1. Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry Analysis Brian Cantwell Department of Aeronautics and Astronautics Stanford University Chapter 16 -Backlund Transformations And Nonlocal Groups

2. 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.

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4. Exact solution of Burgers’ Equation Conserved integral Integrate the equation in space Initial velocity distribution

5. Stanford University Department of Aeronautics and Astronautics Non-dimensionalize the equation The conserved integral becomes Where the Reynolds number is

6. Stanford University Department of Aeronautics and Astronautics Symmetries of the Burgers potential equation Invariance condition Group operators

7. 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 infinite dimensional group Where f is a solution of the heat equation

8. 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

9. Stanford University Department of Aeronautics and Astronautics Let

10. Stanford University Department of Aeronautics and Astronautics The finite transformation of the Burgers potential equation is

11. 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

12. Stanford University Department of Aeronautics and Astronautics The Cole-Hopf transformation. Let U = 0 Let

13. Stanford University Department of Aeronautics and Astronautics The classical single hump solution of Burgers equation. Let The Cole-Hopf transformation gives where

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15. Solitary Waves

16. Stanford University Department of Aeronautics and Astronautics The Great Eastern

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20. The Korteweg de Vries equation is often used to study the relationship between nonlinear convection and dispersion. Begin with the KdV potential equation

21. 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

22. Stanford University Department of Aeronautics and Astronautics The KdV potential equation admits the group with infinitesimal The Lie series is where

23. 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

24. Stanford University Department of Aeronautics and Astronautics The corresponding solution of the KdV equation is the solitary wave

25. Stanford University Department of Aeronautics and Astronautics One can use the group to generate an exact solution for two colliding solitons.

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

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