Asymptotic Nonlinearities in Smectics

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Presentation transcript:

Asymptotic Nonlinearities in Smectics E.A. Brener

Dislocations in smectics

The elastic energy of smectics Equation of equilibrium

Green’s functions

the linear theory is valid Scaling relations For the linear theory is valid

transition takes place at Shear deformation far from the wall near the wall transition takes place at

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

Dislocations in smectics

Dislocations in smectics Linear theory (de Gennes) Nonlinear theory (Brener and Marchenko) At the result of the linear theory is recovered

Details of solution is the solution of the original 3rd order equation subject to the constraint is the solution of the original 3rd order equation

“Nonlinear” Green’s function In the linear approximation

Equation can be reduced to Details of solution Equation can be reduced to

Numerical solution of the equation

Strongly nonlinear regime,