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Stanford University Department of Aeronautics and Astronautics
Introduction to Symmetry Analysis
Brian CantwellDepartment of Aeronautics and Astronautics
Stanford University
Chapter 16 -Backlund TransformationsAnd Nonlocal Groups
Stanford University Department of Aeronautics and Astronautics
Singular behavior of Burgers’ Equation. Work out the steady state solution - invariant under translation in time.
Burgers’ Equation.
Stanford University Department of Aeronautics and Astronautics
C1 =1 / 2C2 =0
ν =2
ν =1 / 2
ν =1 / 8
Stanford University Department of Aeronautics and Astronautics
Exact solution of Burgers’ Equation
Conserved integral
Integrate the equation in space
Initial velocity distribution
Stanford University Department of Aeronautics and Astronautics
Non-dimensionalize the equation
The conserved integral becomes
Where the Reynolds number is
Stanford University Department of Aeronautics and Astronautics
Symmetries of the Burgers potential equation
Invariance condition
Group operators
Stanford University Department of Aeronautics and Astronautics
There is another solution of the invariance condition !!
With the independent variables not transformed,the invariance condition takes the following form
The invariance condition is satisfied by the infinitedimensional group
Where f is a solution of the heat equation
Stanford University Department of Aeronautics and Astronautics
What finite transformation does this correspond to ?To find out we have to sum the Lie series.
Where
First few terms
Stanford University Department of Aeronautics and Astronautics
The finite transformation of the Burgers potential equation is
Stanford University Department of Aeronautics and Astronautics
This group can be used to generate a corresponding transformation of the Burgers equation. Let
The transformation of Burgers equation is
This is an example of a nonlocal group
Stanford University Department of Aeronautics and Astronautics
The Cole-Hopf transformation. Let U = 0
Let
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The classical single hump solution of Burgers equation. Let
The Cole-Hopf transformation gives
where
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The Korteweg de Vries equation
is often used to study the relationship between nonlinear convection and dispersion.
Begin with the KdV potential equation
Stanford University Department of Aeronautics and Astronautics
Invariance condition for the KdV potential equation
Assume an infinitesimal transformation of the form
The invariance condition becomes
Stanford University Department of Aeronautics and Astronautics
The KdV potential equation admits the group with infinitesimal
The Lie series is
where
Stanford University Department of Aeronautics and Astronautics
Summing the Lie series leads to the non-local finite transformation
The simplest propagating solution of the KdV potential equation is
which generates the solution
Stanford University Department of Aeronautics and Astronautics
The corresponding solution of the KdV equation is the solitary wave
Stanford University Department of Aeronautics and Astronautics
One can use the group to generate an exact solution for two colliding solitons.
Stanford University Department of Aeronautics and Astronautics
Singular behavior of Burgers’ Equation
d
dx
u2
2−ν
dudx
⎛
⎝⎜⎞
⎠⎟=0
u2
2−ν
dudx
=C1
dudx
=u2
2ν−C1
ν
u x( ) = 2C1Tanh 2C1 C2 −x2ν
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
Work out the steady state solution - invariant under translation in time
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