Asymptotic Nonlinearities in Smectics

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Asymptotic Nonlinearities in Smectics
E.A. Brener
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Dislocations in smectics
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The elastic energy of smectics
Equation of equilibrium
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Greens functions
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Scaling relations
For
the linear theory is valid
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Shear deformation
far from the wall
near the wall
transition takes place at
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Force applied along the z axis
Far from the defect line one can neglect bending
energy and displacement is
At
transition to the linear theory
u
takes place
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Dislocations in smectics
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Dislocations in smectics
Linear theory (de Gennes)
Nonlinear theory (Brener and Marchenko)
At
the result of the linear theory is recovered
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Details of solution
subject to the constraint
is the solution of the original 3rd order equation
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Nonlinear Greens function
In the linear approximation
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Details of solution
Equation can be reduced to
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Numerical solution of the equation
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Strongly nonlinear regime,
,