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Introduction to Symmetry Analysis

Brian CantwellDepartment of Aeronautics and Astronautics

Stanford University

Chapter 16 -Backlund TransformationsAnd Nonlocal Groups

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Singular behavior of Burgers’ Equation. Work out the steady state solution - invariant under translation in time.

Burgers’ Equation.

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C1 =1 / 2C2 =0

ν =2

ν =1 / 2

ν =1 / 8

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Exact solution of Burgers’ Equation

Conserved integral

Integrate the equation in space

Initial velocity distribution

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Non-dimensionalize the equation

The conserved integral becomes

Where the Reynolds number is

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Symmetries of the Burgers potential equation

Invariance condition

Group operators

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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

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What finite transformation does this correspond to ?To find out we have to sum the Lie series.

Where

First few terms

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Let

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The finite transformation of the Burgers potential equation is

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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

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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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Solitary Waves

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The Great Eastern

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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

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Invariance condition for the KdV potential equation

Assume an infinitesimal transformation of the form

The invariance condition becomes

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The KdV potential equation admits the group with infinitesimal

The Lie series is

where

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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

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The corresponding solution of the KdV equation is the solitary wave

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One can use the group to generate an exact solution for two colliding solitons.

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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

Stanford University Department of Aeronautics and Astronautics

C1 =1 / 2C2 =0

ν =2

ν =1 / 2

ν =1 / 8