Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T23:20:24.207Z Has data issue: false hasContentIssue false
  • English
  • Français

The KP theory and Mach reflection

Published online by Cambridge University Press:  13 July 2016

Yuji Kodama  and
Harry Yeh
Yuji Kodama
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
Harry Yeh*
Affiliation:
School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331, USA
*
†Email address for correspondence: harry@engr.orst.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Interactions of two line solitons in the two-dimensional shallow-water field are studied based on the Kadomtsev–Petviashvili (KP) theory. With the use of the normal form, the extended KP equation with higher-order correction is derived. This extended KP theory improves significantly the predictability of the original KP theory for soliton interactions with finite oblique angles. The previously existing discrepancy between the experiments and the theory in the Mach reflection problem is now resolved by the normal form theory.

JFM classification

Mathematical Foundations: Mathematical Foundations Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 766 - 786
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chakravarty, S. & Kodama, Y. 2009 Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Maths 123, 83–151.Google Scholar
Chakravarty, S. & Kodama, Y. 2014 Construction of KP solitons from wave patterns. J. Phys. A: Math. Theor. 47, 025201.CrossRefGoogle Scholar
Dommermuth, D. & Yue, D. 1987 High-order spectral method for the study of nonlinear gravity waves. reflection of obliquely incident solitary waves. J. Fluid Mech. 184, 267–288.Google Scholar
Freeman, N. C. & Nimmo, J. 1983 Soliton-solutions of the Korteweg–de Vries and Kadomtsev–Petviashvili equations: the Wronskian technique. Phys. Lett. A 95, 1–3.Google Scholar
Funakoshi, M. 1980 Reflection of obliquely incident solitary waves. J. Phys. Soc. Japan 49, 2371–2379.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611–622.Google Scholar
Hiraoka, Y. & Kodama, Y. 2009 Normal form and solitons. In Integrability, Lecture Notes in Physics, vol. 767, pp. 175–214. Springer.Google Scholar
Hirota, R. 2004 The Direct Method in Soliton Theory. Cambridge University Press.Google Scholar
Jia, Y.2014 Numerical study of the KP solitons and higher order Miles theory of the Mach reflection in shallow water. PhD thesis, The Ohio State University.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kao, C.-Y. & Kodama, Y. 2012 Numerical study on the KP equation for non-periodic waves. Maths Comput. Simul. 82, 1185–1218.CrossRefGoogle Scholar
Kato, S., Takagi, T. & Kawahara, M. 1998 A finite element analysis of Mach reflection by using the Boussinesq equation. Intl J. Numer. Math. Fluids 28, 617–631.Google Scholar
Kodama, Y. 1985 Normal forms for weakly dispersive wave equations. Phys. Lett. A 112, 193–196.Google Scholar
Kodama, Y. 1987 On solitary-wave interaction. Phys. Lett. A 123, 276–282.Google Scholar
Kodama, Y. 2004 Young diagrams and N-soliton solutions of the KP equation. J. Phys. A: Math. Gen. 37, 11169–11190.Google Scholar
Kodama, Y. 2010 KP solitons in shallow water. J. Phys. A: Math. Theor. 43, 434004; 54pp.CrossRefGoogle Scholar
Kodama, Y. 2012 KP solitons and Mach reflection in shallow water. In Presented at the Autumn Conference of Mathematical Society of Japan, September 20, 2012, arXiv:1210.0281.Google Scholar
Kodama, Y. & Mikhailov, A. V. 1996 Obstacles to asymptotic integrability. Prog. Nonlinear Diff. Equ. 26, 173–204.Google Scholar
Kodama, Y., Oikawa, M. & Tsuji, H. 2009 Soliton solutions of the KP equation with V-shape initial waves. J. Phys. A: Math. Theor. 42, 312001 (9pp).Google Scholar
Kodama, Y. & Williams, L. 2013 The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244, 979–1032.Google Scholar
Li, W., Yeh, H. & Kodama, Y. 2011 On the Mach reflection of a solitary wave: revisited. J. Fluid Mech. 672, 326–357.Google Scholar
Matveev, V. B. 1979 Darboux transformation and explicit solutions of the Kadomtsev–Petviaschvily equation, depending on functional parameters. Lett. Math. Phys. 3, 213–216.Google Scholar
Melville, W. K. 1980 On the Mach reflexion of a solitary wave. J. Fluid Mech. 98, 285–297.Google Scholar
Miles, J. W. 1977a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.Google Scholar
Miles, J. W. 1977b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.Google Scholar
Perroud, P. H.1957 The solitary wave reflection along a straight vertical wall at oblique incidence. Tech. Rep. 99/3. Institute of Engineering Research, Wave Research Laboratory, University of California, Berkeley.Google Scholar
Satsuma, J. 1979 A Wronskian representation of N-soliton solutions of nonlinear evolution equations. J. Phys. Soc. Japan 46, 356–360.Google Scholar
Tanaka, M. 1993 Mach reflection of a large-amplitude solitary wave. J. Phys. Mech. 248, 637–661.Google Scholar
Yeh, H., Li, W. & Kodama, Y. 2010 Mach reflection and KP solitons in shallow water. Eur. Phys. J. 185, 97–111.Google Scholar