaga G, Radke JD (2019) Tsunamis and rapid coastal remodeling: Linking energy and fractal dimension. Prog. Phys. Geograph.: Earth and Environ. 44: 550–571.

  • Dutykh D (2007) Mathematical modelling of tsunami waves. PhD Thesis. Centre de Mathématiques et de Leurs Applications, École normale supérieure de Cachan.

  • Dyskin AV (2004) Continuum fractal mechanics of the Earth’s crust. Pure Appl Geophys 161:1979–1989

    Article  Google Scholar 

  • Elgar S, Mayer-Kress G (1989) Observations of the fractal dimension of deep and shallow-water ocean surface gravity waves. Phys d Nonlinear Phenomena 27:104–108

    Article  Google Scholar 

  • El-Nabulsi RA (2006) Some geometrical aspects of nonconservative autonomous Hamiltonian dynamical systems. Int J Appl Math Stat 5:50–61

    MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2009) Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems. Chaos, Solitons Fractals 42:52–61

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2011a) The fractional Boltzmann transport equation. Comp Math Appl 62:1568–1575

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2011b) The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars. Appl Math Comp 218:2837–2849

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2017) Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, jerk and snap and oscillatory motions in the rotating fluid tube. Int J Nonlin Mech 93:65–81

    Article  Google Scholar 

  • El-Nabulsi RA (2018a) Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J Stat Phys 172:1617–1640

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2019a) Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic carrier concentrations. J Phys Chem Sol 127:224–230

    Article  Google Scholar 

  • El-Nabulsi RA (2019b) Fractional Navier-Stokes equation from fractional velocity arguments and its implications in fluid flows and microfilaments. Int J Nonlinear Sci Numer Simul 20:449–459

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2019c) Geostrophic flow and wind driven ocean currents based on dimensionality of the space medium. Pure Appl Geophys 176:2739–2750

    Article  Google Scholar 

  • El-Nabulsi RA (2020a) On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc r Soc A 476:20190729

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2021a) On nonlocal fractal laminar steady and unsteady flows. Acta Mech 232:1413–1424

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2021b) Thermal transport equations in porous media from product-like fractal measure. J Therm Stress 44:899–912

    Article  Google Scholar 

  • El-Nabulsi RA (2021c) Superconductivity and nucleation from fractal anisotropy and product-like fractal measure. Proc Roy Soc A 477:20210065

    Article  MathSciNet  Google Scholar 

  • El-Nabulsi RA (2021d) Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal geometry. Phys E: Low Dim Syst Nanostruct 134:114827

    Article  Google Scholar 

  • El-Nabulsi RA (2021e) Quantization of Foster mesoscopic circuit and DC-pumped Josephson parametric amplifier from fractal measure arguments. Phys E: Low Dim Syst Nanostruct 133:114845

    Article  Google Scholar 

  • El-Nabulsi RA (2021f) Position-dependent mass fractal Schrodinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals. Opt Quant Elect 53:503

    Article  Google Scholar 

  • El-Nabulsi RA (2021g) Fractal neutrons diffusion equation: uniformization of heat and fuel burn-up in nuclear reactor. Nucl Eng Des 380:111312

    Article  Google Scholar 

  • El-Nabulsi RA (2021h) Fractal Pennes and Cattaneo-Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth. J R Soc Interface 18:20210564

    Article  Google Scholar 

  • El-Nabulsi RA, Golmankhaneh AK (2021a) On fractional and fractal Einstein’s field equations. Mod Phys Lett A 36:2150030

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA, Golmankhaneh AK (2021b) Dynamics of particles in cold electrons plasma: fractional actionlike variational approach versus fractal spaces approach. Waves Rand Compl Med. https://doi.org/10.1080/17455030.2021c.1909779

  • El-Nabulsi RA, Torres DFM (2008) Fractional actionlike variational problems. J Math Phys 49:053521

    Article  MathSciNet  MATH  Google Scholar 

  • El-Nabulsi RA (2018b) Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors. Eur. Phys. J. P 133: 494.

  • El-Nabulsi RA (2018c) Spectrum of Schrödinger Hamiltonian operator with singular inverted complex and Kratzer's molecular potentials in fractional dimensions. Eur. Phys. J. P 133: 277.

  • El-Nabulsi RA (2020b) Inverse-power potentials with positive-bound energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments. Eur. Phys. J. P135: 693.

  • Falconer KJ (2003) Fractal Geometry-Mathematical Foundations and Applications. Wiley, New York

    Book  MATH  Google Scholar 

  • Gagnon JS, Lovejoy S, Schertzer D (2003) Multifractal surfaces and terrestrial topography. Europhys Lett 62:801–807

    Article  Google Scholar 

  • Gagnon JS, Lovejoy S, Schertzer D (2006) Multifractal Earth Topography Nonlin Processes Geophys 13:541–570

    Article  Google Scholar 

  • Ghez J, Wang YY, Rammal R, Pannetier B, Bellissard J (1987) Band spectrum for an electron on a Sierpinski gasket in a magnetic field. Sol State Comm 64:1291–1294

    Article  Google Scholar 

  • Glimsdal S, Løvholt F, Harbitz CB, Romano F, Lorito S, Orefice S, Brizuela B, Selva J, Hoechner A, Volpe M, Babeyko A, Tonini R, Wronna M, Omira R: A new approximate method for quantifying tsunami maximum inundation height probability. Pure Appl. Geophys.176: 3227–3246.

  • Grimshaw R, Yuan C (2016) Depression and elevation tsunami waves in the framework of the Korteweg-de Vries equation. Nat Hazards 84:S493–S511

    Article  Google Scholar 

  • Grimshaw R, Pelinovsky D, Pelinovsky E (2010a) Homogenization of the variable-speed wave equation. Wave Motion 47:496–507

    Article  MathSciNet  MATH  Google Scholar 

  • Grimshaw R, Pelinovsky E, Talipova T (2010b) Non-reflecting internal wave beam propagation in the deep ocean. J Phys Oceanography 40:802–913

    Article  Google Scholar 

  • Hammack J (1973) A note on tsunamis: their generation and propagation in an ocean of uniform depth. J Fluid Mech 60:769–799

    Article  MATH  Google Scholar 

  • Herzfeld UC, Kim II, Orcutt JA, Fox CG (1993) Fractal geometry and seafloor topography: theoretical concepts versus data analysis for the Juan de Fuca Ridge and the East Pacific Rise. Ann Geophys 11:532–541

    Google Scholar 

  • Herzfeld UC, Kim II, Orcutt JA (1995) Is the ocean floor a fractal? Math Geo 27:421–462

    Article  Google Scholar 

  • Hirabayashi T, Ito K, Yoshi, (1992) Multifractal analysis of earthquakes. Pure Appl Geophys 138:591–610

    Article  Google Scholar 

  • Hirata T, Imoto M (1991) Multifractal analysis of spatial distribution of micro earthquakes in the Kanto region. Geophys Int J 107:155–162

    Article  Google Scholar 

  • Isern-Fontanet J, Turiel A, Garcia-Ladona E, Font J (2007) Microcanonical multifractal formalism: application to the estimation of ocean surface velocities. J Geophys Res 112:C05024

    Google Scholar 

  • Ivanov SS (1994) Estimation of fractal dimension of the global relief. Oceanology 34:94–98

    Google Scholar 

  • Jumarie G (2008) Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl Math Lett 22:378–385

    Article  MathSciNet  MATH  Google Scholar 

  • Kajiura K (1963) The leading wave of a tsunami. Bulletin Earth-quake Research Institute, Tokyo University, vol 41, pp 535–571

  • Khan S, Noor A, Mughal MJ (2013) General solution for waveguide modes in fractional space. Prog Electromag Res M 33:105

    Article  Google Scholar 

  • Korteweg DJ, De Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil Magazine 39:422–443

    MathSciNet  MATH  Google Scholar 

  • Kroger H (2000) Fractal geometry in quantum mechanics, field theory and spin systems. Phys Rep 323:81–181

    Article  MathSciNet  Google Scholar 

  • Lakshmanan M, Rajasekar S (2003) Korteweg-de Vries Equation and Solitons. In: Nonlinear Dynamics. Advanced Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55688-3_12.

  • Lenhardt WW (2000) Fractal Concepts and their Application to Earthquakes in Austria. In: Lehner F.K., Urai J.L. (eds) Aspects of Tectonic Faulting. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59617-9_4.

  • Li J, Ostoja-Starzewski M (2009) Fractal solids, product measures and fractional wave equations. Proc Roy Soc A465:2521–2536

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Ostoja-Starzewski M (2011) Micropolar continuum mechanics of fractal media. Int J Eng Sci 49:1302

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Ostoja-Starzewski M (2020) Thermo-poromechanics of fractal media. Phil Trans Roy Soc A 378:20190288

    Article  MathSciNet  MATH  Google Scholar 

  • Liu J, Yao Y, Liu D, Cai Y, Cai J (2018) Comparison of pore fractal characteristics between marine and continental shares. Fractals 26:1840016

    Article  Google Scholar 

  • Logan BE, Wilkinson DB (1990) Fractal geometry of marine snow and other biological aggregates. Limnology & Ocean 35:130–136

    Article  Google Scholar 

  • Madsen PA, Fuhrman DR, Schaffer HA (2008) On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113: C12012 (1–22 pages).

  • Madsen PA, Fuhrman DR (2008) Run-up of tsunamis and long waves in terms of surf-similarity. Coast Eng 55:209–223

    Article  Google Scholar 

  • Malinverno A (1995) Fractals and Ocean Floor Topography: A Review and a Model. In: Barton C.C., La Pointe P.R. (eds) Fractals in the Earth Sciences. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1397-5_6.

  • Mandelbrot BB (1983) The Fractal Geometry of Nature, W. H. Freeman and Company, New York.

  • Mandelbrot BB (1989) Multifractal Measures, Especially for Geophysicist, in Fractals in Geophysics, Eds C.H. Scholz & B.B. Mandelbrot, pp 5–42.

  • Mareschal JC (1989) Fractal reconstruction of sea-floor topography. Pure Appl Geophys 131:197–210

    Article  Google Scholar 

  • Mashayekhi S, Beerli P (2019) Fractional Coalescent Proc Nat Acad Sci 116:6244–6249

    Article  Google Scholar 

  • Mashayekhi S, Sedaghat S (2021) Fractional model of stem cell population dynamics. Chaos, Solitons Fractals 146:110919

    Article  MathSciNet  Google Scholar 

  • Mashayekhi S, Miles P, Hussaini MY, Oates WS (2008) Fractional viscoelasticity in fractal and non-fractal media: theory, experimental validation, and uncertainty analysis. J Mech Phys Solids 111:134–156

    Article  MathSciNet  MATH  Google Scholar 

  • Mashayekhi S, Hussaini MY, Oates WS (2019) A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation. J Mech Phys Solids 128:137–150

    Article  MathSciNet  Google Scholar 

  • Matsuzaki M (1994) Fractals in earthquakes. Philos Trans Phys Sci Eng 348:449–357

    MATH  Google Scholar 

  • Nottale L (2009) Fractals in the Quantum Theory of Spacetime. In: Meyers R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_228.

  • Ostoja-Starzewski M (2008) On turbulence in fractal porous media. Z Angew Math Phys 59:1111–1117

    Article  MathSciNet  MATH  Google Scholar 

  • Ostoja-Starzewski M (2009) Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech 205:161–170

    Article  MATH  Google Scholar 

  • Ostoja-Starzewski M, Li J, Joumaa H, Demmie PN (2014) From fractal media to continuum mechanics. Zei Ang Math Phys 94:373–401

    MathSciNet  MATH  Google Scholar 

  • Pan H, Zhang W, Jiang W, Wang P, Yang J, Zhang X (2020) Roughness change analysis of sea surface from visible images by fractals. IEEE Access 8:78519–78529

    Article  Google Scholar 

  • Pelinovsky E, Talipova T, Kurkin A, Kharif C (2001) Nonlinear mechanism of tsunami wave generation by atmospheric disturbances. Nat Haz Earth Syst Sci 1:243–250

    Article  Google Scholar 

  • Sears S, Soljacic M, Segev D, Krylov D, Bergman K (2000) Cantor set fractals from solitons. Phys Rev Lett 84:1902

    Article  Google Scholar 

  • Shaw JA, Churnside JH (1997) Fractal laser glints from the ocean surface. J Opt Soc Am 14:1144–1150

    Article  Google Scholar 

  • Shuto N (1985) The Nihonkai-chuubu earthquake tsunami on the north Akita coast. Coastal Eng Jpn Coastal Eng Jpn 28:255–264

    Article  Google Scholar 

  • Soljacic M, Segev M, Menyuk CR (2000) Self-similarity and fractals in soliton-supporting systems. Phys Rev E 61:R1048–R1051

    Article  Google Scholar 

  • Soljacic M, Segev M, Menyuk CR (1999) Self-similarity and fractals driven by soliton dynamics. Quantum Electronics and Laser Science Conference 1999, Baltimore, Maryland United State, 23–26 May (published at Technical Digest. Summaries of papers presented at the Quantum Electronics and Laser Science Conference. Postconference Edition).

  • Sreenivasan KR (1994) Fractals in Fluid Mechanics FRACTALS 02:253–263

    Google Scholar 

  • Stiassnie M (1991) The fractal dimension of the ocean surface. Conference: Proceedings of the International School of Physics Enrico Fermi, 1988, A.R. Osborne, Ed.

  • Tajima T (2018) Computational plasma physics: with applications to fusion and astrophysics. CRC Press, Taylor and Francis Group

    Book  Google Scholar 

  • Tandel P, Patel H, Patel T (2021) Tsunami wave propagation model: a fractional approach. J Ocean Eng Sci. https://doi.org/10.1016/j.joes.2021.10.004

    Article  MATH  Google Scholar 

  • Tarasov VE (2005a) Continuous media model for fractal media. Phys Lett A 336:167–174

    Article  MATH  Google Scholar 

  • Tarasov VE (2005b) Fractional hydrodynamic equations for fractal media. Ann Phys 318:286–307

    Article  MathSciNet  MATH  Google Scholar 

  • Tarasov VE (2014) Flow of fractal fluid in pipes: Non-integer dimensional space approach. Chaos, Solitons Fractals 67:26–37

    Article  MathSciNet  MATH  Google Scholar 

  • Todoechuck JP, Jensen OG (1989) Scaling Geology and Seismic Deconvolution, in Fractals in Geophysics, Eds C.H. Scholz & B.B. Mandelbrot, pp 273–288.

  • Todorovska MI, Trifunac MD (2001) Generation of tsunamis by a slowly spreading uplift of the seafloor. Soil Dyn Earthquake Eng 21:151–167

    Article  Google Scholar 

  • Trifonov EV (2017) Families of exact solutions for linear and nonlinear wave equations with a variable speed of sound and their use in solving initial boundary value problems. Theor Math Phys 192:974–981

    Article  MathSciNet  MATH  Google Scholar 

  • Truong HVP (2011) Dynamic Soil Mass or Added Soil Mass and Dynamic Rock Mass in Foundation Designs and Dynamic Soil Amplification. Proc. of World Congress on Advances in Structural Engineering and Mechanics (ASEM+), Korea: 4893–4901.

  • Turcotte DL (1988) Fractals in fluid mechanics. Ann Rev Fluid Mech 20:5–16

    Article  Google Scholar 

  • Turcotte DL (1989) Fractals in Geology and Geophysics. PAGEOPH 131:171–196

    Article  Google Scholar 

  • Turcotte DL (1992) Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Turiel A, Nieves V, Garcia-Ladona E, Font J, Rio MH, Larnicol G (2009) The multifractal structure of satellite sea surface temperature maps can be used to obtain global maps of streamlines. Ocean Sci Discuss 6:129–151

    Google Scholar 

  • Varley E, Seymour B (1988) A method for obtaining exact solutions to partial differential equations with variable coefficients. Stud Appl Math 78:183–225

    Article  MathSciNet  MATH  Google Scholar 

  • Viana RL, Da Silva EC, Kroetz T, Caldas IL, Roberto M, Sanjuan MAF (2011) Fractal structures in nonlinear plasma physics. Phil Trans R Soc A369:371–395

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Deng Q (2019) Fractal derivative model for Tsunami traveling. Fractals 27:1950017

    Article  MathSciNet  Google Scholar 

  • Wang S, Li X, Xue H, Shen Z, Chen L (2021) Fractal characteristics of shale pore structure and its influence on seepage flow. R Soc Open Sci 8:202271

    Article  Google Scholar 

  • Wu M, Kalinikos BA, Carr LD, Patton CE (2006) Observation of spin-wave solitons fractals in magnetic film active feedback rings. Phys Rev Lett 96:187202

    Article  Google Scholar 

  • Yu DS, Nazaikinskii VE, Tirozzi B (2010) Asymptotic solutions of the 2D wave equations with variable velocity and localized right-hand side. Russ J Math Phys 17:66–76

    Article  MATH  Google Scholar 

  • Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

    Article  MATH  Google Scholar 

  • Zamora-Sillero E, Shapovalov AV (2007) Soliton fractals in the Korteweg-de Vries equation. Phys Rev E 76:046612

    Article  MathSciNet  Google Scholar 

  • Zheng CH (2007) Fractal solitons and complex wave excitations for the general (2+1)-dimensional Korteweg-de Vries system. Far-East J Dyn Syst 9:361–370

    MathSciNet  MATH  Google Scholar 

  • Zubair M, Mughal MJ, Naqvi QA (2010) The wave equation and general plane wave solutions in fractional space. Prog Electromagnet Res Lett 19:137–146

    Article  Google Scholar 

  • Zubair M, Mughal MJ, Naqvi QA (2011) An exact solution of the spherical wave equation in D-dimensional fractional space. J Electromagnet Waves Appl 25:481–1491

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the group of anonymous referees for their useful comments and valuable suggestions.

Funding

The authors would like to thank Chiang Mai University for funding this research.

Author information

Authors and Affiliations

  1. Center of Excellence in Quantum Technology, Faculty of Engineering, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  2. Quantum-Atom Optics Laboratory and Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  3. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

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  1. Rami Ahmad El-Nabulsi
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  2. Waranont Anukool
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Correspondence to Rami Ahmad El-Nabulsi.

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El-Nabulsi, R.A., Anukool, W. Propagation of fractal tsunami solitary waves. J. Ocean Eng. Mar. Energy 9, 255–271 (2023). https://doi.org/10.1007/s40722-022-00266-7

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