Introduction to shallow water waves and mathematical theories

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Introduction to shallow water waves and
mathematical theories

• Zhijun Qiao
• (qiao_at_utpa.edu)
• Department of Mathematics
• The University of Texas Pan American
• Southern Polytechnic State University, Marietta,
  GA
• Mathematics Department
• Nov. 15, 2005

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Outline

• Background
• Solitons and KdV Equation
• Shock Waves and Burgers Equation
• Peakons and Shallow Water Equation
• Multi-peakons
• Hamiltonian Structure and Integrability
• New Peaked and Smooth Solitons
• Conclusions and problems

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Background

• 1844, Scott Russell, Solitary waves
• Study of Solitary Waves began with the
  observations by Scott Russell (1838) over century
  ago. Russell 1st observed a solitary wave while
  riding on horseback beside a narrow barge
  channel. When the boat he was observing stopped,
  Russell noted that it set forth
• a large solitary elevation, a rounded,
  smooth and well defined heap of water, which
  continued its course along the channel apparently
  without change of form or diminution of speed.
  Its height gradually diminished, and after a
  chase of one or two miles I lost it in the
  windings of the channel. Such, in the month of
  August 1834, was my 1st chance interview with
  that singular and beautiful phenomenon.
• No Math Theory at that time.

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This is the way you usually see solitons in
shallow water, scurrying along like mice in the
gutter, at a scale of a only few inches in width
and height.
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Solitons and KdV Equation

• 1895, Korteweg, de Vries, KdV equation
• governs moderately small shallow water
  waves.
• 1965, Kruskal, Zabusky, Soliton
• 1967, Gardner, Greener, Kruskal and Miura,
• Inverse Scattering Transform

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1974, Flaschka, Toda lattice 1974, Novikov,
Stationary system 1975, Delauney-Moser,
Delauney-Moser system 1975, Moser, Confocal
system 1976, Calogero-Moser, N-body problem 1979,
Moser, Moser Constraint method 1982, Knoerrer,
Geodesic flow 1989, Cao Cewen, Nonlinearization
Method 1993, Camassa, Holm, Peakon
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KdVRegular Soliton (1967)
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Two Solitons Interactions
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Two solitons, c1gtc2gt0
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Shock Waves and Burgers Equation

• 1948, JM Burgers

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Examples of fluids include gases and liquids.
Typically, liquids are considered to be
incompressible, whereas gases are considered to
be compressible. Fluid flow is dominated by the
governing equation- NS. There are a bunch of
applications in engineering and mechanics.
Here is the example. There is a well known PDE
that arises in the modeling of various fluid
flows, from shock waves to turbulence, known as
Burgers equation. It is, roughly speaking, a
one-dimensional caricature of the Navier-Stokes
equation, so central to fluid mechanics. Burgers
equation concerns a field u depending on one
spatial coordinate, x, and time, t. The PDE is as
written.
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Burgers Equation Traveling Shock Wave
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plot3d((2x-8t1)/(x-4t),x-10..10,t0..2)a2,
c4,B0Singular Traveling Wave Solution
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plot3d(1tanh(x-2t),x-10..10,t0..3)B0,a2,b0
,c2plot(1tanh(x-21),x-10..10)B0,a2,b0,c
2,t1Shock wave solution for the Burgers
equation
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plot3d(11/tanh(x-2t),x-1..1,t0..0.5)B0,a2,b
0,c2plot(11/tanh(x-2),x1.5..1.99)B0,a2,b
0,c2,t1plot(11/tanh(x-2),x2.01..2.5)B0,a2
,b0,c2,t1
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Peakons and Shallow Water EquationCamassa-Holm
1993
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Multi-Peakons
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Hamiltonian Structures and Integrability
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New Peaked and Smooth Solutions
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Thanks

• Any Questions/Comments?
• Welcome to discuss with me anytime.
• http//www.math.panam.edu/qiao
• My email qiao_at_utpa.edu