Applied Mathematics Seminar

The basic equations of hydrodynamics, magnetohydrodynamics and finite deformation elasticity are intrinsically nonlinear. The extent to which these and other governing equations of nonlinear mechanics naturally admit integrable structure and are accordingly amenable to the powerful techniques of modern soliton theory remains an important open question. Hitherto, whereas solitonic phenomena have been observed in widely diverse areas in nature, the nonlinear equations that describe solitons have typically been derived by approximation or expansion methods. It is well-established that solitonic equations arise naturally out of the geometry of those privileged classes of surfaces that admit invariance under Backlund transformations: ( C.Rogers and W.K.Schief, Backlund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.) Here it is shown how solitonic structure naturally resides in diverse areas of nonlinear continuum mechanics, including, inter alia:

Hydrodynamics and Equilibrium Magnetohydrodynamics
Elastic Shell Membrane Theory
The Kinematics of Fibre-Reinforced Materials