Dmitry Arkhipov and Georgy Khabakhpashev New equations for modeling nonlinear waves interaction on a free surface of fluid shallow layer

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Dmitry Arkhipov and Georgy Khabakhpashev New
equations for modeling nonlinear waves
interaction on a free surface of fluid shallow
layer

• Department of Physical Hydrodynamics
• Institute of Thermophysics SB RAS
• Novosibirsk, Russia

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Moderately long nonlinear plane waves on a
free surface of a liquid layer
G. B. Whitham, Linear and Nonlinear Waves
(1974) L. A. Ostrovsky, A. I. Potapov, Modulated
Waves Theory and Applications (1999) G. A.
Khabakhpashev, Fluid Dynamics (1987) K. Y. Kim,
R. O. Reid, R. E. Whitaker, J. Comp. Phys.
(1988) D. ?. Pelinovsky, Yu. A. Stepanyants,
JETP (1994) R. S. Johnson, J. Fluid Mechanics
(1996)
For plane perturbations above horizontal bottom
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Nonlinear long disturbances of a free
surface of the liquid layer above a
gently sloping bottom

• Basic assumptions of the model

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Main model equations for nonlinear waves,
running at any angles between them
D. G. Arkhipov, G. A. Khabakhpashev, Doklady
Physics (2006)
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New model equation for plane nonlinear
waves in the shallow liquid layer
In the nonlinear term
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Propagation of moderately long nonlinear waves
in the liquid layer above the horizontal
bottom
For perturbations running in one direction
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Test calculations Overtaking interaction of
two plane solitary waves above the horizontal
bottom
h1 3h2
h h / h
x x / h
t 0
t 650
t 472
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Test calculations Exchange interaction of
two plane solitary waves above the horizontal
bottom
h1 2h2
h h / h
x x / h
t 0
t 1400
t 965
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Collision of two nonlinear plane solitary
waves above the horizontal bottom
h1 2h2
h1 h2
t 56
t 51.5
t 103
t 112
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Approximated analytical solutions to the problem
of head-on collision of two solitary waves above
the horizontal bottom
-- modified Boussinesq equation
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Inelastic interaction at head-on collision of
two solitary waves above the horizontal bottom
t 14
h i hi0 sech2 (x x0 Uit ) /Li
h10 h20 0.15 h ,
x0 15 h
t 28
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Collision of two nonlinear plane solitary
waves above the horizontal and uneven
bottoms
h1 h2
h (x) h0 1 sech2 (x / 2L) / 2
t 20
12
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New model equation for axisymmetrical
nonlinear waves in the shallow liquid layer
F. Calogero, A. Degasperis, Spectral Transform
and Solitons Tools to Solve and Investigate
Nonlinear Evolution Equations (1982) KdV
cylindrical eq.
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Evolution of initially bell-type perturbation
above two different bottom profiles
r r / h0
h h0
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Evolution of initially bell-type perturbation
above two different bottom profiles
r r / h0
h h0
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Evolution of initially bell-type perturbation
above two different bottom profiles
r r / h0
h h0
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Evolution of initially bell-type perturbation
above two different bottom profiles
r r / h0
h h0
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Transformation of initially ring-type
disturbance above two different bottom
profiles
r r / h0
h h0
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Transformation of initially ring-type disturbance
above two different bottom profiles
r r / h0
h h0
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Transformation of initially ring-type
disturbance above two different bottom
profiles
r r / h0
h h0
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Transformation of initially ring-type
disturbance above two different bottom
profiles
r r / h0
h h0
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Interaction of initially bell-type and
ring-type perturbations above two different
bottom profiles
h h0
r r / h0
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Principal results
1. New evolution differential equations for
the dynamics description of moderately long
plane and axially-symmetrical nonlinear waves
running towards each other are suggested. 2.
A validity of new equations to the
solution of a number of plane or
axially-symmetrical problems of the nonlinear
wave evolution including the case of a
fluid with variable depth is shown with
the help of numerical experiments. 3.
An analytical solution for the problem of head-on
collision of two solitons was constructed by the
perturbation theory